Definition:Axiomatic Set Theory

'''Axiomatic set theory''' is a system of set theory which differs from so-called naive set theory in that the sets which are allowed to be generated are strictly constrained by the axioms.


Definition:ZFC

'''Zermelo-Fraenkel Set Theory with the Axiom of Choice''' is a system of axiomatic set theory upon which the whole of (at least conventional) mathematics can be based. Its basis consists of a system of Aristotelian logic, appropriately axiomatised, together with the Zermelo-Fraenkel axioms of Set Theory and the (controversial) Axiom of Choice. These are as follows: {{:Definition:Zermelo-Fraenkel Axioms}} === The Axiom of Choice === {{:Axiom:Axiom of Choice}}


Definition:Set Theory

'''Set Theory''' is the branch of mathematics which studies sets.


Definition:Logical Not

The '''logical not''' or '''negation''' operator is a unary connective whose action is to reverse the truth value of the statement on which it operates. :$\neg p$ is defined as: '''$p$ is not true''', or '''It is not the case that $p$ is true'''. Thus the statement $\neg p$ is called the '''negation''' of $p$. $\neg p$ is voiced '''not $p$'''.


Definition:Conjunction

'''Conjunction''' is a binary connective written symbolically as $p \land q$ whose behaviour is as follows: : $p \land q$ is defined as: :'''$p$ is true ''and'' $q$ is true.''' This is called the '''conjunction''' of $p$ and $q$. The statements $p$ and $q$ are known as: : the '''conjuncts''' : the '''members of the conjunction'''. $p \land q$ is voiced: :'''$p$ and $q$'''.


Definition:Disjunction

'''Disjunction''' is a binary connective written symbolically as $p \lor q$ whose behaviour is as follows: : $p \lor q$ is defined as: :'''Either $p$ is true ''or'' $q$ is true ''or possibly both''.''' This is called the '''disjunction''' of $p$ and $q$. $p \lor q$ is voiced: :'''$p$ or $q$'''


Definition:Conditional

The '''conditional''' or '''implication''' is a binary connective: :$p \implies q$ defined as: :'''''If'' $p$ is true, ''then'' $q$ is true.''' This is known as a '''conditional statement'''. A '''conditional statement''' is also known as a '''conditional proposition''' or just a '''conditional'''. $p \implies q$ is voiced: :'''if $p$ then $q$''' or: :'''$p$ implies $q$'''


Definition:Biconditional

The '''biconditional''' is a binary connective: :$p \iff q$ defined as: :$\paren {p \implies q} \land \paren {q \implies p}$ That is: :'''''If'' $p$ is true, ''then'' $q$ is true, ''and if'' $q$ is true, ''then'' $p$ is true'''.


Definition:Parenthesis

'''Parenthesis''' is a syntactical technique to disambiguate the meaning of a logical formula. It allows one to specify that a logical formula should (temporarily) be regarded as being a single entity, being on the same level as a statement variable. Such a formula is referred to as being '''in parenthesis'''. Typically, a formal language, in defining its formal grammar, ensures by means of '''parenthesis''' that all of its well-formed words are uniquely readable. Generally, '''brackets''' are used to indicate that certain formulas are '''in parenthesis'''. The brackets that are mostly used are round ones, the '''left (round) bracket''' $($ and the '''right (round) bracket''' $)$.


Definition:Binding Priority

The '''binding priority''' is the convention defining the order of '''binding strength''' of the individual connectives in a logical formula. '''Binding priorities''' can be overridden by using parenthesis in appropriate places. Parenthesis always takes priority over conventional '''binding priorities'''.


Definition:True

A statement has a truth value of '''true''' {{iff}} what it says matches the way that things are.


Definition:False

A statement has a truth value of '''false''' {{iff}} what it says does not match the way that things are.


Definition:Contingent Statement

A '''contingent statement''' is a statement form which is neither a tautology, nor unsatisfiable, but whose truth value depends upon the truth value of its component substatements. <!-- == Formal Definition == Let $\mathfrak M$ be a collection of models for a particular formal language $\LL$. A well-formed word $\mathbf A$ of $\LL$ is said to be '''contingent (for $\mathfrak M$)''' {{iff}}: :$\exists \MM_1, \MM_2 \in \mathfrak M: \MM_1 \models \mathbf A \land \MM_2 \not \models \mathbf A$ that is, {{iff}} some models in $\mathfrak M$ approve, and others disapprove of $\mathbf A$. == Also known as == {{refactor|Seems to be replaceable by Also see entries. But, will need source review}} In the context of propositional formulas the term '''satisfiable''' is usually used: A propositional formula is '''satisfiable''' if its value is True in at least ''one'' boolean interpretation. A propositional formula is '''not-valid''' or '''falsifiable''' if its value is False in at least ''one'' boolean interpretation. === Set of Logical Formulas === Let $U = \set {P_1, P_2, \ldots, P_n}$ be a set of propositional formulas. Let $U' = \set {p_1, p_2, \ldots, p_m}$ be the set of all the atoms of all the propositional formulas in $U$. (Some of these atoms, and indeed this will most likely be the case, may be in more than one logical formula.) Then $U$ is '''(mutually) satisfiable''' if there exists a boolean interpretation $v$ for all the atoms in $U'$ such that: :$\map v {P_1} = \map v {P_2} = \cdots = \map v {P_n} = \T$. -->


Definition:Natural Deduction

'''Natural deduction''' is a technique for deducing valid sequents from other valid sequents by applying precisely defined proof rules, each of which themselves are either "self-evident" axioms or themselves derived from other valid sequents, by a technique called logical inference.


Definition:Natural Deduction/Proof Rule

A '''proof rule''' is a rule in natural deduction which allows one to infer the validity of propositional formulas from other propositional formulas. === Rule of Substitution === {{:Rule of Substitution}} === Rule of Sequent Introduction === {{:Rule of Sequent Introduction}} === Rule of Theorem Introduction === {{:Rule of Theorem Introduction}}


Definition:Logical Inference

'''Logical inference''' is the process used in natural deduction to deduce the validity of statement forms from other statement forms by use of proof rules. Given a set of logical formulae and the proof rules, we '''(logically) infer''' other formulas.


Definition:Pool of Assumptions

The '''pool of assumptions''', for a formula deduced in a particular proof by natural deduction, is the collection of all the assumptions upon which the formula depends. These consist of premises and possibly some intermediate assumptions. The assumptions in this pool are called, naturally enough, the '''pooled assumptions'''. The '''pooled assumptions''' at the end of a proof (its conclusion) are thus seen to be the premises.


Definition:Aristotelian Logic

'''Aristotelian logic''' is a system of logic which is based upon the philosophy of {{AuthorRef|Aristotle}}. It forms the cornerstone of the entirety of classical logic. The school of '''Aristotelian logic''' consists almost entirely of the relationships between the various categorical syllogisms. This school of philosophy forms the basis of mainstream mathematics, although, for example, mathematicians of the intuitionistic school do not accept the Law of the Excluded middle value.


Definition:Hypothesis

A '''hypothesis''' is a statement that is made whose truth value has not been established, but which is ''believed'' to be true. The underlying idea is that its truth or falsehood is '''about to be investigated'''. A hypothesis is considered by some branches of philosophy as being synonymous with the words '''wild guess'''[http://www.experiencefestival.com/a/Hypothesis/id/183571]. Possibly the most famous still-unproved hypothesis is the Riemann Hypothesis.


Definition:Discharged Assumption

The context of this definition is in the process of deducing the validity of sequents using logical inference in the framework of natural deduction. Assumptions that are made during the course of using certain proof rules remain in force only so long as the use of those rules requires them. Once the appropriate rules have been completed, these are known as '''discharged assumptions''', and are not included in the pool of assumptions on which the conclusion of the rule depends.


Definition:Corollary

A '''corollary''' is a proof which is a direct result, or a direct application, of another proof. It can be considered as being a '''proof for free''' on the back of a proof which has been paid for with blood, sweat and tears.


Definition:Corresponding Conditional

Let $p_1, p_2, p_3, \ldots, p_n \vdash q$ be a sequent of propositional logic. It can be expressed as a theorem as follows: : $\vdash p_1 \implies \left({p_2 \implies \left({p_3 \implies \left({\ldots \implies \left({p_n \implies q}\right) \ldots }\right)}\right)}\right)$ This is known as the sequent's '''corresponding conditional'''. This is proved in the Extended Rule of Implication. {{wtd|This pertains probably to the classical interpretation of propositional calculus, and needs to be reformulated and put in the correct category in due time}} {{SUBPAGENAME}} fpshpqhmzdo2bsytg3qk4e0pvlsyucd


Definition:By Hypothesis

When demonstrating a proof, it is frequently necessary to refer to the specification of the result which is to be proved. This result often contains wording along the lines '''Suppose that ...''' or '''Let ...''' It is convenient to refer back to these specifications during the course of the proof. To do that, the term '''by hypothesis''' is often used.


Definition:Proper Name

A '''proper name''' (or just '''name''') is a symbol or collection of symbols used to identify a ''particular'' object uniquely. In contrast with natural language, a proper name has a wider range than being the particular identifying label attached to a particular single entity (be it a person, or a place, or whatever else). For example: : '''Sloth''' is a '''proper name''' for ''the concept'' of ''being lazy''. : '''Rain''' is a '''proper name''' for ''the meteorological phenomenon of water falling from the sky''.


Definition:Property

A '''property''' is a concept that specifies an aspect of an object. In the phrase ''living thing'', ''living'' is a property that ''thing'' has. It creates a distinction between ''things'' that are ''living'' and ''things'' that are ''not living''.


Definition:Abelian Group

=== Definition 1 === {{:Definition:Abelian Group/Definition 1}} === Definition 2 === {{:Definition:Abelian Group/Definition 2}}


Definition:Commutative

=== Commuting Elements === {{:Definition:Commutative/Elements}} === Commutative Operation === {{:Definition:Commutative/Operation}} === Commuting Set of Elements === {{:Definition:Commutative/Set}} === Commutative Algebraic Structure === {{:Definition:Commutative/Algebraic Structure}}


Definition:Anticommutative

=== Structure with One Operation === {{:Definition:Anticommutative/Structure with One Operation}} === Structure with Two Operations === {{:Definition:Anticommutative/Structure with Two Operations}}


Definition:Normal Subgroup

Let $G$ be a group. Let $N$ be a subgroup of $G$. $N$ is a '''normal subgroup of $G$''' {{iff}}: === Definition 1 === {{:Definition:Normal Subgroup/Definition 1}} === Definition 2 === {{:Definition:Normal Subgroup/Definition 2}} === Definition 3 === {{:Definition:Normal Subgroup/Definition 3}} === Definition 4 === {{:Definition:Normal Subgroup/Definition 4}} === Definition 5 === {{:Definition:Normal Subgroup/Definition 5}} === Definition 6 === {{:Definition:Normal Subgroup/Definition 6}} === Definition 7 === {{:Definition:Normal Subgroup/Definition 7}}


Definition:Idempotence

'''Idempotence''' is a property of an algebraic system, or an element of an algebraic system such that: :$E \circledcirc E = E$ for an object $E$ and an operation $\circledcirc$.


Definition:Subgroup

Let $\struct {G, \circ}$ be an algebraic structure. $\struct {H, \circ}$ is a '''subgroup''' of $\struct {G, \circ}$ {{iff}}: :$(1): \quad \struct {H, \circ}$ is a group :$(2): \quad H$ is a subset of $G$.


Definition:Group

A '''group''' is a semigroup with an identity (that is, a monoid) in which every element has an inverse.


Definition:Center (Abstract Algebra)/Group

The '''center of a group''' $G$, denoted $\map Z G$, is the subset of elements in $G$ that commute with every element in $G$. Symbolically: :$\map Z G = \map {C_G} G = \set {g \in G: g x = x g, \forall x \in G}$ That is, the '''center''' of $G$ is the centralizer of $G$ in $G$ itself.


Definition:Distributive Operation

Let $S$ be a set on which is defined ''two'' binary operations, defined on all the elements of $S \times S$, which we will denote as $\circ$ and $*$. The operation $\circ$ '''is distributive over''' $*$, or '''distributes over''' $*$, {{iff}}: :$\circ$ is right distributive over $*$ and: :$\circ$ is left distributive over $*$.


Definition:Self Distributive

Let $\circ$ be a binary operation on the set $S$. Then $\circ$ is '''self-distributive''' iff: * $\forall a, b, c \in S: \left({a \circ b}\right) \circ c = \left ({a \circ c}\right) \circ \left({b \circ c}\right)$ * $\forall a, b, c \in S: a \circ \left({b \circ c}\right) = \left({a \circ b}\right) \circ \left({a \circ c}\right)$


Definition:Ordered Pair

The definition of a set does not take any account of the order in which the elements are listed. That is, $\set {a, b} = \set {b, a}$, and the elements $a$ and $b$ have the same status - neither is distinguished above the other as being more "important". === Informal Definition === {{:Definition:Ordered Pair/Informal Definition}} === Kuratowski Formalization === {{:Definition:Ordered Pair/Kuratowski Formalization}} === Empty Set Formalization === {{:Definition:Ordered Pair/Empty Set Formalization}} === Wiener Formalization === {{:Definition:Ordered Pair/Wiener Formalization}}


Definition:Cartesian Product

Let $S$ and $T$ be sets or classes. The '''cartesian product''' $S \times T$ of $S$ and $T$ is the set (or class) of ordered pairs $\tuple {x, y}$ with $x \in S$ and $y \in T$: :$S \times T = \set {\tuple {x, y}: x \in S \land y \in T}$


Definition:Relation

Let $S \times T$ be the cartesian product of two sets $S$ and $T$. A '''relation''' on $S \times T$ is an ordered triple: :$\RR = \tuple {S, T, R}$ where $R \subseteq S \times T$ is a subset of the Cartesian product of $S$ and $T$.


Definition:Mapping

Let $S$ and $T$ be sets. Let $S\times T$ be their cartesian product. === Definition 1 === {{:Definition:Mapping/Definition 1}} === Definition 2 === {{:Definition:Mapping/Definition 2}} === Definition 3 === {{:Definition:Mapping/Definition 3}} === Definition 4 === {{:Definition:Mapping/Definition 4}}


Definition:Composition of Relations

Let $\RR_1 \subseteq S_1 \times T_1$ and $\RR_2 \subseteq S_2 \times T_2$ be relations. Then the '''composite of $\RR_1$ and $\RR_2$''' is defined and denoted as: :$\RR_2 \circ \RR_1 := \set {\tuple {x, z} \in S_1 \times T_2: \exists y \in S_2 \cap T_1: \tuple {x, y} \in \RR_1 \land \tuple {y, z} \in \RR_2}$


Definition:Algebraic Structure

An '''algebraic structure''' is an ordered tuple: :$\struct {S, \circ_1, \circ_2, \ldots, \circ_n}$ where $S$ is a set which has one or more binary operations $\circ_1, \circ_2, \ldots, \circ_n$ defined on all the elements of $S \times S$. An '''algebraic structure''' with one (binary) operation is thus an ordered pair which can be denoted $\struct {S, \circ}$ or $\struct {T, *}$, and so on.


Definition:Restriction

=== Restriction of a Relation === {{:Definition:Restriction/Relation}} === Restriction of a Mapping === {{:Definition:Restriction/Mapping}} === Restriction of an Operation === In the same way that a restriction is defined on a relation, it can be defined on a binary operation. {{:Definition:Restriction/Operation}}


Definition:Closure (Abstract Algebra)

=== Algebraic Structures === {{:Definition:Closure (Abstract Algebra)/Algebraic Structure}} === Scalar Product === {{:Definition:Closure (Abstract Algebra)/Scalar Product}} Category:Definitions/Abstract Algebra Category:Definitions/Linear Algebra cx9tvgqnuptsin06zh38wl1671k2lmh


Definition:Magma

A '''magma''' is an algebraic structure $\struct {S, \circ}$ such that $S$ is closed under $\circ$.


Definition:Submagma

Let $\struct {S, \circ}$ be a magma. Let $T \subseteq S$ such that $\struct {T, \circ}$ is a magma. Then $\struct {T, \circ}$ is a '''submagma''' of $\struct {S, \circ}$. This relation can be denoted: : $\struct {T, \circ} \subseteq \struct {S, \circ}$ === Induced Operation === {{:Definition:Operation Induced by Restriction}}


Definition:Cancellable Element

Let $\left ({S, \circ}\right)$ be an algebraic structure.


Definition:Semigroup

: A '''semigroup''' is an algebraic structure which is closed and whose operation is associative.


Definition:Subsemigroup

Let $\struct {S, \circ}$ be an algebraic structure. Let $T \subseteq S$ such that $\struct {T, \circ {\restriction_T} }$, where $\circ {\restriction_T}$ is the restriction of $\circ$ to $T$, is a semigroup. Then $\struct {T, \circ {\restriction_T} }$ is a '''subsemigroup''' of $S$.


Definition:Commutative Semigroup

Let $\left({S, \circ}\right)$ be a semigroup such that the operation $\circ$ is a commutative operation. Then $\left({S, \circ}\right)$ is a '''commutative semigroup'''.


Definition:Regular Representations

Let $\struct {S, \circ}$ be a magma. === Left Regular Representation === {{:Definition:Regular Representations/Left Regular Representation}} === Right Regular Representation === {{:Definition:Regular Representations/Right Regular Representation}}


Definition:Subset Product

Let $\struct {S, \circ}$ be an algebraic structure. We can define an operation on the power set $\powerset S$ as follows: :$\forall A, B \in \powerset S: A \circ_\PP B = \set {a \circ b: a \in A, b \in B}$ This is called the '''operation induced on $\powerset S$ by $\circ$''', and $A \circ_\PP B$ is called the '''subset product''' of $A$ and $B$.


Definition:Cyclic Group

=== Definition 1 === {{:Definition:Cyclic Group/Definition 1}} === Definition 2 === {{:Definition:Cyclic Group/Definition 2}}


Definition:Inverse Relation

Let $\RR \subseteq S \times T$ be a relation. The '''inverse relation to''' (or '''of''') $\RR$ is defined as: :$\RR^{-1} := \set {\tuple {t, s}: \tuple {s, t} \in \RR}$


Definition:Reflexivity

Let $\mathcal R \subseteq S \times S$ be a relation in $S$. === Reflexive === {{:Definition:Reflexive Relation}} === Coreflexive === {{:Definition:Coreflexive Relation}} === Antireflexive === {{:Definition:Antireflexive Relation}} === Non-reflexive === {{Definition:Non-reflexive Relation}}


Definition:Transitivity (Relation Theory)

Let $\mathcal R \subseteq S \times S$ be a relation in $S$.


Definition:Symmetry (Relation)

Let $\mathcal R \subseteq S \times S$ be a relation in $S$. === Symmetric === {{:Definition:Symmetric Relation}} === Asymmetric === {{:Definition:Asymmetric Relation}} === Antisymmetric === {{:Definition:Antisymmetric Relation}} === Non-symmetric === {{:Definition:Non-symmetric Relation}}


Definition:Equivalence Relation

Let $\RR$ be a relation on a set $S$.


Definition:Equivalence Class

Let $S$ be a set. Let $\RR \subseteq S \times S$ be an equivalence relation on $S$. Let $x \in S$. Then the '''equivalence class of $x$ under $\RR$''' is the set: :$\eqclass x \RR = \set {y \in S: \tuple {x, y} \in \RR}$


Definition:Power Set

The '''power set''' of a set $S$ is the set defined and denoted as: :$\powerset S := \set {T: T \subseteq S}$ That is, the set whose elements are all of the subsets of $S$.


Definition:Universal Quantifier

The symbol $\forall$ is called the '''universal quantifier'''. It expresses the fact that, in a particular universe of discourse, all objects have a particular property. That is: :$\forall x:$ means: :'''For all objects $x$, it is true that ...''' In the language of set theory, this can be formally defined: :$\forall x \in S: \map P x := \set {x \in S: \map P x} = S$ where $S$ is some set and $\map P x$ is a propositional function on $S$.


Definition:Existential Quantifier

The symbol $\exists$ is called the '''existential quantifier'''. It expresses the fact that, in a particular universe of discourse, there exists (at least one) object having a particular property. That is: :$\exists x:$ means: :'''There exists at least one object $x$ such that ...''' In the language of set theory, this can be formally defined: :$\exists x \in S: \map P x := \set {x \in S: \map P x} \ne \O$ where $S$ is some set and $\map P x$ is a propositional function on $S$.


Definition:Propositional Expansion

Suppose our universe of discourse consists of the objects $\mathbf X_1, \mathbf X_2, \mathbf X_3, \ldots$ and so on. (There may be an infinite number of objects in this universe.) === Universal Quantifier === {{:Definition:Propositional Expansion/Universal Quantifier}} === Existential Quantifier === {{:Definition:Propositional Expansion/Existential Quantifier}} Category:Definitions/Quantifiers fymh785in8ghd2ze9r80zld20a5k451


Definition:Disjoint Sets

Two sets $S$ and $T$ are '''disjoint''' {{iff}}: :$S \cap T = \O$ That is, '''disjoint sets''' are such that their intersection is the empty set -- they have no elements in common.


Definition:Pairwise Disjoint

=== Set of Sets === {{:Definition:Pairwise Disjoint/Set of Sets}} === Family of Sets === {{:Definition:Pairwise Disjoint/Family}}


Definition:Set Partition

Let $S$ be a set. === Definition 1 === {{:Definition:Set Partition/Definition 1}} === Definition 2 === {{:Definition:Set Partition/Definition 2}}


Definition:Relative Complement

Let $S$ be a set, and let $T \subseteq S$, that is: let $T$ be a subset of $S$. Then the set difference $S \setminus T$ can be written $\relcomp S T$, and is called the '''relative complement of $T$ in $S$''', or the '''complement of $T$ relative to $S$'''. Thus: :$\relcomp S T = \set {x \in S : x \notin T}$


Definition:Identity (Abstract Algebra)

Let $\left({S, \circ}\right)$ be an algebraic structure. === Left Identity === {{:Definition:Identity (Abstract Algebra)/Left Identity}} === Right Identity === {{:Definition:Identity (Abstract Algebra)/Right Identity}} === Two-Sided Identity === {{:Definition:Identity (Abstract Algebra)/Two-Sided Identity}}


Definition:Symmetric Group

Let $S$ be a set. Let $\map \Gamma S$ denote the set of permutations on $S$. Let $\struct {\map \Gamma S, \circ}$ be the algebraic structure such that $\circ$ denotes the composition of mappings. Then $\struct {\map \Gamma S, \circ}$ is called the '''symmetric group on $S$'''. If $S$ has $n$ elements, then $\struct {\map \Gamma S, \circ}$ is often denoted $S_n$.


Definition:Inverse (Abstract Algebra)

Let $\struct {S, \circ}$ be a monoid whose identity is $e_S$. === Left Inverse === {{:Definition:Inverse (Abstract Algebra)/Left Inverse}} === Right Inverse === {{:Definition:Inverse (Abstract Algebra)/Right Inverse}} === Inverse === {{:Definition:Inverse (Abstract Algebra)/Inverse}}


Definition:Symmetric Difference

The '''symmetric difference''' between two sets $S$ and $T$ is written $S * T$ and is defined as: === Definition 1 === {{:Definition:Symmetric Difference/Definition 1}} === Definition 2 === {{:Definition:Symmetric Difference/Definition 2}} === Definition 3 === {{:Definition:Symmetric Difference/Definition 3}} === Definition 4 === {{:Definition:Symmetric Difference/Definition 4}} === Definition 5 === {{:Definition:Symmetric Difference/Definition 5}}


Definition:Set Complement

The '''set complement''' (or, when the context is established, just '''complement''') of a set $S$ in a universe $\mathbb U$ is defined as: :$\map \complement S = \relcomp {\mathbb U} S = \mathbb U \setminus S$ See the definition of Relative Complement for the definition of $\relcomp {\mathbb U} S$.


Definition:Left Hand Side

In an equation: :$\text {Expression $1$} = \text {Expression $2$}$ the term $\text {Expression $1$}$ is the '''left hand side'''.


Definition:Right Hand Side

In an equation: :$\text {Expression $1$} = \text {Expression $2$}$ the term $\text {Expression $2$}$ is the '''right hand side'''.


Definition:Domain (Set Theory)

=== Relation === {{:Definition:Domain (Set Theory)/Relation}} === Mapping === The term '''domain''' is usually seen when the relation in question is actually a mapping. {{:Definition:Domain (Set Theory)/Mapping}} === Binary Operation === {{:Definition:Domain (Set Theory)/Binary Operation}}


Definition:Range of Relation

Let $\RR \subseteq S \times T$ be a relation, or (usually) a mapping (which is, of course, itself a relation). The '''range''' of $\RR$, denoted  is defined as one of two things, depending on the source. On {{ProofWiki}} it is denoted $\Rng \RR$, but this may non-standard. === Range as Codomain === The '''range''' of a relation $\RR \subseteq S \times T$ can be defined as the set $T$. As such, it is the same thing as the term '''codomain''' of $\RR$. === Range as Image === The '''range''' of a relation $\RR \subseteq S \times T$ can also be defined as: :$\Rng \RR = \set {t \in T: \exists s \in S: \tuple {s, t} \in \RR}$ Defined like this, it is the same as what is defined as the '''image set''' of $\RR$.


Definition:Image (Set Theory)

== Relation == {{:Definition:Image (Set Theory)/Relation}} == Mapping == {{:Definition:Image (Set Theory)/Mapping}}


Definition:Trivial Relation

The '''trivial relation''' is the relation $\RR \subseteq S \times T$ in $S$ to $T$ such that ''every'' element of $S$ relates to ''every'' element in $T$: :$\RR: S \times T: \forall \tuple {s, t} \in S \times T: \tuple {s, t} \in \RR$


Definition:Null Relation

The '''null relation ''' is a relation $\RR$ in $S$ to $T$ such that $\RR$ is the empty set: :$\RR \subseteq S \times T: \RR = \O$ That is, ''no'' element of $S$ relates to ''any'' element in $T$: :$\RR: S \times T: \forall \tuple {s, t} \in S \times T: \neg s \mathrel \RR t$


Definition:Diagonal Relation

Let $S$ be a set. The '''diagonal relation on $S$''' is a relation $\Delta_S$ on $S$ such that: :$\Delta_S = \set {\tuple {x, x}: x \in S} \subseteq S \times S$ Alternatively: :$\Delta_S = \set {\tuple {x, y}: x, y \in S: x = y}$


Definition:Proper Subset

If a set $S$ is a subset of another set $T$, that is, $S \subseteq T$, and also: :$S \ne T$ :$S \ne \O$ then $S$ is referred to as a '''proper subset''' of $T$. The set $T$ '''properly contains''', or '''strictly contains''', the set $S$. If $S \subseteq T$ and $S \ne T$, then the notation $S \subsetneqq T$ is used. If we wish to refer to a set which we specifically require not to be empty, we can denote it like this: :$\O \subsetneqq S$ and one which we want to specify as possibly being empty, we write: :$\O \subseteq S$ Thus for $S$ to be a '''proper subset''' of $T$, we can write it as $\O \subsetneqq S \subsetneqq T$. === Proper Superset === {{:Definition:Proper Subset/Proper Superset}}


Definition:Complement of Relation

Let $\RR \subseteq S \times T$ be a relation. The '''complement of $\RR$''' is the relative complement of $\RR$ with respect to $S \times T$: :$\relcomp {S \times T} \RR := \set {\tuple {s, t} \in S \times T: \tuple {s, t} \notin \RR}$ If the sets $S$ and $T$ are implicit, then $\map \complement \RR$ can be used.


Definition:Extension of Relation

Let: : $\mathcal R_1 \subseteq X \times Y$ be a relation on $X \times Y$ : $\mathcal R_2 \subseteq S \times T$ be a relation on $S \times T$ : $X \subseteq S$ : $Y \subseteq T$ : $\mathcal R_2 \restriction_{X \times Y}$ be the restriction of $\mathcal R_2$ to $X \times Y$. Let $\mathcal R_2 \restriction_{X \times Y} = \mathcal R_1$. Then $\mathcal R_2$ '''extends''' or '''is an extension of''' $\mathcal R_1$.


Definition:Agreement

=== Relations === {{:Definition:Agreement/Relations}} === Mappings === The concept is usually seen in the context of mappings: {{:Definition:Agreement/Mappings}} Category:Definitions/Relation Theory Category:Definitions/Mapping Theory qv7evisfim8za8vixzcutf0u8sgo5sj


Definition:Combinable

=== Relations === {{:Definition:Combinable/Relations}} === Mappings === The concept is usually seen in the context of mappings: {{:Definition:Combinable/Mappings}} Category:Definitions/Relation Theory Category:Definitions/Mapping Theory 6xigoakjky7ou6jg6gg0lnsowcl5esl


Definition:Union Relation

Let: :$(1): \quad \mathcal R_1 \subseteq S_1 \times T_1$ be a relation on $S_1 \times T_1$ :$(2): \quad \mathcal R_2 \subseteq S_2 \times T_2$ be a relation on $S_2 \times T_2$ Let $\mathcal R_1$ and $\mathcal R_2$ be combinable, that is, that they agree on $S_1 \cap S_2$. Then the '''union relation''' (or '''combined relation''') $\mathcal R$ of $\mathcal R_1$ and $\mathcal R_2$ is: :$\mathcal R \subseteq \left({S_1 \cup S_2}\right) \times \left({T_1 \cup T_2}\right): \mathcal R \left({s}\right) = \begin{cases}  \mathcal R_1 \left({s}\right) : & s \in S_1 \\  \mathcal R_2 \left({s}\right) : & s \in S_2 \end{cases}$


Definition:Relational Structure

A '''relational structure''' is an ordered pair $\struct {S, \RR}$, where: :$S$ is a set :$\RR$ is an endorelation on $S$.


Definition:Many-to-One Relation

A relation $\RR \subseteq S \times T$ is '''many-to-one''' {{iff}}: :$\forall x \in \Dom \RR: \forall y_1, y_2 \in \Cdm \RR: \tuple {x, y_1} \in \RR \land \tuple {x, y_2} \in \RR \implies y_1 = y_2$ That is, every element of the domain of $\RR$ relates to no more than one element of its codomain.


Definition:One-to-Many Relation

A relation $\RR \subseteq S \times T$ is '''one-to-many''' {{iff}}: :$\RR \subseteq S \times T: \forall y \in \Img \RR: \tuple {x_1, y} \in \RR \land \tuple {x_2, y} \in \RR \implies x_1 = x_2$ That is, every element of the image of $\RR$ is related to by exactly one element of its domain. Note that the condition on $t$ concerns the elements in the ''image'', not the codomain. Thus a '''one-to-many relation''' may leave some element(s) of the codomain unrelated.


Definition:One-to-One Relation

A relation $\RR \subseteq S \times T$ is '''one-to-one''' if it is both many-to-one and one-to-many. That is, every element of the domain of $\RR$ relates to no more than one element of its codomain, and every element of the image is related to by exactly one element of its domain.


Definition:Many-to-Many Relation

A relation $\mathcal R \subseteq S \times T$ is '''many-to-many''' if it is neither many-to-one nor one-to-many. That is, there is no restriction to the number of elements relating to or being related to by any individual element. {{SUBPAGENAME}} 9pp6l117o3zdptpdhqwbjgxldvc5l89


Definition:Graph (Graph Theory)

A '''graph''' is intuitively defined as a pair consisting of a set of vertices and a set of edges. 400px


Definition:Quotient Set

Let $\RR$ be an equivalence relation on a set $S$. For any $x \in S$, let $\eqclass x \RR$ be the $\RR$-equivalence class of $x$. The '''quotient set of $S$ induced by $\RR$''' is the set $S / \RR$ of $\RR$-classes of $\RR$: :$S / \RR := \set {\eqclass x \RR: x \in S}$


Definition:Proper Coloring

=== Proper Vertex Coloring === A '''proper (vertex) $k$-coloring''' of a simple graph $G = \left({V, E}\right)$ is defined as a vertex coloring from a set of $k$ colors such that no two adjacent vertices share a common color. That is, a $k$-coloring of the graph $G = \left({V, E}\right)$ is a mapping $c: V \to \left\{{1, 2, \ldots k}\right\}$ such that: :$\forall e = \left\{{u, v}\right\} \in E: c \left({u}\right) \ne \left({v}\right)$ === Proper Edge Coloring === A '''proper (edge) $k$-coloring''' of a simple graph $G = \left({V, E}\right)$ is defined as an edge coloring from a set of $k$ colors such that no two adjacent edges share a common color. That is, a $k$-coloring of the graph $G = \left({V, E}\right)$ is a mapping $c: E \to \left\{{1, 2, \ldots k}\right\}$ such that: :$\forall v \in V: \forall e = \left\{{u_k, v}\right\} \in E: c \left\{{u_i, v}\right\} \ne c \left\{{u_j, v}\right\}$ {{SUBPAGENAME}} gso9y1cj4k3eofv3qpouxubl0xume6b


Definition:Chromatic Number

The '''chromatic number''' $\map \chi G$ of a graph $G$ is the smallest positive integer $k$ such that there exists a proper vertex $k$-coloring of $G$.


Definition:Graph of Mapping

Let $S$ and $T$ be sets. Let $f: S \to T$ be a mapping. The '''graph''' of $f$ is the relation  $\RR \subseteq S \times T$ defined as $\RR = \set {\tuple {x, \map f x}: x \in S}$ Alternatively, this can be expressed: :$G_f = \set {\tuple {s, t} \in S \times T: \map f s = t}$ where $G_f$ is the '''graph of $f$'''.


Definition:Composition of Mappings

Let $S_1$, $S_2$ and $S_3$ be sets. Let $f_1: S_1 \to S_2$ and $f_2: S_2 \to S_3$ be mappings such that the domain of $f_2$ is the same set as the codomain of $f_1$. === Definition 1 === {{:Definition:Composition of Mappings/Definition 1}} === Definition 2 === {{:Definition:Composition of Mappings/Definition 2}} === Definition 3 === {{:Definition:Composition of Mappings/Definition 3}}


Definition:Inverse of Mapping

Let $S$ and $T$ be sets. Let $f: S \to T$ be a mapping The '''inverse''' of $f$ is its inverse relation, defined as: :$f^{-1} := \set {\tuple {t, s}: \map f s = t}$ That is: :$f^{-1} := \set {\tuple {t, s}: \tuple {s, t} \in f}$ That is, $f^{-1} \subseteq T \times S$ is the relation which satisfies: :$\forall s \in S: \forall t \in T: \tuple {t, s} \in f^{-1} \iff \tuple {s, t} \in f$


Definition:Empty Mapping

Let $T$ be a set. Then the mapping whose domain is the empty set and whose codomain is $T$ is called the '''empty mapping''': :$\O \subseteq \O \times T = \O$


Definition:Identity Mapping

The '''identity mapping''' of a set $S$ is the mapping $I_S: S \to S$ defined as: :$I_S = \set {\tuple {x, y} \in S \times S: x = y}$ or alternatively: :$I_S = \set {\tuple {x, x}: x \in S}$ That is: :$I_S: S \to S: \forall x \in S: \map {I_S} x = x$


Definition:Inverse Mapping

Let $S$ and $T$ be sets.


Definition:Injection

=== Definition 1 === {{:Definition:Injection/Definition 1}} === Definition 2 === {{:Definition:Injection/Definition 2}} === Definition 3 === {{:Definition:Injection/Definition 3}} === Definition 4 === {{:Definition:Injection/Definition 4}} === Definition 5 === {{:Definition:Injection/Definition 5}} === Definition 6 === {{:Definition:Injection/Definition 6}}


Definition:Left Cancellable Mapping

A mapping $f: Y \to Z$ is '''left cancellable''' (or '''left-cancellable''') {{iff}}: :$\forall X: \forall \struct {g_1, g_2: X \to Y}: f \circ g_1 = f \circ g_2 \implies g_1 = g_2$ That is, for any set $X$, if $g_1$ and $g_2$ are mappings from $X$ to $Y$: :If $f \circ g_1 = f \circ g_2$ :then $g_1 = g_2$.


Definition:Inclusion Mapping

Let $T$ be a set. Let $S\subseteq T$ be a subset. The '''inclusion mapping''' $i_S: S \to T$ is the mapping defined as: :$i_S: S \to T: \forall x \in S: \map {i_S} x = x$


Definition:Surjection

Let $S$ and $T$ be sets or classes. Let $f: S \to T$ be a mapping from $S$ to $T$.


Definition:Right Cancellable Mapping

A mapping $f: X \to Y$ is '''right cancellable''' (or '''right-cancellable''') {{iff}}: :$\forall Z: \forall \paren {h_1, h_2: Y \to Z}: h_1 \circ f = h_2 \circ f \implies h_1 = h_2$ That is, {{iff}} for any set $Z$: :If $h_1$ and $h_2$ are mappings from $Y$ to $Z$ :then $h_1 \circ f = h_2 \circ f$ implies $h_1 = h_2$.


Definition:Bijection

=== Definition 1 === {{:Definition:Bijection/Definition 1}} === Definition 2 === {{:Definition:Bijection/Definition 2}} === Definition 3 === {{:Definition:Bijection/Definition 3}} === Definition 4 === {{:Definition:Bijection/Definition 4}} === Definition 5 === {{:Definition:Bijection/Definition 5}}


Definition:Permutation

A bijection $f: S \to S$ from a set $S$ to itself is called a '''permutation on''' (or '''of''') '''$S$'''.


Definition:Inverse Mapping/Definition 2

Let $f: S \to T$ and $g: T \to S$ be mappings. Let: :$g \circ f = I_S$ :$f \circ g = I_T$ where: :$g \circ f$ and $f \circ g$ denotes the composition of $f$ with $g$ in either order :$I_S$ and $I_T$ denote the identity mappings on $S$ and $T$ respectively. That is, $f$ and $g$ are both left inverse mappings and right inverse mappings of each other. Then: :$g$ is '''the inverse (mapping) of $f$''' :$f$ is '''the inverse (mapping) of $g$'''.


Definition:Set Equivalence

Let $S$ and $T$ be sets. Then $S$ and $T$ are '''equivalent''' {{iff}}: :there exists a bijection $f: S \to T$ between the elements of $S$ and those of $T$. That is, if they have the '''same cardinality'''. This can be written $S \sim T$. If $S$ and $T$ are not '''equivalent''' we write $S \nsim T$.


Definition:Injective Restriction

Let $f: S \to T$ be a mapping which is not injective. Then an '''injective restriction''' of $f$ is a restriction $g {\restriction_{S'}}: S' \to T$ of $f$ which is an injection. Category:Definitions/Injections Category:Definitions/Restrictions bm55fhzivy70xxq051q4xt7egqr3kda


Definition:Surjective Restriction

Let $f: S \to T$ be a mapping which is not surjective. Then a '''surjective restriction''' of $f$ is a restriction $g: S' \to T'$ of $f$ such that $g \restriction_{S'}: S' \to T'$ is a surjection.


Definition:Bijective Restriction

Let $f: S \to T$ be a mapping which is not bijective. A '''bijective restriction''' of $f$ is a restriction $f {\restriction_{S' \times T'} }: S' \to T'$ of $f$ such that $f {\restriction_{S' \times T'} }$ is a bijection.


Definition:Set of All Mappings

Let $S$ and $T$ be sets. The '''set of (all) mappings from $S$ to $T$''' is: :$T^S := \set {f \subseteq S \times T: f: S \to T \text { is a mapping} }$


Definition:Projection (Mapping Theory)

Let $S_1, S_2, \ldots, S_j, \ldots, S_n$ be sets. Let $\displaystyle \prod_{i \mathop = 1}^n S_i$ be the Cartesian product of $S_1, S_2, \ldots, S_n$. For each $j \in \set {1, 2, \ldots, n}$, the '''$j$th projection on $\displaystyle S = \prod_{i \mathop = 1}^n S_i$''' is the mapping $\pr_j: S \to S_j$ defined by: :$\map {\pr_j} {s_1, s_2, \ldots, s_j, \ldots, s_n} = s_j$ for all $\tuple {s_1, s_2, \ldots, s_n} \in S$.


Definition:Quotient Mapping

Let $\RR \subseteq S \times S$ be an equivalence on a set $S$. Let $\eqclass s \RR$ be the $\RR$-equivalence class of $s$. Let $S / \RR$ be the quotient set of $S$ determined by $\RR$. Then $q_\RR: S \to S / \RR$ is the '''quotient mapping induced by $\RR$''', and is defined as: :$q_\RR: S \to S / \RR: \map {q_\RR} s = \eqclass s \RR$


Definition:Equivalence Relation Induced by Mapping

Let $f: S \to T$ be a mapping. Let $\RR_f \subseteq S \times S$ be the relation defined as: :$\tuple {s_1, s_2} \in \RR_f \iff \map f {s_1} = \map f {s_2}$ Then $\RR_f$ is known as the '''equivalence (relation) induced by $f$'''.


Definition:Well-Defined

=== Well-Defined Mapping === {{:Definition:Well-Defined/Mapping}} === Well-Defined Relation === The concept can be generalized to include the general relation $\RR: S \to T$. {{:Definition:Well-Defined/Relation}} === Well-Defined Operation === {{:Definition:Well-Defined/Operation}}


Definition:Dominate (Set Theory)

Let $S$ and $T$ be sets. === Definition 1 === {{:Definition:Dominate (Set Theory)/Definition 1}} === Definition 2 === {{:Definition:Dominate (Set Theory)/Definition 2}} === Strictly Dominated === {{:Definition:Dominate (Set Theory)/Strictly Dominate}}


Definition:Ordering

Let $S$ be a set. === Definition 1 === {{:Definition:Ordering/Definition 1}} === Definition 2 === {{:Definition:Ordering/Definition 2}}


Definition:Partially Ordered Set

A '''partially ordered set''' is a relational structure $\left({S, \preceq}\right)$ such that $\preceq$ is a partial ordering. The '''partially ordered set''' $\left({S, \preceq}\right)$ is said to be '''partially ordered by $\preceq$'''.


Definition:Maximal

Let $\struct {S, \preceq}$ be an ordered set. Let $T \subseteq S$ be a subset of $S$. === Ordered Set === {{:Definition:Maximal/Ordered Set}} === Maximal Set === {{:Definition:Maximal/Set}}


Definition:Minimal

=== Relation === {{:Definition:Minimal/Relation}} === Ordered Set === Let $\struct {S, \preceq}$ be an ordered set. Let $T \subseteq S$ be a subset of $S$. {{:Definition:Minimal/Ordered Set}} === Minimal Set === {{:Definition:Minimal/Set}}


Definition:Non-Comparable

Let $\struct {S, \RR}$ be a relational structure. Two elements $x, y \in S, x \ne y$ are '''non-comparable''' if neither $x \mathrel \RR y$ nor $y \mathrel \RR x$. If $x$ and $y$ are not non-comparable then they are '''comparable''', but the latter term is not so frequently encountered.


Definition:Order Isomorphism

=== Definition 1 === {{:Definition:Order Isomorphism/Definition 1}} === Definition 2 === {{:Definition:Order Isomorphism/Definition 2}} Two ordered sets $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ are '''(order) isomorphic''' if there exists such an '''order isomorphism''' between them. $\struct {S, \preceq_1}$ is described as '''(order) isomorphic to''' (or '''with''') $\struct {T, \preceq_2}$, and vice versa. This may be written $\struct {S, \preceq_1} \cong \struct {T, \preceq_2}$. Where no confusion is possible, it may be abbreviated to $S \cong T$. === Well-Ordered Sets === When $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ are well-ordered sets, the condition on the order preservation can be relaxed: {{:Definition:Order Isomorphism/Wosets}}


Definition:Immediate Predecessor Element

Let $\struct {S, \preceq}$ be an ordered set. Let $a, b \in S$. Then $a$ is an '''immediate predecessor (element)''' to $b$ {{iff}}: :$(1): \quad a \prec b$ :$(2): \quad \neg \exists c \in S: a \prec c \prec b$ That is, there exists no element strictly between $a$ and $b$ in the ordering $\preceq$. That is: :$a \prec b$ and $\openint a b = \O$ where $\openint a b$ denotes the open interval from $a$ to $b$. We say that '''$a$ ''immediately'' precedes $b$'''.


Definition:Immediate Successor Element

Let $\struct {S, \preceq}$ be an ordered set. Let $a, b \in S$. Then $a$ is an '''immediate successor (element)''' to $b$ {{iff}} $b$ is an immediate predecessor (element) to $a$. That is, {{iff}}: :$(1): \quad b \prec a$ :$(2): \quad \nexists c \in S: b \prec c \prec a$ That is, there exists no element strictly between $b$ and $a$ in the ordering $\preceq$. That is: :$a \prec b$ and $\openint a b = \O$ where $\openint a b$ denotes the open interval from $a$ to $b$. We say that '''$a$ ''immediately'' succeeds $b$'''.


Definition:Lattice Ordering

Let $\left({S, \preceq}\right)$ be a lattice. Then the ordering $\preceq$ is referred to as a '''lattice ordering'''.


Definition:Complete Lattice

=== Definition 1 === {{:Definition:Complete Lattice/Definition 1}} === Definition 2 === {{:Definition:Complete Lattice/Definition 2}}


Definition:Total Ordering

Let $\mathcal R \subseteq S \times S$ be a relation on a set $S$.


Definition:Totally Ordered Set

Let $\struct {S, \preceq}$ be a relational structure. Then $\struct {S, \preceq}$ is a '''totally ordered set''' {{iff}} $\preceq$ is a total ordering.


Definition:Chain (Set Theory)

Let $\struct {S, \preceq}$ be an ordered set. A '''chain in $S$''' is a totally ordered subset of $S$. Thus a totally ordered set is itself a '''chain''' in its own right. === Length === {{Definition:Length of Chain}}


Definition:Maximal Chain

Let $\left({S, \preceq}\right)$ be an ordered set. Let $\left({T, \preceq}\right) \subseteq \left({S, \preceq}\right)$ be a chain in $\left({S, \preceq}\right)$ such that there is no other chain in $\left({S, \preceq}\right)$ which has $\left({T, \preceq}\right)$ as a proper subset. Then $\left({T, \preceq}\right)$ is a '''maximal chain in $S$'''. Category:Definitions/Order Theory 0e8z7g0a1z5089hyq1817vtbb4royah


Definition:Order Embedding

Let $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ be ordered sets. Let $\phi: S \to T$ be a mapping. === Definition 1 === {{:Definition:Order Embedding/Definition 1}} === Definition 2 === {{:Definition:Order Embedding/Definition 2}} === Definition 3 === {{:Definition:Order Embedding/Definition 3}} === Definition 4 === {{:Definition:Order Embedding/Definition 4}}


Definition:Strictly Increasing

=== Ordered Sets === {{:Definition:Strictly Increasing/Mapping}} === Real Functions === This definition continues to hold when $S = T = \R$. {{:Definition:Strictly Increasing/Real Function}} === Sequences === {{:Definition:Strictly Increasing Sequence}}


Definition:Strictly Decreasing

=== Ordered Sets === {{:Definition:Strictly Decreasing/Mapping}} === Real Functions === This definition continues to hold when $S = T = \R$. {{:Definition:Strictly Decreasing/Real Function}} === Sequences === {{:Definition:Strictly Decreasing Sequence}}


Definition:Strictly Monotone

=== Ordered Sets === {{:Definition:Strictly Monotone/Mapping}} === Real Functions === This definition continues to hold when $S = T = \R$: {{:Definition:Strictly Monotone/Real Function}} === Sequences === {{:Definition:Strictly Monotone/Sequence}}


Definition:Increasing

=== Ordered Sets === {{:Definition:Increasing/Mapping}} === Real Functions === This definition continues to hold when $S = T = \R$. {{:Definition:Increasing/Real Function}} === Sequences === {{:Definition:Increasing/Sequence}}


Definition:Decreasing

=== Ordered Sets === {{:Definition:Decreasing/Mapping}} === Real Functions === This definition continues to hold when $S = T = \R$; thus: {{:Definition:Decreasing/Real Function}} === Sequences === {{:Definition:Decreasing Sequence}}


Definition:Monotone (Order Theory)

=== Ordered Sets === {{:Definition:Monotone (Order Theory)/Mapping}} === Real Functions === {{:Definition:Monotone (Order Theory)/Real Function}} === Sequences === {{:Definition:Monotone (Order Theory)/Sequence}}


Definition:Well-Ordered Set

Let $\struct {S, \preceq}$ be an ordered set. Then $\struct {S, \preceq}$ is a '''well-ordered set''' if the ordering $\preceq$ is a well-ordering. That is, if ''every'' non-empty subset of $S$ has a smallest element: :$\forall T \in \powerset S: \exists a \in T: \forall x \in T: a \preceq x$ where $\powerset S$ denotes the power set of $S$. Or, such that $\preceq$ is a well-founded total ordering.


Definition:Entropic

=== Entropic Operation === {{:Definition:Entropic Operation}} === Entropic Structure === {{:Definition:Entropic Structure}}


Definition:Constant Operation

Let $S$ be a set. For a given $c \in S$, a '''constant operation''' on $S$ is defined as: :$\forall x, y \in S: x \left[{c}\right] y = c$


Definition:Left Operation

Let $S$ be a set. For any $x, y \in S$, the '''left operation''' on $S$ is the binary operation defined as: :$\forall x, y \in S: x \leftarrow y = x$


Definition:Right Operation

Let $S$ be a set. For any $x, y \in S$, the '''right operation''' on $S$ is the binary operation defined as: :$\forall x, y \in S: x \rightarrow y = y$


Definition:Max Operation

Let $\struct {S, \preceq}$ be a totally ordered set. The '''max operation''' is the binary operation on $\left({S, \preceq}\right)$ defined as: :$\forall x, y \in S: \map \max {x, y} = \begin{cases} y & : x \preceq y \\ x & : y \preceq x \end{cases}$ === General Definition === {{:Definition:Max Operation/General Definition}}


Definition:Min Operation

Let $\struct {S, \preceq}$ be a totally ordered set. The '''min operation''' is the binary operation on $\struct {S, \preceq}$ defined as: :$\forall x, y \in S: \map \min {x, y} = \begin{cases} x & : x \preceq y \\ y & : y \preceq x \end{cases}$


Definition:Underlying Set

=== Abstract Algebra === {{:Definition:Underlying Set/Abstract Algebra}} === Relational Structure === {{:Definition:Underlying Set/Relational Structure}} === Metric Space === {{:Definition:Underlying Set/Metric Space}} === Topological Space === {{:Definition:Underlying Set/Topological Space}} Category:Definitions/Category Theory a7mb18de9mttrahnxjpc0qw88z2jsxf


Definition:Free Semigroup

A semigroup which has a non-commutative product in which no product can ever be expressed more simply in terms of the other elements is called a '''free semigroup'''. {{wtd|add formal definition}} Category:Definitions/Semigroups 5x3mhh7kk70r8suh77znqcgbbq4ia1w


Definition:Monoid

A '''monoid''' is a semigroup with an identity element.


Definition:Commutative Monoid

A monoid whose operation is commutative is a '''commutative monoid'''.


Definition:Free Idempotent Monoid

Let $\left({S, \circ}\right)$ be a monoid which is generated by a set of elements, and such that: :$\forall x \in S: x = x \circ x$ Then $\left({S, \circ}\right)$ is a '''free idempotent monoid'''. Category:Definitions/Monoids e9oprbgcas2ctxk70wfg41gmz4it0i7


Definition:Submonoid

A '''submonoid''' of a monoid $\struct {S, \circ}$ is a monoid $\struct {T, \circ}$ such that $T \subseteq S$.


Definition:Inverse Semigroup

An '''inverse semigroup''' is a semigroup $\struct {S, \circ}$ such that: :$\forall a \in S: \exists! b \in S: a = a \circ b \circ a, b = b \circ a \circ b$


Definition:Invertible Element

Let $\struct {S, \circ}$ be an algebraic structure which has an identity $e_S$. If $x \in S$ has an inverse, then $x$ is said to be '''invertible for $\circ$'''. That is, $x$ is '''invertible''' {{iff}}: :$\exists y \in S: x \circ y = e_S = y \circ x$ === Invertible Operation === {{:Definition:Invertible Operation}}


Definition:Trivial Group

A '''trivial group''' is a group with only one element $e$.


Definition:Quasigroup

A '''quasigroup''' is a magma $\struct {S, \circ}$ which has the Latin square property. That is, such that $\forall a \in S$, the left and right regular representations $\lambda_a$ and $\rho_a$ are permutations on $S$.


Definition:Algebra Loop

An '''algebra loop''' $\struct {S, \circ}$ is a quasigroup with an identity element.


Definition:Relation Compatible with Operation

Let $\struct {S, \circ}$ be a closed algebraic structure. Let $\RR$ be a relation on $S$. Then $\RR$ is '''compatible with $\circ$''' {{iff}}: :$\forall x, y, z \in S: x \mathrel \RR y \implies \paren {x \circ z} \mathrel \RR \paren {y \circ z}$ :$\forall x, y, z \in S: x \mathrel \RR y \implies \paren {z \circ x} \mathrel \RR \paren {z \circ y}$


Definition:Congruence Relation

Let $\struct {S, \circ}$ be an algebraic structure. Let $\mathcal R$ be an equivalence relation on $S$. Then $\mathcal R$ is a '''congruence relation for $\circ$''' {{iff}}: :$\forall x_1, x_2, y_1, y_2 \in S: \paren {x_1 \mathrel {\mathcal R} x_2} \land \paren {y_1 \mathrel {\mathcal R} y_2} \implies \paren {x_1 \circ y_1} \mathrel {\mathcal R} \paren {x_2 \circ y_2}$


Definition:Universally Compatible Relation

A relation $\RR$ is '''universally compatible''' on a set $S$ {{iff}} it is compatible with every closed operation that can be defined on $S$. Category:Definitions/Abstract Algebra twhewcuarnswdtdw7zeva9pgo97zit0


Definition:Universally Congruent

A equivalence $\RR$ is '''universally congruent''' on a set $S$ {{iff}} it is a congruence for every closed operation that can be defined on $S$. Category:Definitions/Abstract Algebra 9z1lpx8mkuih5wpyapcmr1bebin32g8


Definition:Operation Induced on Quotient Set

Let $\struct {S, \circ}$ be an algebraic structure. Let $\mathcal R$ be a congruence relation on $\struct {S, \circ}$. Let $S / \mathcal R$ be the quotient set of $S$ by $\mathcal R$. The '''operation $\circ_\mathcal R$ induced on $S / \mathcal R$ by $\circ$''' is defined as: :$\eqclass x {\mathcal R} \circ_\mathcal R \eqclass y {\mathcal R} = \eqclass {x \circ y} {\mathcal R}$


Definition:Quotient Structure

Let $\left({S, \circ}\right)$ be an algebraic structure. Let $\mathcal R$ be a congruence relation on $\left({S, \circ}\right)$. Let $S / \mathcal R$ be the quotient set of $S$ by $\mathcal R$. Let $\circ_\mathcal R$ be the operation induced on $S / \mathcal R$ by $\circ$. The '''quotient structure defined by $\mathcal R$''' is the algebraic structure: : $\left({S / \mathcal R, \circ_\mathcal R}\right)$


Definition:External Direct Product

Let $\left({S, \circ_1}\right)$ and $\left({T, \circ_2}\right)$ be algebraic structures. The '''(external) direct product''' $\left({S \times T, \circ}\right)$ of two algebraic structures $\left({S, \circ_1}\right)$ and $\left({T, \circ_2}\right)$ is the set of ordered pairs: :$\left({S \times T, \circ}\right) = \left\{{\left({s, t}\right): s \in S, t \in T}\right\}$ where the operation $\circ$ is defined as: :$\left({s_1, t_1}\right) \circ \left({s_2, t_2}\right) = \left({s_1 \circ_1 s_2, t_1 \circ_2 t_2}\right)$


Definition:Algebraic Substructure

Let $\left({S, \circ}\right)$ be an algebraic structure. Let $T \subseteq S$. Then $\left({T, \circ}\right)$ is an '''algebraic substructure''' of $\left({S, \circ}\right)$.


Definition:Internal Direct Product

Let $\left({S_1, \circ {\restriction_{S_1}} }\right), \left({S_2, \circ {\restriction_{S_2}} }\right)$ be closed algebraic substructures of an algebraic structure $\left({S, \circ}\right)$ where $\circ {\restriction_{S_1}}, \circ {\restriction_{S_2}}$ are the operations induced by the restrictions of $\circ$ to $S_1, S_2$ respectively. The structure $\left({S, \circ}\right)$ is the '''internal direct product of $S_1$ and $S_2$''' if the mapping: :$C: S_1 \times S_2 \to S: C \left({\left({s_1, s_2}\right)}\right) = s_1 \circ s_2$ is an isomorphism from the cartesian product $\left({S_1, \circ {\restriction_{S_1}}}\right) \times \left({S_2, \circ {\restriction_{S_2}}}\right)$ onto $\left({S, \circ}\right)$. The operation $\circ$ on $S$ is the operation induced on $S$ by $\circ {\restriction_{S_1}}$ and $\circ {\restriction_{S_2}}$.


Definition:Homomorphism (Abstract Algebra)

Let $\struct {S, \circ}$ and $\struct {T, *}$ be magmas. Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be a mapping from $\struct {S, \circ}$ to $\struct {T, *}$. Let $\circ$ have the morphism property under $\phi$, that is: :$\forall x, y \in S: \map \phi {x \circ y} = \map \phi x * \map \phi y$ Then $\phi$ is a '''homomorphism'''.


Definition:Morphism Property

Let $\phi: \left({S, \circ}\right) \to \left({T, *}\right)$ be a mapping from one algebraic structure $\left({S, \circ}\right)$ to another $\left({T, *}\right)$. Then $\circ$ has the '''morphism property''' under $\phi$ iff: :$\forall x, y \in S: \phi \left({x \circ y}\right) = \phi \left({x}\right) * \phi \left({y}\right)$


Definition:Epimorphism (Abstract Algebra)

A homomorphism which is a surjection is described as '''epic''', or called an '''epimorphism'''.


Definition:Endomorphism

An '''endomorphism''' is a homomorphism from an algebraic structure into itself.


Definition:Monomorphism (Abstract Algebra)

A homomorphism which is an injection is descibed as '''monic''', or called a '''monomorphism'''.


Definition:Isomorphism (Abstract Algebra)

An '''isomorphism''' is a homomorphism which is a bijection. That is, it is a mapping which is both a monomorphism and an epimorphism.


Definition:Automorphism (Abstract Algebra)

An '''automorphism''' is an isomorphism from an algebraic structure to itself.


Let $\struct {S_1, \circ_1}$ and $\struct {S_2, \circ_2}$ be algebraic structures with identities $e_1, e_2$ respectively. The following mappings: :$\inj_1: \struct {S_1, \circ_1} \to \struct {S_1, \circ_1} \times \struct {S_2, \circ_2}: \forall x \in S_1: \map {\inj_1} x = \tuple {x, e_2}$ :$\inj_2: \struct {S_2, \circ_2} \to \struct {S_1, \circ_1} \times \struct {S_2, \circ_2}: \forall x \in S_2: \map {\inj_2} x = \tuple {e_1, x}$ are called the '''canonical injections'''. === General Definition === {{:Definition:Canonical Injection (Abstract Algebra)/General Definition}}


Definition:Inverse Completion

Let $\struct {S, \circ}$ be a semigroup. Let $\struct {C, \circ} \subseteq \struct {S, \circ}$ be the subsemigroup of cancellable elements of $\struct {S, \circ}$. Let $\struct {T, \circ'}$ be a semigroup defined such that: :$(1): \quad \struct {S, \circ}$ is a subsemigroup of $\struct {T, \circ'}$ :$(2): \quad$ Every element of $C$ has an inverse in $T$ for $\circ'$ :$(3): \quad \gen {S \cup C^{-1} } = \struct {T, \circ'}$ where: :$\gen {S \cup C^{-1} }$ denotes the semigroup generated by $S \cup C^{-1}$ :$C^{-1}$ denotes the inverse of $C$. Then $\struct {T, \circ'}$ is called an '''inverse completion''' of $\struct {S, \circ}$.


Definition:Inverse of Subset

=== Monoid === {{Definition:Inverse of Subset/Monoid}} === Group === {{Definition:Inverse of Subset/Group}}


Definition:Automorphism Group/Group

Let $\struct {S, *}$ be an algebraic structure. Let $\mathbb S$ be the set of automorphisms of $S$. Then the algebraic structure $\struct {\mathbb S, \circ}$, where $\circ$ denotes composition of mappings, is called the '''automorphism group''' of $S$. The structure $\struct {S, *}$ is usually a group. However, this is not necessary for this definition to be valid. The '''automorphism group of $S$''' is denoted on {{ProofWiki}} as $\Aut S$.


Definition:Inner Automorphism

Let $G$ be a group. Let $x \in G$. Let the mapping $\kappa_x: G \to G$ be defined such that: :$\forall g \in G: \map {\kappa_x} g = x g x^{-1}$ $\kappa_x$ is called the '''inner automorphism of $G$ (given) by $x$'''.


Definition:Proper Subgroup

Let $\struct {G, \circ}$ be a group. Then $\struct {H, \circ}$ is a '''proper subgroup of $\struct {G, \circ}$''' {{iff}}: : $(1): \quad \struct {H, \circ}$ is a subgroup of $\struct {G, \circ}$ : $(2): \quad H \ne G$, i.e. $H \subset G$. The notation $H < G$, or $G > H$, means: : '''$H$ is a proper subgroup of $G$'''. If $H$ is a subgroup of $G$, but it is not specified whether $H = G$ or not, then we write $H \le G$, or $G \ge H$. === Non-Trivial Proper Subgroup === {{:Definition:Proper Subgroup/Non-Trivial}}


Definition:Trivial Subgroup

Let $\struct {G, \circ}$ be a group whose identity is $e$. Then the algebraic structure $\struct {\set e, \circ}$ is called '''the trivial subgroup''' of $\struct {G, \circ}$.


Definition:Conjugate (Group Theory)

Let $\left({G, \circ}\right)$ be a group. === Conjugate of an Element === {{:Definition:Conjugate (Group Theory)/Element}} === Conjugate of a Set === {{:Definition:Conjugate (Group Theory)/Subset}} Category:Definitions/Conjugacy Category:Definitions/Group Theory 6rvsq6cjwk0uolx4vxip5h1ihz56stg


Definition:Conjugacy Class

The equivalence classes into which the conjugacy relation divides its group into are called '''conjugacy classes'''. The '''conjugacy class''' of an element $x \in G$ can be denoted $\conjclass x$.


Definition:Kernel (Abstract Algebra)

=== Kernel of Magma Homomorphism === {{:Definition:Kernel of Magma Homomorphism}} === Kernel of Group Homomorphism === {{:Definition:Kernel of Group Homomorphism}} === Kernel of Ring Homomorphism === {{:Definition:Kernel of Ring Homomorphism}} === Kernel of Linear Transformation === {{:Definition:Kernel of Linear Transformation}} === Kernel of Homomorphism of Differential Complexes === {{:Definition:Kernel of Homomorphism of Differential Complexes}} Category:Definitions/Abstract Algebra 0imy2su25o141xy7qlspg0qh3nw1sf7


Definition:Entropic Structure

An '''entropic structure''' is an algebraic structure $\struct {S, \circ}$ such that $\circ$ is an entropic operation.


Definition:Ringoid (Abstract Algebra)

A '''ringoid''' is a triple $\struct {S, *, \circ}$ where: :$S$ is a set :$*$ and $\circ$ are binary operations on $S$ :the operation $\circ$ distributes over $*$. That is: :$\forall a, b, c \in S: a \circ \paren {b * c} = \paren {a \circ b} * \paren {a \circ c}$ :$\forall a, b, c \in S: \paren {a * b} \circ c = \paren {a \circ c} * \paren {b \circ c}$


Definition:Semiring (Abstract Algebra)

A '''semiring''' is a ringoid $\struct {S, *, \circ}$ in which: :$(1): \quad \struct {S, *}$ forms a semigroup :$(2): \quad \struct {S, \circ}$ forms a semigroup. That is, such that $\struct {S, *, \circ}$ has the following properties: {{begin-axiom}} {{axiom | n = \text A 0         | q = \forall a, b \in S         | m = a * b \in S         | c = Closure under $*$ }} {{axiom | n = \text A 1         | q = \forall a, b, c \in S         | m = \paren {a * b} * c = a * \paren {b * c}         | c = Associativity of $*$ }} {{axiom | n = \text M 0         | q = \forall a, b \in S         | m = a \circ b \in S         | c = Closure under $\circ$ }} {{axiom | n = \text M 1         | q = \forall a, b, c \in S         | m = \paren {a \circ b} \circ c = a \circ \paren {b \circ c}         | c = Associativity of $\circ$ }} {{axiom | n = \text D         | q = \forall a, b, c \in S         | m = a \circ \paren {b * c} = \paren {a \circ b} * \paren {a \circ c} }} {{axiom | m = \paren {a * b} \circ c = \paren {a \circ c} * \paren {b \circ c}         | c = $\circ$ is distributive over $*$ }} {{end-axiom}} These are called the '''semiring axioms'''.


Definition:Cancellable Semiring

A '''cancellable semiring''' is a semiring $\left({S, *, \circ}\right)$ in which all the elements of $S$ are cancellable for $*$.


Definition:Ring (Abstract Algebra)

A '''ring''' $\struct {R, *, \circ}$ is a semiring in which $\struct {R, *}$ forms an abelian group. That is, in addition to $\struct {R, *}$ being closed, associative and commutative under $*$, it also has an identity, and each element has an inverse.


Definition:Ring Zero

Let $\struct {R, +, \circ}$ be a ring. The identity for ring addition is called the '''ring zero''' (of $\struct {R, +, \circ}$). It is denoted $0_R$ (or just $0$ if there is no danger of ambiguity).


Definition:Ring Negative

Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$. Let $x \in R$. The inverse of $x$ with respect to the addition operation $+$ in the additive group $\struct {R, +}$ of $R$ is referred to as the '''(ring) negative''' of $x$ and is denoted $-x$. That is, the '''(ring) negative''' of $x$ is the element $-x$ of $R$ such that: :$x + \paren {-x} = 0_R$


Definition:Commutative Ring

A '''commutative ring''' is a ring $\struct {R, +, \circ}$ in which the ring product $\circ$ is commutative.


Definition:Unity (Abstract Algebra)/Ring

Let $\struct {R, +, \circ}$ be a ring. If the semigroup $\struct {R, \circ}$ has an identity, this identity is referred to as '''the unity of the ring $\struct {R, +, \circ}$'''. It is (usually) denoted $1_R$, where the subscript denotes the particular ring to which $1_R$ belongs (or often $1$ if there is no danger of ambiguity).


Definition:Trivial Ring

A ring $\struct {R, +, \circ}$ is a '''trivial ring''' {{iff}}: :$\forall x, y \in R: x \circ y = 0_R$


Definition:Null Ring

A ring with one element is called '''the null ring'''. That is, the '''null ring''' is $\struct {\set {0_R}, +, \circ}$, where ring addition and the ring product are defined as: :$0_R + 0_R = 0_R$ :$0_R \circ 0_R = 0_R$


Definition:Ring with Unity

Let $\struct {R, +, \circ}$ be a non-null ring. Then $\struct {R, +, \circ}$ is a '''ring with unity''' {{iff}} the multiplicative semigroup $\struct {R, \circ}$ has an identity element. Such an identity element is known as a unity.


Definition:Unit of Ring

Let $\struct {R, +, \circ}$ be a ring with unity whose unity is $1_R$. === Definition 1 === {{:Definition:Unit of Ring/Definition 1}} === Definition 2 === {{:Definition:Unit of Ring/Definition 2}}


Definition:Group of Units

=== Group of Units of Monoid === {{:Definition:Group of Units/Monoid}} === Group of Units of Ring === {{:Definition:Group of Units/Ring}} Category:Definitions/Ring Theory Category:Definitions/Monoids n24ojfgrw9lt9htgvmq9928d9jdx6sg


Definition:Division Product

Let $\struct {R, +, \circ}$ be a commutative ring with unity. Let $\struct {U_R, \circ}$ be the group of units of $\struct {R, +, \circ}$. Then we define the following notation: :$\forall x \in U_R, y \in R$, we have: ::$\dfrac y x := y \circ \paren {x^{-1} } = \paren {x^{-1} } \circ y$ $\dfrac y x$ is a '''division product''', and $\dfrac y x$ is voiced '''$y$ divided by $x$'''. We also write (out of space considerations) $y / x$ for $\dfrac y x$. This notation is usually used when $\struct {R, +, \circ}$ is a field.


Definition:Proper Element of Ring

A non-zero element of a ring which does not have a product inverse is called a '''proper element'''.


Definition:Zero Divisor

=== Rings === {{:Definition:Zero Divisor/Ring}} === Commutative Rings === The definition is usually made when the ring in question is commutative: {{:Definition:Zero Divisor/Commutative Ring}} === Algebras === {{:Definition:Zero Divisor/Algebra}}


Definition:Division Ring

A '''division ring''' is a ring with unity $\struct {R, +, \circ}$ with the following properties: === Definition 1 === {{:Definition:Division Ring/Definition 1}} === Definition 2 === {{:Definition:Division Ring/Definition 2}} === Definition 3 === {{:Definition:Division Ring/Definition 3}}


Definition:Field (Abstract Algebra)

A '''field''' is a non-trivial division ring whose ring product is commutative. Thus, let $\struct {F, +, \times}$ be an algebraic structure. Then $\struct {F, +, \times}$ is a '''field''' {{iff}}: :$(1): \quad$ the algebraic structure $\struct {F, +}$ is an abelian group :$(2): \quad$ the algebraic structure $\struct {F^*, \times}$ is an abelian group where $F^* = F \setminus \set 0$ :$(3): \quad$ the operation $\times$ distributes over $+$.


Definition:Skew Field

A '''skew field''' is a division ring whose ring product is specifically '''not''' commutative.


Definition:Integral Domain

=== Definition 1 === {{:Definition:Integral Domain/Definition 1}} === Definition 2 === {{:Definition:Integral Domain/Definition 2}}


Definition:Subring

Let $\struct {R, +, \circ}$ be an algebraic structure with two operations. A '''subring of $\struct {R, +, \circ}$''' is a subset $S$ of $R$ such that $\struct {S, +_S, \circ_S}$ is a ring.


Definition:Subdomain

Let $\left({R, +, \circ}\right)$ be an algebraic structure with two operations. A '''subdomain of $\left({R, +, \circ}\right)$''' is a subset $S$ of $R$ such that $\left({S, +_S, \circ_S}\right)$ is an integral domain.


Definition:Division Subring

Let $\struct {R, +, \circ}$ be a ring. A '''division subring''' of $\struct {R, +, \circ}$ is a subset $S$ of $R$ such that $\struct {S, +_S, \circ_S}$ is a division ring.


Definition:Centralizer

=== Centralizer of a Group Element === {{:Definition:Centralizer/Group Element}} === Centralizer of a Subset of a Group === {{:Definition:Centralizer/Group Subset}} === Centralizer of a Subgroup === {{:Definition:Centralizer/Subgroup}} === Centralizer of a Ring Subset === {{:Definition:Centralizer/Ring Subset}}


Definition:Ideal of Ring

Let $\struct {R, +, \circ}$ be a ring. Let $\struct {J, +}$ be a subgroup of $\struct {R, +}$. Then $J$ is an '''ideal of $R$''' {{iff}}: :$\forall j \in J: \forall r \in R: j \circ r \in J \land r \circ j \in J$ that is, {{iff}}: :$\forall r \in R: J \circ r \subseteq J \land r \circ J \subseteq J$


Definition:Congruence Modulo an Ideal

Let $\left({R, +, \circ}\right)$ be a ring, and let $J$ be an ideal of $R$. The notation: :$a \equiv b \pmod J$ is used to mean: :$a + \left({-b}\right) \in J$


Definition:Quotient Ring

Let $\struct {R, +, \circ}$ be a ring. Let $J$ be an ideal of $R$. Let $R / J$ be the (left) coset space of $R$ modulo $J$ with respect to $+$. Define an operation $+$ on $R / J$ by: :$\forall x, y: \paren {x + J} + \paren {y + J} := \paren {x + y} + J$ Also, define the operation $\circ$ on $R / J$ by: :$\forall x, y: \paren {x + J} \circ \paren {y + J} := \paren {x \circ y} + J$ The algebraic structure $\struct {R / J, +, \circ}$ is called the '''quotient ring of $R$ by $J$'''.


Definition:Divisor (Algebra)

=== Ring with Unity === {{:Definition:Divisor (Algebra)/Ring with Unity}} === Natural Numbers === {{:Definition:Divisor (Algebra)/Natural Numbers}} === Integers === As the set of integers form an integral domain, the concept '''divides''' is fully applicable to the integers. {{:Definition:Divisor (Algebra)/Integer}} === Gaussian Integers === As the set of Gaussian integers form an integral domain, the concept '''divides''' is also fully applicable to the Gaussian integers. {{:Definition:Divisor (Algebra)/Gaussian Integer}} === Real Numbers === The concept of '''divisibility''' can also be applied to the real numbers $\R$. {{:Definition:Divisor (Algebra)/Real Number}}


Definition:Multiple

=== Integral Domain === {{:Definition:Multiple/Integral Domain}} === Integers === As the set of integers form an integral domain, the concept of being a '''multiple''' is fully applicable to the integers. {{:Definition:Multiple/Integer}}


Definition:Associate

{{transclude:Definition:Associate/Integral Domain   |title = Integral Domain   |section = def   |link = true   |increase = 1   |header = 3 }} === Integers === As the integers form an integral domain, the definition can be applied directly to the set of integers $\Z$: {{:Definition:Associate/Integers}} === Commutative and Unitary Ring === The concept of associatehood can also be applied to the general commutative and unitary ring, even though there may be (proper) zero divisors in the latter: {{:Definition:Associate/Commutative and Unitary Ring}}


Definition:Trivial Factorization

Let $\struct {D, +, \circ}$ be an integral domain. Let $\struct {U_D, \circ}$ be the group of units of $\struct {D, +, \circ}$. A factorization in $\struct {D, +, \circ}$ of the form $x = u \circ y$, where $u \in U_D$ (that is, where $x$ is an associate of $y$) is called a '''trivial factorization'''. === Non-Trivial Factorization === {{:Definition:Trivial Factorization/Non-Trivial Factorization}}


Definition:Irreducible Element of Ring

Let $\struct {D, +, \circ}$ be an integral domain whose zero is $0_D$. Let $\struct {U_D, \circ}$ be the group of units of $\struct {D, +, \circ}$. Let $x \in D: x \notin U_D, x \ne 0_D$, that is, $x$ is non-zero and not a unit. === Definition 1 === {{:Definition:Irreducible Element of Ring/Definition 1}} === Definition 2 === {{:Definition:Irreducible Element of Ring/Definition 2}}


Definition:Tidy Factorization

Let $\struct {D, +, \circ}$ be an integral domain whose unity is $1_D$. Let $\struct {U_D, \circ}$ be the group of units of $\struct {D, +, \circ}$. Any factorization of $x \in D$ can always be '''tidied''' into the form: :$x = u \circ y_1 \circ y_2 \circ \cdots \circ y_n$ where $u \in \struct {U_D, \circ}$, and may be $1_D$, and $y_1, y_2, \ldots, y_n$ are all non-zero and non-units. This is done by forming the ring product of all units of a factorization into one unit, and rearranging all the remaining factors as necessary. Such a factorization is called '''tidy'''.


Definition:Equivalent Factorizations

Let $\struct {D, +, \circ}$ be an integral domain. Let $x$ be a non-zero non-unit element of $D$. Let there be two tidy factorizations of $x$: :$x = u_y \circ y_1 \circ y_2 \circ \cdots \circ y_m$ :$x = u_z \circ z_1 \circ z_2 \circ \cdots \circ z_n$ These two factorizations are '''equivalent''' if one of the following equivalent statements holds:   :$(1): \quad$ There exists a bijection $\pi: \set {1, \ldots, m} \to \set {1, \ldots, n}$ such that $y_i$ and $z_{\map \pi i}$ are associates of each other for each $i \in \set {1, \ldots, m}$. :$(2): \quad$ The multisets of principal ideals $\multiset {\ideal {y_i}: i = 1, \ldots, m}$ and $\multiset {\ideal {z_i}: i = 1, \ldots, n}$ are equal. The equivalence of the definitions is shown by part $(3)$ of Principal Ideals in Integral Domain.


Definition:Complete Factorization

Let $\struct {D, +, \circ}$ be an integral domain. Let $x$ be a non-zero non-unit element of $D$. A '''complete factorization''' of $x$ in $D$ is a tidy factorization: :$x = u \circ y_1 \circ y_2 \circ \cdots \circ y_n$ such that: :$u$ is a unit of $D$ :all of $y_1, y_2, \ldots, y_n$ are irreducible in $D$.


Definition:Unique Factorization Domain

Let $\struct {D, +, \circ}$ be an integral domain. If, for all $x \in D$ such that $x$ is non-zero and not a unit of $D$: :$(1): \quad x$ possesses a complete factorization in $D$ :$(2): \quad$ Any two complete factorizations of $x$ in $D$ are equivalent then $D$ is a '''unique factorization domain'''.


Definition:Subfield

Let $\struct {S, *, \circ}$ be an algebraic structure with $2$ operations. Let $T$ be a subset of $S$ such that $\struct {T, *, \circ}$ is a field. Then $\struct {T, *, \circ}$ is a '''subfield''' of $\struct {S, *, \circ}$.


Definition:Congruence Modulo Subgroup

Let $G$ be a group. Let $H$ be a subgroup of $G$. We can use $H$ to define relations on $G$ as follows: === Left Congruence Modulo Subgroup === {{:Definition:Congruence Modulo Subgroup/Left Congruence}} === Right Congruence Modulo Subgroup === {{:Definition:Congruence Modulo Subgroup/Right Congruence}}


Definition:Coset

=== Left Coset === {{:Definition:Coset/Left Coset}} === Right Coset === {{:Definition:Coset/Right Coset}}


Definition:Coset Space

Let $G$ be a group, and let $H$ be a subgroup of $G$. === Left Coset Space === {{:Definition:Coset Space/Left Coset Space}} === Right Coset Space === {{:Definition:Coset Space/Right Coset Space}} === Note === If we are (as is usual) concerned at a particular time with only the left or the right coset space, then the superscript is usually dropped and the notation $G / H$ is used for both the left and right coset space. If, in addition, $H$ is a normal subgroup of $G$, then $G / H^l = G / H^r$ and the notation $G / H$ is then unambiguous anyway.


Definition:Quotient Group

Let $G$ be a group. Let $N$ be a normal subgroup of $G$. Then the left coset space $G / N$ is a group, where the group operation is defined as: :$\paren {a N} \paren {b N} = \paren {a b} N$ $G / N$ is called the '''quotient group of $G$ by $N$'''.


Definition:Quotient Epimorphism

=== Group === {{:Definition:Quotient Epimorphism/Group}} === Ring === {{:Definition:Quotient Epimorphism/Ring}}


Definition:Central Subgroup

Let $G$ be a group. Let $H$ be a subgroup of $G$ which is a subset of the center of $G$. Then $H$ is a '''central subgroup of $G$'''.


Definition:Normalizer

Let $G$ be a group. Let $S$ be a subset of $G$. Then the '''normalizer of $S$ in $G$''' is the set $\map {N_G} S$ defined as: :$\map {N_G} S := \set {a \in G: S^a = S}$ where $S^a$ is the $G$-conjugate of $S$ by $a$.


Definition:Ordered Ring

Let $\struct {R, +, \circ}$ be a ring. Let $\preceq$ be an ordering compatible with the ring structure of $\struct {R, +, \circ}$. Then $\struct {R, +, \circ, \preceq}$ is an '''ordered ring'''.


Definition:Positive

Let $\struct {R, +, \circ, \le}$ be an ordered ring whose zero is $0_R$. Then $x \in R$ is '''positive''' {{iff}} $0_R \le x$. The set of all '''positive elements of $R$''' is denoted: :$R_{\ge 0_R} := \set {x \in R: 0_R \le x}$


Definition:Strictly Positive

Let $\struct {R, +, \circ, \le}$ be an ordered ring whose zero is $0_R$. Then $x \in R$ is '''strictly positive''' {{iff}} $0_R \le x$ and $x \ne 0_R$. One may write (more conveniently) that $0_R < x$ or $x > 0_R$ to express that $x$ is '''strictly positive'''. Thus, the set of all '''strictly positive''' elements of $R$ can be denoted: :$R_{>0_R} := \set {x \in R: x > 0_R}$


Definition:Ordering Compatible with Ring Structure

{{begin-axiom}} {{axiom | n = \text {OR} 1         | lc= $\preccurlyeq$ is compatible with $+$:         | q = \forall a, b, c \in  R         | ml= a \preccurlyeq b         | mo= \implies         | mr= \paren {a + c} \preccurlyeq \paren {b + c} }} {{axiom | n = \text {OR} 2         | lc= Product of Positive Elements is Positive         | q = \forall a, b \in R         | ml= 0_R \preccurlyeq x, 0_R \preccurlyeq y         | mo= \implies         | mr= 0_R \preccurlyeq x \circ y }} {{end-axiom}}


Definition:Ring of Endomorphisms

Let $\struct {G, \oplus}$ be an abelian group. Let $\mathbb G$ be the set of all group endomorphisms of $\struct {G, \oplus}$. Let $*: \mathbb G \times \mathbb G \to \mathbb G$ be the operation defined as: :$\forall u, v \in \mathbb G: u * v = u \circ v$ where $u \circ v$ is defined as composition of mappings. Then $\struct {\mathbb G, \oplus, *}$ is called the '''ring of endomorphisms''' of the abelian group $\struct {G, \oplus}$.


Definition:Field of Quotients

Let $D$ be an integral domain. Let $F$ be a field. === Definition 1 === {{:Definition:Field of Quotients/Definition 1}} === Definition 2 === {{:Definition:Field of Quotients/Definition 2}} === Definition 3 === {{:Definition:Field of Quotients/Definition 3}} === Definition 4 === {{:Definition:Field of Quotients/Definition 4}}


Definition:Ordered Field

Let $\left({R, +, \circ, \preceq}\right)$ be an ordered ring. Let $\left({R, +, \circ}\right)$ be a field. Then $\left({R, +, \circ, \preceq}\right)$ is an '''ordered field'''.


Definition:Ordered Structure

An '''ordered structure''' $\left({S, \circ, \preceq}\right)$ is a set $S$ such that: : $(1): \quad \left({S, \circ}\right)$ is an algebraic structure : $(2): \quad \left({S, \preceq}\right)$ is an ordered set : $(3): \quad \preceq$ is compatible with $\circ$.


Definition:Zero (Number)

The number '''zero''' is defined as being the cardinal of the empty set.


Definition:Multiplication

'''Multiplication''' is the basic operation $\times$ everyone is familiar with. For example: :$3 \times 4 = 12$ :$13 \cdotp 2 \times 7 \cdotp 7 = 101 \cdotp 64$


Definition:Subtraction

=== Natural Numbers === {{:Definition:Subtraction/Natural Numbers}} === Integers === {{:Definition:Subtraction/Integers}} === Rational Numbers === {{:Definition:Subtraction/Rational Numbers}} === Real Numbers === {{:Definition:Subtraction/Real Numbers}} === Complex Numbers === {{:Definition:Subtraction/Complex Numbers}} === Extended Real Subtraction === {{:Definition:Extended Real Subtraction}}


Definition:Finite

=== Finite Cardinal === {{:Definition:Finite Cardinal}} === Finite Set === {{:Definition:Finite Set}} === Finite Extended Real Number === {{:Definition:Finite Extended Real Number}}


Definition:Infinite Set

A set which is not finite is called '''infinite'''. That is, it is a set for which there is no bijection between it and any $\N_n$, where $\N_n$ is the the set of all elements of $n$ less than $n$, no matter how big we make $n$.


Definition:Cardinality

Two sets (either '''finite''' or '''infinite''') which are '''equivalent''' are said to have the same '''cardinality'''. The '''cardinality''' of a set $S$ is written $\card S$.


Definition:Countable Set

Let $S$ be a set. === Definition 1 === {{:Definition:Countable Set/Definition 1}} === Definition 2 === {{:Definition:Countable Set/Definition 2}} === Definition 3 === {{:Definition:Countable Set/Definition 3}}


Definition:Uncountable Set

A set which is infinite but not countable is described as '''uncountable'''.


Definition:Ordered Tuple

Let $n \in \N$ be a natural number. Let $\N^*_n$ be the first $n$ non-zero natural numbers: :$\N^*_n := \set {1, 2, \ldots, n}$


Definition:Operation

An '''operation''' is an object, identified by a symbol, which can be interpreted as a process which, from a number of objects, creates a new object.


Definition:Iterated Binary Operation

=== Indexed Iteration === {{Definition:Indexed Iterated Binary Operation}} === Iteration over Finite Set === {{:Definition:Iterated Binary Operation over Finite Set}} === Iteration over Set with Finite Support === {{Definition:Iterated Binary Operation over Set with Finite Support}}


Definition:Summation

Let $\struct {S, +}$ be an algebraic structure where the operation $+$ is an operation derived from, or arising from, the addition operation on the natural numbers. Let $\tuple {a_1, a_2, \ldots, a_n} \in S^n$ be an ordered $n$-tuple in $S$. === Definition by Index === {{:Definition:Summation/Indexed}} === Definition by Inequality === {{:Definition:Summation/Inequality}} === Definition by Propositional Function === {{:Definition:Summation/Propositional Function}}


Definition:Product Notation (Algebra)

Let $\struct {S, \times}$ be an algebraic structure where the operation $\times$ is an operation derived from, or arising from, the multiplication operation on the natural numbers. Let $\tuple {a_1, a_2, \ldots, a_n} \in S^n$ be an ordered $n$-tuple in $S$. === Definition by Index === {{:Definition:Product Notation (Algebra)/Index}}


Definition:Addition

'''Addition''' is the basic operation $+$ everyone is familiar with. For example: :$2 + 3 = 5$ :$47 \cdotp 3 + 191\cdotp 4 = 238 \cdotp 7$


Definition:Power of Element

=== Magma === {{:Definition:Power of Element/Magma}} === Magma with Identity === {{:Definition:Power of Element/Magma with Identity}} === Semigroup === {{:Definition:Power of Element/Semigroup}} === Monoid === {{:Definition:Power of Element/Monoid}} === Group === {{:Definition:Power of Element/Group}} === Ring === {{:Definition:Power of Element/Ring}}


Definition:Indexing Set

Let $I$ and $S$ be sets. Let $x: I \to S$ be a mapping. Let $x_i$ denote the image of an element $i \in I$ of the domain $I$ of $x$. Let $\family {x_i}_{i \mathop \in I}$ denote the set of the images of all the element $i \in I$ under $x$. When a mapping is used in this context, the domain $I$ of $x$ is called the '''indexing set''' of the terms $\family {x_i}_{i \mathop \in I}$. === Index === {{:Definition:Indexing Set/Index}} === Indexed Set === {{:Definition:Indexing Set/Indexed Set}} === Indexing Function === {{:Definition:Indexing Set/Function}} === Family === {{:Definition:Indexing Set/Family}} === Term === {{:Definition:Indexing Set/Term}} === Family of Distinct Elements === {{:Definition:Indexing Set/Family of Distinct Elements}} === Family of Sets === {{:Definition:Indexing Set/Family of Sets}} === Family of Subsets === {{:Definition:Indexing Set/Family of Subsets}}


Definition:Integer

The numbers $\set {\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots}$ are called the '''integers'''. This set is usually denoted $\Z$. An individual element of $\Z$ is called '''an integer'''.


Definition:Additive Group of Integers

The '''additive group of integers''' $\struct {\Z, +}$ is the set of integers under the operation of addition.


Definition:Rational Number

A number in the form $\dfrac p q$, where both $p$ and $q$ are integers ($q$ non-zero), is called a '''rational number'''. The set of all '''rational numbers''' is usually denoted $\Q$. Thus: :$\Q = \set {\dfrac p q: p \in \Z, q \in \Z_{\ne 0} }$


Definition:Irrational Number

An '''irrational number''' is a real number which is not rational. That is, an '''irrational number''' is one that can not be expressed in the form $\dfrac p q$ such that $p$ and $q$ are both integers. The set of '''irrational numbers''' can therefore be expressed as $\R \setminus \Q$, where: :$\R$ is the set of real numbers :$\Q$ is the set of rational numbers :$\setminus$ denotes set difference.


Definition:Factorial

Let $n \in \Z_{\ge 0}$ be a positive integer. === Definition 1 === {{:Definition:Factorial/Definition 1}} === Definition 2 === {{:Definition:Factorial/Definition 2}}


Definition:Binomial Coefficient

=== Definition 1 === {{:Definition:Binomial Coefficient/Integers/Definition 1}} === Definition 2 === {{:Definition:Binomial Coefficient/Integers/Definition 2}} === Definition 3 === {{:Definition:Binomial Coefficient/Integers/Definition 3}}


Definition:Number

There are five main classes of number: :$(1): \quad$ The natural numbers: $\N = \set {0, 1, 2, 3, \ldots}$ :$(2): \quad$ The integers: $\Z = \set {\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots}$ :$(3): \quad$ The rational numbers: $\Q = \set {p / q: p, q \in \Z, q \ne 0}$ :$(4): \quad$ The real numbers: $\R = \set {x: x = \sequence {s_n} }$ where $\sequence {s_n}$ is a Cauchy sequence in $\Q$ :$(5): \quad$ The complex numbers: $\C = \set {a + i b: a, b \in \R, i^2 = -1}$


Definition:Absolute Value

=== Definition 1 === {{:Definition:Absolute Value/Definition 1}} === Definition 2 === {{:Definition:Absolute Value/Definition 2}}


Definition:Remainder

Let $a, b \in \Z$ be integers such that $b \ne 0$. From the Division Theorem, we have that: :$\forall a, b \in \Z, b \ne 0: \exists! q, r \in \Z: a = q b + r, 0 \le r < \left|{b}\right|$ The value $r$ is defined as the '''remainder of $a$ on division by $b$''', or the '''remainder of $\dfrac a b$'''. === Real Arguments === When $x, y \in \R$ the '''remainder''' is still defined: {{:Definition:Remainder/Real}}


Definition:Even Integer

=== Definition 1 === {{:Definition:Even Integer/Definition 1}} === Definition 2 === {{:Definition:Even Integer/Definition 2}} === Definition 3 === {{:Definition:Even Integer/Definition 3}}


Definition:Odd Integer

=== Definition 1 === {{:Definition:Odd Integer/Definition 1}} === Definition 2 === {{:Definition:Odd Integer/Definition 2}} === Definition 3 === {{:Definition:Odd Integer/Definition 3}} === Euclid's Definition === {{EuclidSaid}} :''{{:Definition:Euclid's Definitions - Book VII/7 - Odd Number}}'' {{EuclidDefRefNocat|VII|7|Odd Number}}


Definition:Integer Combination

Let $a, b \in \Z$. An integer $n$ of the form: :$n = p a + q b: p, q \in \Z$ is an '''integer combination''' of $a$ and $b$.


Definition:Common Divisor

=== Integral Domain === {{:Definition:Common Divisor/Integral Domain}} === Integers === The definition is usually applied when the integral domain in question is the set of integers $\Z$, thus: {{:Definition:Common Divisor/Integers}} === Real Numbers === The definition can also be applied when the integral domain in question is the real numbers $\R$, thus: {{:Definition:Common Divisor/Real Numbers}}


Definition:Coprime

=== GCD Domain === {{:Definition:Coprime/GCD Domain}} === Euclidean Domain === {{:Definition:Coprime/Euclidean Domain}} === Integers === {{:Definition:Coprime/Integers}}


Definition:Integers Modulo m

Let $m \in \Z$ be an integer. The '''integers modulo $m$''' are the set of least positive residues of the set of residue classes modulo $m$: :$\Z_m = \set {0, 1, \ldots, m - 1}$


Definition:Prime Number

=== Definition 1 === {{:Definition:Prime Number/Definition 1}} === Definition 2 === {{:Definition:Prime Number/Definition 2}} === Definition 3 === {{:Definition:Prime Number/Definition 3}} === Definition 4 === {{:Definition:Prime Number/Definition 4}} === Definition 5 === {{:Definition:Prime Number/Definition 5}} === Definition 6 === {{:Definition:Prime Number/Definition 6}} === Definition 7 === {{:Definition:Prime Number/Definition 7}} === Euclid's Definition === {{EuclidSaid}} :''{{:Definition:Euclid's Definitions - Book VII/11 - Prime Number}}'' {{EuclidDefRefNocat|VII|11|Prime Number}}


Definition:Composite Number

A '''composite number''' $c$ is a positive integer that has more than two positive divisors. That is, an integer greater than $1$ which is not prime is defined as composite.


Definition:Sylow p-Subgroup

Let $p$ be prime. Let $G$ be a finite group whose order is denoted by $\left\vert{G}\right\vert$. === Definition 1 === {{:Definition:Sylow p-Subgroup/Definition 1}} === Definition 2 === {{:Definition:Sylow p-Subgroup/Definition 2}} === Definition 3 === {{:Definition:Sylow p-Subgroup/Definition 3}}


Definition:Prime Decomposition

Let $n > 1 \in \Z$. From the Fundamental Theorem of Arithmetic, $n$ has a unique factorization of the form: {{begin-eqn}} {{eqn | l = n       | r = \prod_{p_i \mathop \divides n} {p_i}^{k_i}       | c =  }} {{eqn | r = {p_1}^{k_1} {p_2}^{k_2} \cdots {p_r}^{k_r}       | c =  }} {{end-eqn}} where: :$p_1 < p_2 < \cdots < p_r$ are distinct primes :$k_1, k_2, \ldots, k_r$ are (strictly) positive integers. This unique expression is known as the '''prime decomposition of $n$'''.


Definition:Möbius Function

Let $n \in \Z_{>0}$, that is, a strictly positive integer. The '''Möbius function''' is the function $\mu: \Z_{>0} \to \Z_{>0}$ defined as: :$\map \mu n = \begin{cases} 1 & : n = 1 \\ 0 & : \exists p \in \mathbb P: p^2 \divides n\\ \left({-1}\right)^k & : n = p_1 p_2 \ldots p_k: p_i \in \mathbb P \end{cases}$


Definition:Square

=== Geometry === {{:Definition:Quadrilateral/Square}} === Abstract Algebra === {{:Definition:Square/Mapping}}


Definition:Fibonacci Number

The '''Fibonacci numbers''' are a sequence $\sequence {F_n}$ of integers which is formally defined recursively as: :$F_n = \begin {cases} 0 & : n = 0 \\ 1 & : n = 1 \\ F_{n - 1} + F_{n - 2} & : \text {otherwise} \end {cases}$ for all $n \in \Z_{\ge 0}$.


Definition:Cauchy Sequence

=== Metric Space === {{:Definition:Cauchy Sequence/Metric Space}} === Normed Vector Space === {{:Definition:Cauchy Sequence/Normed Vector Space}} === Normed Division Ring === {{:Definition:Cauchy Sequence/Normed Division Ring}}


Definition:Real Number

{{:Definition:Real Number/Number Line Definition}}


Definition:Real Interval

Informally, the set of all real numbers between any two given real numbers $a$ and $b$ is called a '''(real) interval'''.


Definition:Mediant

Let $r, s \in \Q$, i.e. let $r, s$ be rational numbers. Let $r$ and $s$ be expressed as $r = \dfrac a b, s = \dfrac c d$ where $a, b, c, d$ are integers such that $b > 0, d > 0$ (this is always possible by Divided by Positive Element of Field of Quotients). Then the '''mediant''' of $r$ and $s$ is $\dfrac {a + c} {b + d}$.


Definition:Close Packed

Let $\left({S, \preceq}\right)$ be an ordered set. Then $\left({S, \preceq}\right)$ is defined as '''close packed (in itself)''' {{iff}} between every two elements of $S$ there exists another element of $S$: : $\forall a, b \in S: a \prec b \implies \exists c \in S: a \prec c \prec b$ === Close Packed Subset === {{:Definition:Close Packed/Subset}}


Definition:Floor Function

Let $x$ be a real number. Informally, the '''floor function of $x$''' is the greatest integer less than or equal to $x$. === Definition 1 === {{Definition:Floor Function/Definition 1}} === Definition 2 === {{Definition:Floor Function/Definition 2}} === Definition 3 === {{Definition:Floor Function/Definition 3}}


Definition:Ceiling Function

Let $x$ be a real number. Informally, the '''ceiling function of $x$''' is the smallest integer greater than or equal to $x$. === Definition 1 === {{Definition:Ceiling Function/Definition 1}} === Definition 2 === {{Definition:Ceiling Function/Definition 2}} === Definition 3 === {{Definition:Ceiling Function/Definition 3}}


Definition:Real Function

A '''real function''' is a mapping or function whose domain and codomain are subsets of the set of real numbers $\R$.


Definition:Implicit Function

Consider a (real) function of two independent variables $z = \map f {x, y}$. Let a relation between $x$ and $y$ be expressed in the form $\map f {x, y} = 0$ defined on some interval $\mathbb I$. If there exists a function: :$y = \map g x$ defined on $\mathbb I$ such that: :$\forall x \in \mathbb I: \map f {x, \map g x} = 0$ then the relation $\map f {x, y} = 0$ defines $y$ as an '''implicit function''' of $x$.


Definition:Order of Structure

The '''order''' of an algebraic structure $\struct {S, \circ}$ is the cardinality of its underlying set, and is denoted $\order S$. Thus, for a finite set $S$, the '''order of $\struct {S, \circ}$''' is the number of elements in $S$.


Definition:Word (Abstract Algebra)

Let $\struct {G, \circ}$ be a magma. Let $S \subseteq G$ be a subset. A '''word in $S$''' is the product of a finite number of elements of $S$. The '''set of words in $S$''' is denoted $\map W S$: :$\map W S := \set {s_1 \circ s_2 \circ \cdots \circ s_n: n \in \N_{>0}: s_i \in S, 1 \le i \le n}$


Definition:Truth Table

A '''truth table''' is a tabular array that represents the computation of a truth function, that is, a function of the form: :$f : \mathbb B^k \to \mathbb B$ where: :$k$ is a non-negative integer :$\mathbb B$ is a set of truth values, usually $\set {0, 1}$ or $\set {T, F}$.


Definition:Index of Subgroup

Let $G$ be a group. Let $H$ be a subgroup of $G$. The '''index of $H$ (in $G$)''', denoted $\index G H$, is the cardinality of the left (or right) coset space $G / H$. === Finite Index === {{:Definition:Index of Subgroup/Finite}} === Infinite Index === {{:Definition:Index of Subgroup/Infinite}}


Definition:Parametric Operator

Let $A$ be an index set. Let $\Omega$ be a indexed family of operators $\left({\Omega_{\alpha} }\right)_A = \left\{ {\Omega_\alpha: \alpha \in A}\right\}$ indexed by the index $\alpha$. {{explain|Definition of "operator" in this context.}} Then $\Omega$ is a '''parametric operator''' with '''parameter $\alpha$''' in the '''parameter set $A$'''. Category:Definitions/Set Theory 2qv55f8elqvw9z1t799m5q40r3h0865


Definition:Order of Group Element

Let $G$ be a group whose identity is $e_G$. Let $x \in G$ be an element of $G$. === Definition 1 === {{:Definition:Order of Group Element/Definition 1}} === Definition 2 === {{:Definition:Order of Group Element/Definition 2}} === Definition 3 === {{:Definition:Order of Group Element/Definition 3}}


Definition:Derangement

A '''derangement''' is a permutation $f: S \to S$ from a set $S$ to itself where $\map f s \ne s$ for any $s \in S$.  If $S$ is finite, the number of '''derangements''' is denoted by $D_n$ where $n = \card S$ (the cardinality of $S$.)


Definition:Hamiltonian Group

A group $G$ is described as '''Hamiltonian''' {{iff}} $G$ is a non-abelian group such that every subgroup is normal.


Definition:Simple Group

A group $G$ is '''simple''' {{iff}} it has only $G$ and the trivial group as normal subgroups. That is, if the composition length of $G$ is $1$.


Definition:Group Direct Product

Let $\struct {G, \circ_1}$ and $\struct {H, \circ_2}$ be groups. Let $G \times H: \set {\tuple {g, h}: g \in G, h \in H}$ be their cartesian product. The '''(external) direct product''' of $\struct {G, \circ_1}$ and $\struct {H, \circ_2}$ is the group $\struct {G \times H, \circ}$ where the operation $\circ$ is defined as: :$\tuple {g_1, h_1} \circ \tuple {g_2, h_2} = \tuple {g_1 \circ_1 g_2, h_1 \circ_2 h_2}$ This is usually referred to as the '''group direct product''' of $G$ and $H$.


Definition:Independent Subgroups

Let $G$ be a group whose identity is $e$. Let $\left \langle {H_n} \right \rangle$ be a sequence of subgroups of $G$. === Definition 1 === {{:Definition:Independent Subgroups/Definition 1}} === Definition 2 === {{:Definition:Independent Subgroups/Definition 2}}


Definition:Group Presentation

Let $G$ be a group. A '''presentation for $G$''' is a triple $\tuple {S, R, f}$ where: :$S$ is a set :$R$ is a set of relations on $S$ :$f: \gen {S \mid R} \to G$ is a group isomorphism from the group defined by $\struct {S, R}$.


Definition:Infinite Cyclic Group

=== Definition 1 === {{:Definition:Infinite Cyclic Group/Definition 1}} === Definition 2 === {{:Definition:Infinite Cyclic Group/Definition 2}}


Definition:Free Group

=== Definition 1 === {{:Definition:Free Group/Definition 1}} === Definition 2 === {{:Definition:Free Group/Definition 2}}


Definition:Ring of Integers Modulo m

Let $m \in \Z: m \ge 2$. Let $\Z_m$ be the set of integers modulo $m$. Let $+_m$ and $\times_m$ denote addition modulo $m$ and multiplication modulo $m$ respectively. The algebraic structure $\struct {\Z_m, +_m, \times_m}$ is '''the ring of integers modulo $m$'''.


Definition:Symmetry Mapping

A '''symmetry mapping''' (or just '''symmetry''') of a geometric figure is a bijection from the figure to itself which preserves the distance between points. In other words, it is a self-congruence. Intuitively and informally, a '''symmetry''' is a movement of the figure so that it looks exactly the same after it has been moved.


Definition:Geometric Figure

A '''geometric figure''' is intuitively defined as a set of points and lines in space. {{EuclidSaid}} :''{{:Definition:Euclid's Definitions - Book I/14 - Geometric Figure}}'' {{EuclidDefRefNocat|I|14|Geometric Figure}} The boundary may or may not be included in a particular figure. If this is important (and in the study of topology it usually is), then whether it is included or not needs to be specified. === Plane Figure === {{:Definition:Geometric Figure/Plane Figure}} === Three-Dimensional Figure === {{:Definition:Geometric Figure/Three-Dimensional Figure}} === Rectilineal Figure === {{:Definition:Geometric Figure/Rectilineal Figure}}


Definition:Permutation on n Letters

Let $\N_k$ be used to denote the initial segment of natural numbers: : $\N_k = \closedint 1 k = \set {1, 2, 3, \ldots, k}$ A '''permutation on $n$ letters''' is a '''permutation''': : $\pi: \N_n \to \N_n$ The usual symbols for denoting a general '''permutation''' are $\pi$ (not to be confused with the famous circumference over diameter), $\rho$ and $\sigma$.


Definition:Fixed Element of Permutation

Let $S$ be a set. Let $\pi: S \to S$ be a permutation on $S$. Let $x \in S$. $x$ is '''fixed by $\pi$''' {{iff}}: :$\map \pi x = x$


Definition:Cyclic Permutation

Let $S_n$ denote the symmetric group on $n$ letters. Let $\rho \in S_n$ be a permutation on $S$. Then $\rho$ is a '''cyclic permutation of length $k$''' {{iff}} there exists $k \in \Z: k > 0$ and $i \in \Z$ such that: :$(1): \quad k$ is the smallest such that $\map {\rho^k} i = i$ :$(2): \quad \rho$ fixes each $j$ not in $\set {i, \map \rho i, \ldots, \map {\rho^{k - 1} } i}$.


Definition:Disjoint Permutations

Let $S_n$ denote the symmetric group on $n$ letters. Let $\pi, \rho \in S_n$ both be permutations on $S_n$. Then $\pi$ and $\rho$ are '''disjoint''' {{iff}}: :$(1): \quad i \notin \Fix \pi \implies i \in \Fix \rho$ :$(2): \quad i \notin \Fix \rho \implies i \in \Fix \pi$ That is, each element moved by $\pi$ is fixed by $\rho$ and (equivalently) each element moved by $\rho$ is fixed by $\pi$. That is, {{iff}} their supports are disjoint sets. We may say that: : $\pi$ is '''disjoint from $\rho$''' : $\rho$ is '''disjoint from $\pi$''' : $\pi$ and $\rho$ are '''(mutually) disjoint'''. Note of course that it is perfectly possible for $i \in \Fix \pi$ and also $i \in \Fix \rho$, that is, there may well be elements fixed by more than one of a pair of '''disjoint permutations'''.


Definition:Group Action

Let $X$ be a set. Let $\struct {G, \circ}$ be a group whose identity is $e$. === Left Group Action === {{:Definition:Group Action/Left Group Action}} === Right Group Action === {{:Definition:Group Action/Right Group Action}} The group $G$ thus '''acts on''' the set $X$. The group $G$ can be referred to as the '''group of transformations''', or a '''transformation group'''.


Definition:Orbit (Group Theory)

Let $G$ be a group acting on a set $X$. === Definition 1 === {{:Definition:Orbit (Group Theory)/Definition 1}} === Definition 2 === {{:Definition:Orbit (Group Theory)/Definition 2}}


Definition:Effective Transformation Group

Let $G$ be a group whose identity is $e$. Let $X$ be a set. Let $\phi: G \times X \to X$ be a group action. Then $G$ is an '''effective transformation group for $\phi$''' {{iff}} $\phi$ is faithful.


Definition:Transposition

Let $S$ be a set. A '''transposition''' on $S$ is a $2$-cycle. That is, a '''transposition''' is a permutation $\rho$ on $S$ which exchanges, or '''transposes''', exactly two elements of $S$. Thus if $\rho$ is a '''transposition''' which '''transposes''' two elements $r, s \in S$, it follows from the definition of fixed elements that: :$\Fix \rho = S \setminus \set {r, s}$


Definition:R-Algebraic Structure

Let $\struct {R, +_R, \times_R}$ be a ring. Let $\struct {S, \ast_1, \ast_2, \ldots, \ast_n}$ be an algebraic structure with $n$ operations. Let $\circ: R \times S \to S$ be a binary operation. Then $\struct {S, \ast_1, \ast_2, \ldots, \ast_n, \circ}_R$ is an '''$R$-algebraic structure with $n$ operations'''. If the number of operations in $S$ is either understood or general, it is just called an '''$R$-algebraic structure''', and the structure can be denoted $\struct {S, \circ}_R$.