Definition:Right-Truncatable Prime

A '''right-truncatable prime''' is a prime number which remains prime when any number of digits are removed from the right hand end.


Definition:Hilbert Proof System/Instance 2

This instance of a Hilbert proof system is used in: * {{BookReference|Introduction to Symbolic Logic|1959|A.H. Basson|author2 = D.J. O'Connor|ed = 3rd|edpage = Third Edition}} Let $\mathcal L$ be the language of propositional logic. $\mathscr H$ has the following axioms and rules of inference: === Axioms === Let $p, q, r$ be propositional variables. Then the following WFFs are axioms of $\mathscr H$: {{begin-axiom}} {{axiom|n = A1        |lc = Rule of Idempotence        |m = (p \lor p) \implies p }} {{axiom|n = A2        |lc = Rule of Addition        |m = q \implies (p \lor q) }} {{axiom|n = A3        |lc = Rule of Commutation        |m = (p \lor q) \implies (q \lor p) }} {{axiom|n = A4        |lc = Factor Principle        |m = (q \implies r) \implies \left({ (p \lor q) \implies (p \lor r)}\right) }} {{end-axiom}} === Rules of Inference === ==== $RST \, 1$: Rule of Uniform Substitution ==== {{refactor|This rule should have its own dedicated page like the others. But separate from the Rule of Substitution, which is too vague at this point (might be a subpage though). Also review source flow afterwards}} Any WFF $\mathbf A$ may be substituted for any propositional variable $p$ in a $\mathscr H_2$-theorem $\mathbf B$. The resulting theorem can be denoted $\mathbf B \paren{ \mathbf A \mathbin{//} p }$. See the Rule of Substitution. ==== $RST \, 2$: Rule of Substitution by Definition ==== The following expressions are regarded definitional abbreviations: {{begin-axiom}} {{axiom|n = 1        |lc = Conjunction        |ml = \mathbf A \land \mathbf B        |mo = {{=}}_{\text{def} }        |mr = \neg \left({ \neg \mathbf A \lor \neg \mathbf B }\right) }} {{axiom|n = 2        |lc = Conditional        |ml = \mathbf A \implies \mathbf B        |mo = {{=}}_{\text{def} }        |mr = \neg \mathbf A \lor \mathbf B }} {{axiom|n = 3        |lc = Biconditional        |ml = \mathbf A \iff \mathbf B        |mo = {{=}}_{\text{def} }        |mr = (\mathbf A \implies \mathbf B) \land (\mathbf B \implies \mathbf A) }} {{end-axiom}} ==== $RST \, 3$: Rule of Detachment ==== If $\mathbf A \implies \mathbf B$ and $\mathbf A$ are theorems of $\mathscr H$, then so is $\mathbf B$. That is, Modus Ponendo Ponens. ==== $RST \, 4$: Rule of Adjunction ==== If $\mathbf A$ and $\mathbf B$ are theorems of $\mathscr H$, then so is $\mathbf A \land \mathbf B$. That is, the Rule of Conjunction. (This rule can be proved from the other three and so is only a convenience.)


Definition:Hardy-Ramanujan Number

The $n$th '''Hardy-Ramanujan number''' $\operatorname {Ta} \left({n}\right)$ is the smallest positive integer which can be expressed as the sum of $2$ cubes in $n$ different ways.


Definition:Cube Number/Sequence

The sequence of positive cube numbers begins: :$1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, \ldots$


Definition:Summation/Finite Support

Let $G$ be an abelian group. Let $S$ be a set. Let $f: S \to G$ be a mapping. Let the support $\operatorname{Supp} \left({f}\right)$ be finite. Let $g$ be the restriction of $f$ to $\operatorname{Supp}f$. The '''summation of $f$ over $S$''', denoted $\displaystyle \sum_{s \mathop \in S} f \left({s}\right)$, is the summation over the finite set $\operatorname{Supp} \left({f}\right)$ of $g$: :$\displaystyle \sum_{s \mathop \in S} f \left({s}\right) = \sum_{s \mathop \in \operatorname {Supp} \left({f}\right)} g \left({s}\right)$


Definition:Independent Proof System

Let $\mathcal L$ be a logical language. Let $\mathscr P$ be a proof system. Then $\mathscr P$ is '''independent''' if it is not possible to derive one axiom or rule of inference of $\mathscr P$ from the others. {{stub|Deliberately vague, as the context in which this definition was originally couched was equally vague. Needs to be made rigorous, or merged with pages building a more comprehensive treatment.}}


Definition:Sum of Finite Set

Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$. Let $S \subset \mathbb A$ be a finite subset. The '''sum of $S$''' is the summation of the identity mapping on $S$ over $S$: :$\displaystyle \sum S = \sum_{s \mathop \in S} s$ Category:Definitions/Summations 2qv4z6kzqhyodq18cepnvy97xmbqzvb


Definition:Unital Magma

A '''unital magma''' is a magma that has an identity element.


Definition:Commutative Magma

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


Definition:N-Ary Operation Induced by Binary Operation

Let $\struct {G, \oplus}$ be a magma. Let $n \ge 1$ be a natural number. Let $G^n$ be the $n$th cartesian power of $G$. The '''$n$-ary operation induced by $\oplus$''' is the $n$-ary operation $\oplus_n: G^n \to G$ defined as: :$\map {\oplus_n} f = \displaystyle \bigoplus_{i \mathop = 1}^n \map f i$ where $\bigoplus$ denotes indexed iteration of $f$ from $1$ to $n$.


Definition:Constant

A '''constant''' is a name for an object (usually a number, but the concept has wider applications) which ''does not change'' during the context of a logical or mathematical argument. A constant can be considered as an operator which takes no operands. A constant can also be considered as a variable whose domain is a singleton.


Definition:Polynomial Ring/Indeterminate

==== Single indeterminate ==== Let $\left({S, \iota, X}\right)$ be a polynomial ring over $R$. The '''indeterminate''' of $\left({S, \iota, X}\right)$ is the term $X$. ==== Multiple Indeterminates ==== Let $I$ be a set. Let $\left({S, \iota, f}\right)$ be a polynomial ring over $R$ in $I$ '''indeterminates'''. The '''indeterminates''' of $\left({S, \iota, f}\right)$ are the elements of the image of the family $f$.


Definition:Polynomial Ring/Universal Property

A '''polynomial ring over $R$''' is a pointed $R$-algebra $(S, \iota, X)$ that satisfies the following universal property: :For every pointed $R$-algebra $(A, \kappa, a)$ there exists a unique pointed algebra homomorphism $h : S\to A$, called evaluation homomorphism. This is known as the '''universal property of a polynomial ring'''.


Definition:Polynomial Algebra

Let $R$ be a ring. Let $\Z \sqbrk X$ be the polynomial ring over the ring of integers $\Z$. The '''polynomial algebra in one indeterminate''' over $R$ is the tensor product $R \otimes_\Z \Z \sqbrk X$. {{expand}}


Let $R$ be a commutative ring with unity. === Definition 1 === {{:Definition:Unital Associative Commutative Algebra/Definition 1}} === Definition 2 === {{:Definition:Unital Associative Commutative Algebra/Definition 2}} == Also known as == A '''unital associative commutative algebra''' over $R$ is also known as an '''$R$-algebra''', as is a general algebra over $R$. == Equivalence of Definitions == While, strictly speaking, the above definitions do define different objects, they are equivalent in the following sense: * An algebra $\left({A, *}\right)$ over $R$ that is unital, associative and commutative and whose underlying module is unitary, is identified with the ring under $R$ equal to its underlying ring together with its canonical mapping $R \to A$. The algebra is '''viewed as a ring'''. * A ring under $R$, $(A, f)$, is identified with the algebra defined by $f$. The algebra is '''viewed as an algebra'''. For the detailed statements, see Equivalence of Definitions of Unital Associative Commutative Algebra.


Definition:Unital Ring Homomorphism

Let $R$ and $S$ be rings with unities $1_R$ and $1_S$. A '''unital ring homomorphism''' is a ring homomorphism $f: R \to S$ such that $\map f {1_R} = 1_S$.


Definition:Unital Associative Commutative Algebra/Definition 1

A '''unital associative commutative algebra''' over $R$ is an algebra $\left({A, *}\right)$ over $R$ that is unital, associative and commutative and whose underlying module is unitary.


Definition:Unital Associative Commutative Algebra/Definition 2

A '''unital associative commutative algebra''' over $R$ is a ring under $A$, that is, an ordered pair $(A, f)$ where: :$A$ is a commutative ring with unity :$f : R \to A$ is a unital ring homomorphism <!--whose image is contained in the center of $A$-->


Definition:Algebra Defined by Ring Homomorphism

Let $R$ be a commutative ring. Let $\struct {S, +, *}$ be a ring. Let $f : R \to S$ be a ring homomorphism. Let the image of $f$ be a subset of the center of $S$. Let $S_R$ be the module defined by the ring homomorphism $f$. The '''algebra defined by $f$''' is the algebra over $R$ defined as: :$\struct {S_R, *}$


Definition:Underlying Ring of Associative Algebra

Let $R$ be a commutative ring. Let $(A, *)$ be an associative algebra over $R$. Let $A = (M, +, \circ)$ be the underlying module of $(A, *)$. The '''underlying ring of $(A, *)$''' is the ring $(A, +, *)$.


Definition:Pointed Algebra over Ring

Let $R$ be a commutative ring. A '''pointed algebra over $R$''' is an ordered pair $(A, a)$ where: :$A$ is an algebra over $R$ :$a$ is an element of $A$


Definition:Polynomial Ring/Embedding

Let the ordered triple $(S, \iota, X)$ be a polynomial ring over $R$ in one indeterminate $X$. The unital ring homomorphism $\iota$ is called the '''canonical embedding into the polynomial ring'''. === Multiple Indeterminates === Let $I$ be a set. Let $(S, \iota, X)$ be a polynomial ring over $R$ in $I$ indeterminates. The unital ring homomorphism $\iota$ is called the '''canonical embedding into the polynomial ring'''.


Definition:Monoid Ring/Canonical Mapping

Let $e_1$ be the canonical basis element. The '''canonical mapping to $R \left[{G}\right]$''' is the mapping $R \to R \left[{G}\right]$ which sends $r$ to $r * e_1$.


Definition:Unital Subalgebra/Definition 1

The subalgebra $\left({B_R, *}\right)$ is a '''unital subalgebra''' of $A_R$ {{iff}} $1_A \in B$.


Definition:Unital Subalgebra/Definition 2

The subalgebra $\left({B_R, *}\right)$ is a '''unital subalgebra''' of $A_R$ {{iff}}: :$(1) \quad B_R$ is unital :$(2) \quad$Its unit is $1_A$. That is, a '''unital subalgebra''' of $A_R$ must not only have a unit, but that unit must also be the same unit as that of $A_R$.


Definition:Field of Rational Fractions

Let $R$ be an integral domain.


Definition:Continuous Mapping (Topology)/Point/Open Sets

The mapping $f$ is '''continuous at (the point) $x$''' (with respect to the topologies $\tau_1$ and $\tau_2$) {{iff}}: :For every neighborhood $N$ of $\map f x$ in $T_2$, there exists a neighborhood $M$ of $x$ in $T_1$ such that $f \sqbrk M \subseteq N$.


Definition:Continuous Mapping (Topology)/Point/Filters

The mapping $f$ is '''continuous at (the point) $x$''' {{iff}} for any filter $\mathcal F$ on $T_1$ that converges to $x$, the corresponding image filter $f \left({\mathcal F}\right)$ converges to $f \left({x}\right)$.


Definition:Continuous Mapping (Topology)/Everywhere/Pointwise

The mapping $f$ is '''continuous everywhere''' (or simply '''continuous''') {{iff}} $f$ is continuous at every point $x \in S_1$.


Definition:Open Neighborhood/Topology

Let $T = \struct {S, \tau}$ be a topological space. Let $A \subseteq S$ be a subset of $S$. Let $N_A$ be a neighborhood of $A$. If $N_A \in \tau$, that is, if $N_A$ is itself open in $T$, then $N_A$ is called an '''open neighborhood'''.


Definition:Continuous Real Function/Everywhere

Let $f: \R \to \R$ be a real function. Then $f$ is '''everywhere continuous''' {{iff}} $f$ is continuous at every point in $\R$.


Let $f$ be a real function defined on a closed interval $\left[{a \,.\,.\, b}\right]$. ==== Definition 1 ==== {{Definition:Continuous Real Function/Closed Interval/Definition 1}} ==== Definition 2 ==== {{Definition:Continuous Real Function/Closed Interval/Definition 2}}


Definition:Continuous Real Function/Closed Interval/Definition 2

The function $f$ is '''continuous on $\left[{a \,.\,.\, b}\right]$''' {{iff}} it is continuous at every point of $\left[{a \,.\,.\, b}\right]$.


Definition:Indexed Iterated Binary Operation

Let $\struct {G, *}$ be a magma. Let $a, b \in \Z$ be integers. Let $\closedint a b$ be the integer interval between $a$ and $b$. Let $f: \closedint a b \to G$ be a mapping. The '''indexed iteration of $*$ of $f$ from $a$ to $b$''' is recursively defined and denoted: :$\ds \prod_{k \mathop = a}^b \map f k = \begin {cases} \map f a & : b = a \\ \paren {\ds \prod_{k \mathop = a}^{b - 1} \map f k} * \map f b & : b > a \end {cases}$ For each ordered $n$-tuple $\tuple {a_1, a_2, \ldots, a_n} \in S^n$, the '''composite''' of $\tuple {a_1, a_2, \ldots, a_n}$ for $\oplus$ is the value at $\tuple {a_1, a_2, \ldots, a_n}$ of the $n$-ary operation defined by $\oplus$. This '''composite''' is recursively defined and denoted: {{begin-eqn}} {{eqn | l = \bigoplus_{k \mathop = 1}^n a_k       | r = \map {\oplus_n} {a_1, a_2, \ldots, a_n}       | c =  }} {{eqn | r = \begin {cases} a & : n = 1 \\ \map {\oplus_m} {a_1, \ldots, a_m} \oplus a_{m + 1} & : n = m + 1 \end {cases}       | c =  }} {{eqn | r = \paren {\paren {\cdots \paren {\paren {a_1 \oplus a_2} \oplus a_3} \oplus \cdots} \oplus a_{n - 1} } \oplus a_n       | c =  }} {{end-eqn}}


Definition:Iterated Binary Operation over Finite Set

Let $\struct {G, *}$ be a commutative semigroup. Let $S$ be a finite non-empty set. Let $f: S \to G$ be a mapping. Let $n \in \N$ be the cardinality of $S$. Let $g: \N_{<n} \to S$ be a bijection, where $\N_{<n}$ is an initial segment of the natural numbers. The '''iteration of $*$ of $f$ over $S$''', denoted $\displaystyle \prod_{s \mathop \in S} \map f s$, is the indexed iteration of $*$ of the composition $f \circ g$ over $\N_{<n}$: :$\displaystyle \prod_{s \mathop \in S} \map f s = \displaystyle \prod_{i \mathop = 0}^{n - 1} \map f {\map g i}$


Definition:Iterated Binary Operation over Set with Finite Support

Let $\struct {G, *}$ be a commutative monoid. Let $S$ be a set. Let $f: S \to G$ be a mapping. Let the support $\operatorname {Supp} f$ be finite. The '''iteration of $*$ of $f$ over $S$''', denoted $\displaystyle \prod_{s \mathop \in S} \map f s$, is the iteration over the finite set $\operatorname{Supp} f$ of $f$: :$\displaystyle \prod_{s \mathop \in S} \map f s = \prod_{s \mathop \in \operatorname {Supp} f} \map f s$


Definition:Ring of Sequences of Finite Support

Let $\struct {R, +, \circ}$ be a ring. Let $P \sqbrk R$ be the set of all sequences in $R$ whose domain is $\N$: :$P \sqbrk R = \set {\sequence {r_0, r_1, r_2, \ldots} }$ and whose support is finite. Let the operations $\oplus$ and $\odot$ on $P \sqbrk R$ be defined as follows: {{:Definition:Operations on Polynomial Ring of Sequences}} The '''ring of sequences of finite support over $R$''' is the ring $\struct {P \sqbrk R, \oplus, \odot}$.


Definition:Consistent (Logic)/Proof System

Then $\mathscr P$ is '''consistent''' {{iff}}: :There exists a logical formula $\phi$ such that $\not \vdash_{\mathscr P} \phi$ That is, some logical formula $\phi$ is '''not''' a theorem of $\mathscr P$.


Definition:Consistent (Logic)/Proof System/Propositional Logic/Definition 1


Definition:Consistent (Logic)/Proof System/Propositional Logic/Definition 2

Suppose that in $\mathscr P$, the Rule of Explosion (Variant 3) holds. Then $\mathscr P$ is '''consistent''' {{iff}}: :For every logical formula $\phi$, not ''both'' of $\phi$ and $\neg \phi$ are theorems of $\mathscr P$


Definition:Consistent (Logic)/Set of Formulas

Let $\FF$ be a collection of logical formulas. Then $\FF$ is '''consistent for $\mathscr P$''' {{iff}}: :There exists a logical formula $\phi$ such that $\FF \nvdash_{\mathscr P} \phi$. That is, some logical formula $\phi$ is '''not''' a provable consequence of $\FF$.


Definition:Left-Truncatable Prime

A '''left-truncatable prime''' is a prime number which remains prime when any number of digits are removed from the left hand end. Zeroes are excluded, in order to eliminate, for example, prime numbers of the form $10^n + 3$ for arbitrarily large $n$.


Definition:Irreducible Component

Let $T = \left({S, \tau}\right)$ be a topological space. A subspace $Y \subset T$ is an '''irreducible component of $T$''' {{iff}}: : $Y$ is irreducible : $Y$ is not a proper subset of an irreducible subspace of $T$. That is, {{iff}} $Y$ is maximal in the ordered set of irreducible subspaces of $T$, ordered by inclusion.


Definition:Irreducible Space/Definition 1

A topological space $T = \struct {S, \tau}$ is '''irreducible''' {{iff}} there exists no cover of $T$ by two proper closed sets of $T$.


Definition:Irreducible Space/Definition 2

A topological space $T = \struct {S, \tau}$ is '''irreducible''' {{iff}} there is no finite cover of $T$ by proper closed sets of $T$.


Definition:Irreducible Space/Definition 3

A topological space $T = \struct {S, \tau}$ is '''irreducible''' {{iff}} every two non-empty open sets of $T$ have non-empty intersection: :$\forall U, V \in \tau: U, V \ne \O \implies U \cap V \ne \O$


Definition:Irreducible Space/Definition 4

A topological space $T = \struct {S, \tau}$ is '''irreducible''' {{iff}} every non-empty open set of $T$ is (everywhere) dense in $T$.


Definition:Irreducible Space/Definition 5

A topological space $T = \struct {S, \tau}$ is '''irreducible''' {{iff}} the interior of every proper closed set of $T$ is empty.


Definition:Krull Dimension

=== Krull Dimension of a Ring === {{Definition:Krull Dimension of Ring}} === Krull Dimension of a Topological Space === {{Definition:Krull Dimension of Topological Space}} {{NamedforDef|Wolfgang Krull|cat = Krull}} Category:Definitions/Algebraic Geometry t98gwppos4spvq3h1086xm9c80ofd9a


Definition:Length of Chain

Let $T$ be a chain in $S$. Let $T$ be finite and non-empty. The '''length''' of the chain $T$ is its cardinality minus $1$.


Definition:Krull Dimension of Topological Space

Let $T$ be a topological space. Its '''Krull dimension''' $\map {\dim_{\mathrm {Krull} } } T$ is the supremum of lengths of chains of closed irreducible sets of $T$, ordered by inclusion. Thus, the '''Krull dimension''' is $\infty$ if there exist arbitrarily long chains.


Definition:Noetherian Topological Space

=== Definition 1 === {{:Definition:Noetherian Topological Space/Definition 1}} === Definition 2 === {{:Definition:Noetherian Topological Space/Definition 2}}


Definition:Set Ordered by Inclusion

Let $S$ be a set. Then '''$S$ ordered by inclusion''' is the ordered set $\struct {S, \subseteq}$, where $\subseteq$ is the inclusion relation on $S$.


Definition:Total Ring of Fractions

Let $A$ be a commutative ring with unity. The '''total ring of fractions''' of $A$ is the localization of $A$ at the set of its regular elements.


Definition:Regular Element of Ring

Let $A$ be a commutative ring with unity. Let $a\in A$. Then $a$ is '''regular''' {{iff}} it is not a zero divisor.


Definition:Monomial of Polynomial Ring

=== Multiple Variables === {{Definition:Monomial of Polynomial Ring/Multiple Variables}}


Definition:Pointed Set

A '''pointed set''' is an ordered pair $\struct {A, a}$ where: :$A$ is a set :$a \in A$ is an element.


Definition:Pointed Mapping

Let $\struct {A, a}$ and $\struct {B, b}$ be pointed sets. A '''pointed mapping''' $f: \struct {A, a} \to \struct {B, b}$ is a mapping $f : A \to B$ such that $\map f a = b$.


Definition:Multiple Pointed Set

Let $I$ be a set. An '''$I$-pointed set''' is an ordered pair $\left({A, f}\right)$ where: :$A$ is a set :$f : I \to A$ is a mapping


Definition:Multiple Pointed Algebra

Let $R$ be a commutative ring with unity. Let $I$ be a set. An '''$I$-pointed algebra over $R$''' is an ordered pair $(A, f)$ where: :$A$ is an algebra over $R$ :$f : I \to A$ is a mapping


Definition:Square Pyramorphic Number

A '''square pyramorphic number''' is a square pyramidal number $P_n$ whose decimal representation ends in $n$.


Definition:Square Pyramorphic Number/Sequence

The sequence of square pyramorphic numbers, for $n \in \Z_{\ge 0}$, begins: {{begin-eqn}} {{eqn | l = P_1       | r = 1 }} {{eqn | l = P_5       | r = 55 }} {{eqn | l = P_{25}       | r = 5525 }} {{eqn | l = P_{40}       | r = 22 \, 140 }} {{eqn | l = P_{65}       | r = 93 \, 665 }} {{eqn | l = P_{80}       | r = 1 \, 043 \, 280 }} {{end-eqn}}


Definition:Additive Arithmetic Function

Let $f : \N \to \C$ be an arithmetic function. Then $f$ is '''additive''' {{iff}}: :$m \perp n \implies \map f {m n} = \map f m + \map f n$


Definition:Multiplicative Function on UFD

Let $R$ be a unique factorization domain. Let $f : R \to \C$ be a complex-valued function. Then $f$ is '''multiplicative''' {{iff}}: :For all coprime $x, y\in R$: $f \left({x y}\right) = f \left({x}\right) f \left({y}\right)$


Definition:Constructed Semantics

Let $\mathcal L$ be a formal language. A '''constructed semantics''' for $\mathcal L$ is a formal semantics which is invented ''solely'' for proving a property about $\mathcal L$ or other entities related to $\mathcal L$.


Definition:Additive Function on UFD

Let $R$ be a unique factorization domain. Let $f : R \to \C$ be a complex-valued function. Then $f$ is '''additive''' {{iff}}: :For all coprime $x,y\in R$, $f(xy) = f(x) + f(y)$


Definition:Multiplicative Arithmetic Function

Let $f : \N \to \C$ be an arithmetic function. Then $f$ is '''multiplicative''' {{iff}}: :$m \perp n \implies \map f {m n} = \map f m \map f n$ where $m \perp n$ denotes that $m$ is coprime to $n$.


Definition:Constructed Semantics/Instance 1

Let $\mathcal L_0$ be the language of propositional logic. The constructed semantics $\mathscr C_1$ for $\mathcal L_0$ is used for the following results: * Hilbert Proof System Instance 2 is Consistent === Structures === Define the structures of $\mathscr C_1$ as mappings $v$ by the Principle of Recursive Definition, as follows. Let $\mathcal P_0$ be the vocabulary of $\mathcal L_0$. Let a mapping $v: \mathcal P_0 \to \{ 1, 2 \}$ be given. Next, regard the following as definitional abbreviations: {{begin-axiom}} {{axiom | n  = 1         | lc = Conjunction         | ml = \mathbf A \land \mathbf B         | mo = {{=}}_{\text{def} }         | mr = \neg \left({ \neg \mathbf A \lor \neg \mathbf B }\right) }} {{axiom | n  = 2         | lc = Conditional         | ml = \mathbf A \implies \mathbf B         | mo = {{=}}_{\text{def} }         | mr = \neg \mathbf A \lor \mathbf B }} {{axiom | n  = 3         | lc = Biconditional         | ml = \mathbf A \iff \mathbf B         | mo = {{=}}_{\text{def} }         | mr = \left({\mathbf A \implies \mathbf B}\right) \land \left({\mathbf B \implies \mathbf A}\right) }} {{end-axiom}} It only remains to define $v \left({ \neg \phi }\right)$ and $v \left({ \phi \lor \psi}\right)$ recursively, by: {{begin-eqn}} {{eqn | l = v \left({\neg \phi }\right)       | o = :=       | r = \begin{cases} 1 & : \text{if $v \left({\phi}\right) = 2$} \\ 2 & : \text{if $v \left({\phi}\right) = 1$}\end{cases} }} {{eqn | l = v \left({ \phi \lor \psi }\right)       | o = :=       | r = \begin{cases} 1 & : \text{if $v \left({\phi}\right) = v \left({\psi}\right) = 1$} \\ 2 & : \text{otherwise}\end{cases} }} {{end-eqn}} === Validity === Define validity in $\mathscr C_1$ by declaring: :$\models_{\mathscr C_1} \phi$ {{iff}} $v \left({\phi}\right) = 2$


Definition:Floor Function/Definition 2

The '''floor function of $x$''', denoted $\floor x$, is defined as the greatest element of the set of integers: :$\set {m \in \Z: m \le x}$ where $\le$ is the usual ordering on the real numbers.


Definition:Ceiling Function/Definition 1

The '''ceiling function of $x$''' is defined as the infimum of the set of integers no smaller than $x$: :$\ceiling x := \inf \set {m \in \Z: x \le m}$ where $\le$ is the usual ordering on the real numbers.


Definition:Ceiling Function/Definition 2

The '''ceiling function of $x$''', denoted $\ceiling x$, is defined as the smallest element of the set of integers: :$\set {m \in \Z: x \le m}$ where $\le$ is the usual ordering on the real numbers.


Definition:Ceiling Function/Definition 3

The '''ceiling function of $x$''' is the unique integer $\ceiling x$ such that: :$\ceiling x - 1 < x \le \ceiling x$


Definition:Functor Category

Let $C$ and $D$ be categories. The '''functor category''' $\operatorname{Funct}(C, D)$ is the category with: {{DefineCategory |ob = covariant functors $C \to D$ |mor = natural transformations |comp = vertical composition of natural transformations |id = identity natural transformations }} {{definition wanted|the contravariant case}}


Definition:Generated Subgroup

Let $G$ be a group. Let $S \subset G$ be a subset. === Definition 1 === {{Definition:Generated Subgroup/Definition 1}} === Definition 2 === {{Definition:Generated Subgroup/Definition 2}} === Definition 3 === {{Definition:Generated Subgroup/Definition 3}}


Definition:Generated Subgroup/Definition 1

The '''subgroup generated by $S$''' is the smallest subgroup containing $S$.


Definition:Generated Subgroup/Definition 3

Let $S^{-1}$ be the set of inverses of $S$. The '''subgroup generated by $S$''' is the set of words on the union $S \cup S^{-1}$.


Definition:Generator of Ring

Let $R$ be a ring. Let $S \subset R$ be a subset. Then $S$ is a '''generator''' of $R$ {{iff}} $R$ is the subring generated by $S$.


Definition:Generator of Ring Extension

Let $f : A \to B$ be a ring extension of commutative rings with unity. Let $S \subset B$ be a subset. Then $S$ is a '''generator of $B$ over $A$''' {{iff}} $B$ is the ring extension generated by $S$.


Definition:Generated Normal Subgroup

Let $G$ be a group. Let $S \subset G$ be a subset. === Definition 1 === {{Definition:Generated Normal Subgroup/Definition 1}} === Definition 2 === {{Definition:Generated Normal Subgroup/Definition 2}} === Definition 3 === {{Definition:Generated Normal Subgroup/Definition 3}}


Definition:Generated Normal Subgroup/Definition 1

The '''normal subgroup generated by $S$''', denoted $\gen {S^G}$,  is the intersection of all normal subgroups of $G$ containing $S$.


Definition:Generated Normal Subgroup/Definition 2

The '''normal subgroup generated by $S$''', denoted $\gen {S^G}$, is the subgroup generated by the set of conjugates of $S$: :$S^G = \set {g^{−1}sg: g \in G, s \in S}$


Definition:Generator of Ideal of Ring

Let $R$ be a commutative ring.  Let $I \subset R$ be an ideal. Let $S \subset I$ be a subset. Then $S$ is a '''generator''' of $I$ {{iff}} $I$ is the ideal generated by $S$.


Definition:Riemann Zeta Function/Zero/Nontrivial

The '''nontrivial zeroes''' of the Riemann zeta function $\zeta$ are the zeroes of $\zeta$ that are not trivial.


Definition:Formal Derivative of Dirichlet Series

Let $(a_n) : \N \to \C$ be an arithmetic function. Let $F(s)=\displaystyle \sum_{n\mathop=1}^\infty\frac{a_n}{n^s}$ be the corresponding formal Dirichlet series. The '''formal derivative''' of $F(s)$ is the formal Dirichlet series: :$\displaystyle \sum_{n\mathop=1}^\infty\frac{a_n\cdot \log n}{n^s}$


Definition:Abscissa of Convergence

Let $f \left({s}\right)$ be a Dirichlet series The '''abscissa of convergence''' of $f$ is the extended real number $\sigma_0 \in \overline \R$ defined by: :$\sigma_0 = \displaystyle \inf \left\{ {\operatorname{Re} \left({s}\right) : s \in \C, f \left({s}\right) \text{converges} }\right\}$ where $\inf \varnothing = +\infty$.


Definition:Dirichlet Inverse of Arithmetic Function

Let $f : \N \to \C$ be an arithmetic function. Let $\varepsilon$ be the identity arithmetic function. A '''Dirichlet inverse''' of $f$ is an arithmetic function $g$ such that $g*f = \varepsilon$, where $*$ denotes Dirichlet convolution. That is, an inverse of $f$ in the ring of arithmetic functions.


Definition:Formal Product of Dirichlet Series

Let $f,g : \N \to \C$ be arithmetic functions. Let $F,G$ be their formal Dirichlet series. The '''formal product''' of $F$ and $G$ is the formal Dirichlet series: :$\displaystyle \sum_{n \mathop = 1}^\infty \frac{(f*g)(n)}{n^s}$ where $*$ denotes Dirichlet convolution.


Definition:Ring of Arithmetic Functions

Let $\AA$ be the set of all arithmetic functions. Let $*$ denote Dirichlet convolution, and $+$ the pointwise sum of functions. The '''ring of arithmetic functions''' is the ring $\struct {\AA, +, *}$.


Definition:Consistent (Logic)/Set of Formulas/Propositional Logic/Definition 1

Then $\FF$ is '''consistent for $\mathscr P$''' {{iff}}: :There exists a logical formula $\phi$ such that $\FF \not \vdash_{\mathscr P} \phi$ That is, some logical formula $\phi$ is '''not''' a $\mathscr P$-provable consequence of $\FF$.


Definition:Consistent (Logic)/Set of Formulas/Propositional Logic/Definition 2

Suppose that in $\mathscr P$, the Rule of Explosion (Variant 3) holds. Then $\FF$ is '''consistent for $\mathscr P$''' {{iff}}: :For every logical formula $\phi$, not ''both'' of $\phi$ and $\neg \phi$ are $\mathscr P$-provable consequences of $\FF$


Definition:Pole of Complex Function

Let $U \subset \C$ be an open subset. Let $f : U \to \C$ be a holomorphic function. Let $p\in \C$ be an isolated singularity of $f$. === Definition 1 === The point $p$ is a '''pole''' of $f$ {{iff}} the Laurent expansion of $f$ around $p$ has the form: :$f(z) = \displaystyle \sum_{k = -n}^\infty a_k(z-p)^k$ === Definition 2 === The point $p$ is a '''pole''' of $f$ {{iff}} there exists a natural number $m>0$ such that: :$\displaystyle\lim_{z \to p}(z-p)^m f(z) \in \C\setminus\{0\}$ === Definition 3 === The point $p$ is a '''pole''' of $f$ {{iff}} the improper limit: :$\displaystyle\lim_{z\to p}|f(z)| = \infty$.


Definition:Complex Disk

Let $a \in \C$ be a complex number. Let $R>0$ be a real number. === Open disk === {{Definition:Complex Disk/Open}} === Closed disk === {{Definition:Complex Disk/Closed}}


Definition:Complex Disk/Open

The '''open (complex) disk of center $a$ and radius $R$''' is the set: :$\map B {a, R} = \set {z \in \C: \cmod {z - a} < R}$ where $\cmod {\, \cdot \,}$ denotes complex modulus.


Definition:Complex Disk/Closed

The '''closed (complex) disk of center $a$ and radius $R$''' is the set: :$\map B {a, R} = \set {z \in \C: \cmod {z - a} \le R}$ where $\cmod {\, \cdot \,}$ denotes complex modulus.


Definition:Unit Ball

Let $V$ be a normed vector space with norm $\norm {\, \cdot \,}$. The '''closed unit ball''' of $V$, denoted $\operatorname {ball} V$, is the set: :$\set {v \in V: \norm v_V \mathop \le 1}$


Definition:Design

A '''design''' is an ordered pair $\tuple {X, \mathcal B}$ where: :$X$ is a non-empty finite set of points :$\mathcal B$ is a multiset of blocks. Category:Definitions/Design Theory 37g472ap1lcm782m0j6gjg3gk9wulrl


Definition:Point (Design Theory)

Let $X$ be non-empty finite set. If $X$ is to be analyzed in the context of Design Theory, then its elements are referred to as '''points'''. In application, the '''points''' in $X$ are often variables. Category:Definitions/Design Theory 9d5dhk8qpe9vg081c8xwfxi2o0ravo0


Definition:Block

Let $X$ be a set of points. Let $\mathcal B$ be a multiset of subsets of $X$. Then the elements of $\mathcal B$ are called '''blocks'''. Category:Definitions/Design Theory 1p8idw86cxu109u4kfn62pj0gl2shmq


Definition:Real Interval/Definition 2

A '''real interval''' is a subset of $\R$ that is one of the following real interval types: * closed bounded interval * open bounded interval * left half-open bounded interval * right half-open bounded interval * closed and bounded on the right, also known as a closed unbounded below real interval * open and bounded on the right, also known as an open unbounded below real interval * closed and bounded on the left, also known as a closed unbounded above real interval * open and bounded on the left, also known as an open unbounded above real interval * unbounded interval without endpoints


Definition:Real Interval Types

It is usual to define intervals in terms of inequalities. These are in the form of a pair of brackets, either round or square, enclosing the two endpoints of the interval separated by two dots. Whether the bracket at either end is round or square depends on whether the end point is '''inside''' or '''outside''' the interval, as specified in the following. Let $a, b \in \R$ be real numbers. === Bounded Intervals === ==== Open Interval ==== {{:Definition:Real Interval/Open}} ==== Half-Open Interval ==== {{:Definition:Real Interval/Half-Open}} ==== Closed Interval ==== {{:Definition:Real Interval/Closed}} ==== Bounded Interval ==== {{:Definition:Real Interval/Bounded}} === Unbounded Intervals === {{:Definition:Real Interval/Unbounded}} === Other Intervals === ==== Empty Interval ==== {{:Definition:Real Interval/Empty}} ==== Singleton Interval ==== {{:Definition:Real Interval/Singleton}} ==== Unit Interval ==== {{:Definition:Real Interval/Unit Interval}}


Definition:Real Interval/Half-Open/Right

The '''right half-open (real) interval''' from $a$ to $b$ is the subset: :$\hointr a b := \set {x \in \R: a \le x < b}$


Definition:Real Interval/Half-Open/Left

The '''left half-open (real) interval''' from $a$ to $b$ is the subset: :$\hointl a b := \set {x \in \R: a < x \le b}$


Definition:Compact Real Interval

A '''compact real interval''' is a closed interval of the form: :$[a\,.\,.\,b]$ with $a,b\in \R$.


Let $A \subseteq \R$ be a subset of the real numbers. Let $f: A \to \R$ be a real function. Let $f$ be discontinuous at $a\in A$. === Definition 1  === {{Definition:Removable Discontinuity of Real Function/Definition 1}} === Definition 2  === {{Definition:Removable Discontinuity of Real Function/Definition 2}}


Definition:Removable Discontinuity of Real Function/Definition 1

The point $a$ is a '''removable discontinuity''' of $f$ {{iff}} the limit $\displaystyle \lim_{x \mathop \to a} \map f x$ exists.


Definition:Removable Discontinuity of Real Function/Definition 2

The point $a$ is a '''removable discontinuity''' of $f$ {{iff}} there exists $b \in \R$ such that the function $f_b$ defined by: :$\map {f_b} x = \begin {cases} \map f x &: x \ne a \\ b &: x = a \end {cases}$ is continuous at $a$.


Definition:Discontinuous Mapping/Real Function/Point

Let $A \subseteq \R$ be a subset of the real numbers. Let $f : A \to \R$ be a real function. Let $a\in A$. Then $f$ is '''discontinuous''' at $a$ {{iff}} $f$ is not continuous at $a$.


=== At a Point === {{Definition:Discontinuous Mapping/Real Function/Point}}


Definition:Discontinuous Mapping/Topological Space/Point

Let $T_1 = \left({A_1, \tau_1}\right)$ and $T_2 = \left({A_2, \tau_2}\right)$ be topological spaces. Let $f: A_1 \to A_2$ $x \in T_1$ be a mapping from $A_1$ to $A_2$. Then by definition $f$ is continuous at $x$ if for every neighborhood $N$ of $f \left({x}\right)$ there exists a neighborhood $M$ of $x$ such that $f \left({M}\right) \subseteq N$. Therefore, $f$ is discontinuous at $x$ if for some neighbourhood $N$ of $f \left({x}\right)$ and every neighbourhood $M$ of $x$, $f \left({M}\right) \nsubseteq N$. The point $x$ is called a '''discontinuity of $f$'''.


Definition:Polynomial Function/Real/Definition 1

A '''real polynomial function''' on $S$ is a function $f: S \to \R$ for which there exist: :a natural number $n\in \N$ :real numbers $a_0, \ldots, a_n \in \R$ such that for all $x \in S$: :$\map f x = \displaystyle \sum_{k \mathop = 0}^n a_k x^k$ where $\sum$ denotes indexed summation.


Definition:Polynomial Function/Complex/Definition 1

A '''complex polynomial function''' on $S$ is a function $f : S \to \C$ for which there exist: * a natural number $n\in \N$ * complex numbers $a_0, \ldots, a_n \in \C$ such that for all $z \in S$: :$\map f z = \displaystyle \sum_{k \mathop = 0}^n a_k z^k$ where $\displaystyle \sum$ denotes indexed summation.


Definition:Polynomial Function/Complex/Definition 2

Let $\C \sqbrk X$ be the polynomial ring in one variable over $\C$. Let $\C^S$ be the ring of mappings from $S$ to $\C$. Let $\iota \in \C^S$ denote the inclusion $S \hookrightarrow \C$. A '''complex polynomial function''' on $S$ is a function $f: S \to \C$ which is in the image of the evaluation homomorphism $\C \sqbrk X \to \C^S$ at $\iota$.


Definition:Polynomial Function/Real/Definition 2

Let $\R \sqbrk X$ be the polynomial ring in one variable over $\R$. Let $\R^S$ be the ring of mappings from $S$ to $\R$. Let $\iota \in \R^S$ denote the inclusion $S \hookrightarrow \R$. A '''real polynomial function''' on $S$ is a function $f: S \to \R$ which is in the image of the evaluation homomorphism $\R \sqbrk X \to \R^S$ at $\iota$.


Definition:Polynomial Function/Ring

Let $R$ be a commutative ring with unity. Let $S \subset R$ be a subset. === Definition 1 === {{Definition:Polynomial Function/Ring/Definition 1}} === Definition 2 === {{Definition:Polynomial Function/Ring/Definition 2}}


Definition:Polynomial Function/Ring/Definition 1

A '''polynomial function''' on $S$ is a mapping $f : S \to R$ for which there exist: : a natural number $n \in \N$ : $a_0, \ldots, a_n \in R$ such that for all $x\in S$: :$\map f x = \displaystyle \sum_{k \mathop = 0}^n a_k x^k$ where $\sum$ denotes indexed summation.


Definition:Polynomial Function/Ring/Definition 2

Let $R \sqbrk X$ be the polynomial ring in one variable over $R$. Let $R^S$ be the ring of mappings from $S$ to $R$. Let $\iota \in R^S$ denote the inclusion $S \hookrightarrow R$. A '''polynomial function''' on $S$ is a mapping $f : S \to R$ which is in the image of the evaluation homomorphism $R \sqbrk X \to R^S$ at $\iota$.


Definition:Polynomial in Ring Element/Definition 1

A '''polynomial in $x$ over $S$''' is an element $y \in R$ for which there exist: :a natural number $n \in \N$ :$a_0, \ldots, a_n \in S$ such that: :$y = \displaystyle \sum_{k \mathop = 0}^n a_k x^k$ where: :$\displaystyle \sum$ denotes indexed summation :$x^k$ denotes the $k$th power of $x$


Definition:Polynomial in Ring Element/Definition 2

Let $S[X]$ be the polynomial ring in one variable over $S$. A '''polynomial in $x$ over $S$''' is an element that is in the image of the evaluation homomorphism $S[X]\to R$ at $x$.


Definition:Monomial of Polynomial Ring/One Variable

Let $R \sqbrk X$ be a polynomial ring over $R$ in one indeterminate $X$. A '''monomial''' of $R \sqbrk X$ is an element that is a power of $X$.


Let $I$ be a set. Let $R \sqbrk {\family {X_i}_{i \mathop \in I} }$ be a polynomial ring in $I$ variables $\family {X_i}_{i \mathop \in I}$. Let $y \in R \sqbrk {\family {X_i}_{i \mathop \in I} }$. A '''monomial''' of $R \sqbrk {\family {X_i}_{i \mathop \in I} }$ is an element that is a product of variables; specifically: === Definition 1 === {{Definition:Monomial of Polynomial Ring/Multiple Variables/Definition 1}} === Definition 2 === {{Definition:Monomial of Polynomial Ring/Multiple Variables/Definition 2}}


=== At a Point === {{Definition:Discontinuous Mapping/Topological Space/Point}}


Definition:Strong Fibonacci Pseudoprime

A '''strong Fibonacci pseudoprime''' is a Carmichael number which also satisfies one of the following conditions: === Type I === {{:Definition:Strong Fibonacci Pseudoprime/Type I}} === Type II === {{:Definition:Strong Fibonacci Pseudoprime/Type II}}


Definition:Strong Fibonacci Pseudoprime/Type I

A '''strong Fibonacci pseudoprime of type I''' is a Carmichael number $N = \displaystyle \prod p_i$ such that an even number of the prime factors $p_i$ are of the form $4 m - 1$ where: {{begin-eqn}} {{eqn | n = 1       | l = 2 \paren {p_i + 1}       | o = \divides       | r = \paren {N - 1}       | c = for those $p_i$ of the form $4 m - 1$ }} {{eqn | n = 2       | l = \paren {p_i + 1}       | o = \divides       | r = \paren {N \pm 1}       | c = for those $p_i$ of the form $4 m + 1$ }} {{end-eqn}}


Definition:Strong Fibonacci Pseudoprime/Type II

A '''strong Fibonacci pseudoprime of type II''' is a Carmichael number $N = \displaystyle \prod p_i$ such that an odd number of the prime factors $p_i$ are of the form $4 m - 1$ where: :$2 \paren {p_i + 1} \divides \paren {N - p_i}$ for all $p_i$


Definition:Cunningham Chain/First Kind

A '''Cunningham chain of the first kind''' is a (finite) sequence $\left({p_1, p_2, \ldots, p_n}\right)$ such that: :$(1): \quad \forall i \in \left\{ {1, 2, \ldots, n - 1}\right\}: p_{i + 1} = 2 p_i + 1$ :$(2): \quad p_i$ is prime for all $i \in \left\{ {1, 2, \ldots, n - 1}\right\}$ :$(3): \quad n$ is not prime such that $2 n + 1 = p_1$ :$(4): \quad 2 p_n + 1$ is not prime. Thus: : each term except the last is a Sophie Germain prime : each term except the first is a safe prime.


Definition:Aurifeuillian Factorization

An '''Aurifeuillian factorization''' is an operation to find the prime factors of integers of various forms based upon the identity: :$a^2 + b^2 = \left({a + \sqrt {2a b} + b}\right) \left({a - \sqrt{2ab} + b}\right)$


Definition:Polydivisible Number

=== Definition 1 === {{:Definition:Polydivisible Number/Definition 1}} === Definition 2 === {{:Definition:Polydivisible Number/Definition 2}} === Sequence of Polydivisible Numbers === {{:Definition:Polydivisible Number/Sequence}}


Definition:Fischer-Griess Monster

The '''Fischer-Griess Monster''' is the automorphism group of the Griess algebra.


Definition:Vigintillion

=== Short Scale === {{:Definition:Vigintillion/Short Scale}} === Long Scale === {{:Definition:Vigintillion/Long Scale}}


Definition:Number-Naming System

There are various '''number-naming systems''' for naming large numbers (that is: greater than $1 \, 000 \, 000$). === Short Scale === {{:Definition:Number-Naming System/Short Scale}} === Long Scale === {{:Definition:Number-Naming System/Long Scale}}


Definition:Number-Naming System/Short Scale

The '''short scale system''' is the number-naming system which uses: :the word '''million''' for $10^6 = 1 \, 000 \, 000$ :the Latin-derived prefixes '''bi-''', '''tri-''', '''quadri-''', '''quint-''', etc. for each further multiple of $1 \, 000$, appended to the root '''-(i)llion''', corresponding to the indices $2$, $3$, $4$, $5$, $\ldots$ Thus: {{begin-axiom}} {{axiom | lc= one billion:         | m = = 1 \, 000 \, 000 \, 000         | mm= = 10^9 = 10^{2 \times 3 + 3} }} {{axiom | lc= one trillion         | m = = 1 \, 000 \, 000 \, 000 \, 000         | mm= = 10^{12} = 10^{3 \times 3 + 3} }} {{axiom | lc= one quadrillion         | m = = 1 \, 000 \, 000 \, 000 \, 000 \, 000         | mm= = 10^{15} = 10^{4 \times 3 + 3} }} {{axiom | lc= one quintillion         | m = = 1 \, 000 \, 000 \, 000 \, 000 \, 000 \, 000         | mm= = 10^{18} = 10^{5 \times 3 + 3} }} {{end-axiom}} Thus '''one $n$-illion''' equals $1000 \times 10^{3 n}$ or $10^{3 n + 3}$


Definition:Number-Naming System/Long Scale

The '''long scale system''' is the number-naming system which uses: :the word '''million''' for $10^6 = 1 \, 000 \, 000$ :the Latin-derived prefixes '''bi-''', '''tri-''', '''quadri-''', '''quint-''', etc. for each further multiple of $1 \, 000 \, 000$, appended to the root '''-(i)llion''', corresponding to the indices $2$, $3$, $4$, $5$, $\ldots$ Thus: {{begin-axiom}} {{axiom | lc= one billion:         | m = = 1 \, 000 \, 000 \, 000 \, 000         | mm= = 10^{12} = 10^{2 \times 6} }} {{axiom | lc= one trillion         | m = = 1 \, 000 \, 000 \, 000 \, 000 \, 000 \, 000         | mm= = 10^{18} = 10^{3 \times 6} }} {{axiom | lc= one quadrillion         | m = = 1 \, 000 \, 000 \, 000 \, 000 \, 000 \, 000 \, 000 \, 000         | mm= = 10^{24} = 10^{4 \times 6} }} {{axiom | lc= one quintillion         | m = = 1 \, 000 \, 000 \, 000 \, 000 \, 000 \, 000 \, 000 \, 000 \, 000 \, 000         | mm= = 10^{30} = 10^{5 \times 6} }} {{end-axiom}} Thus '''one $n$-illion''' equals $10^{6 n}$. Additional terms are occasionally found to fill some of the gaps, but these are rare nowadays: {{begin-axiom}} {{axiom | lc= one milliard:         | m = = 1 \, 000 \, 000 \, 000         | mm= = 10^9 }} {{axiom | lc= one billiard         | m = = 1 \, 000 \, 000 \, 000 \, 000 \, 000         | mm= = 10^{15} }} {{end-axiom}}


Definition:Vigintillion/Also known as

Some sources use the word '''vigillion''' for '''vigintillion'''.


Definition:Vigintillion/Short Scale

'''Vigintillion''' is a name for $10^{63}$ in the short scale system: :'''One vigintillion''' $= 10^{3 \times 20 + 3}$


Definition:Vigintillion/Long Scale

'''Vigintillion''' is a name for $10^{120}$ in the long scale system: :'''One vigintillion''' $= 10^{6 \times 20}$


Definition:Multiindex Power Notation

Let $M$ be a commutative monoid. Let $I$ be a set. Let $\mathbf x = (x_i)_{i\in I}$ be a family of elements of $M$. Let $\mathbf a = (a_i)_{i\in I}$ be a family of natural numbers of finite support. We denote $\mathbf x^{\mathbf a} = \displaystyle \prod_{i \in I} x_i^{a_i}$ where: * $x_i^{a_i}$ is the $a_i$th power of $x_i$ * $\prod$ denotes product with finite support Category:Definitions/Polynomial Theory bfvaevrkwgbfdcr1qv3ob8rjtqyee8y


Definition:Restriction of Scalars of Module

Let $R$ and $S$ be commutative rings with unity. Let $f : R \to S$ be a ring homomorphism. Let $M$ be an $S$-module with ring representation $\rho$. The '''restriction of scalars''' of $M$ to $R$ via $f$ is the $R$-module with ring representation $\rho \circ f$.


Definition:Centillion

=== Short Scale === {{:Definition:Centillion/Short Scale}} === Long Scale === {{:Definition:Centillion/Long Scale}}


Definition:Centillion/Short Scale

'''Centillion''' is a name for $10^{303}$ in the short scale system: :'''One centillion''' $= 10^{3 \times 100 + 3}$


Definition:Milli-Millillion/Short Scale

'''Milli-millillion''' is a name for $10^{3 \, 000 \, 003}$ in the short scale system: :'''One milli-millillion''' $= 10^{3 \times 1 \, 000 \, 000 + 3}$


Definition:Module Structure of Ring

Let $R$ be a commutative ring with unity. Then '''$R$ as an $R$-module''' is the $R$-module $R$ with ring action: :$R\times R \to R$ equal to the ring product of $R$.


Definition:Koti

'''Koti''' is a Jaina word for a '''hundred-hundred-thousand''' ($10 \, 000 \, 000$).


Definition:Pakoti

'''Pakoti''' is a Jaina word for a koti multiplied by a koti: that is: :$10 \, 000 \, 000 \times 10 \, 000 \, 000$


Definition:Asankhyeya

'''Asankhyeya''' is a Jaina word for a koti to the power of $20$: that is: :$10 \, 000 \, 000^{20} = 10^{140}$


Definition:Module Structure of Polynomial Ring

Let $R$ be a commutative ring with unity. Let $R \sqbrk X$ be a polynomial ring in one variable $X$ over $R$. The '''$R$-module structure''' on $R \sqbrk X$ is the module structure as an algebra over $R$.


Definition:Ring Representation

Let $R$ be a ring. Let $M$ be an abelian group. A '''ring representation''' of $R$ on $M$ is a ring homomorphism from $R$ to the endomorphism ring $\map {\operatorname {End} } M$. === Unital Ring Representation === {{:Definition:Ring Representation/Unital}}


Definition:Constant Polynomial/Definition 1

The polynomial $P$ is a '''constant polynomial''' {{iff}} its coefficients of $x^k$ are zero for $k \ge 1$.


Definition:Constant Polynomial/Definition 2

The polynomial $P$ is a '''constant polynomial''' {{iff}} $P$ is either the zero polynomial or has degree $0$.


Definition:Constant Polynomial/Definition 3

The polynomial $P$ is a '''constant polynomial''' {{iff}} it is in the image of the canonical embedding $R \to R \left[{x}\right]$.


Definition:Minimal Polynomial/Definition 1

The '''minimal polynomial''' of $\alpha$ over $K$ is the unique monic polynomial $f \in K \sqbrk x$ of smallest degree such that $\map f \alpha = 0$.


Definition:Minimal Polynomial/Definition 2

The '''minimal polynomial''' of $\alpha$ over $K$ is the unique irreducible, monic polynomial $f \in K \left[{x}\right]$ such that $f \left({\alpha}\right) = 0$.


Definition:Minimal Polynomial/Definition 3

The '''minimal polynomial''' of $\alpha$ over $K$ is the unique monic polynomial $f \in K \left[{x}\right]$ that generates the kernel of the evaluation homomorphism $K \left[{x}\right] \to L$ at $\alpha$. That is, such that for all $g \in K \left[{x}\right]$: :$g \left({\alpha}\right) = 0$ {{iff}} $f$ divides $g$.


Definition:Integral Element of Algebra

Let $A$ be a commutative ring with unity. Let $f : A \to B$ be a commutative $A$-algebra. Let $b\in B$. === Definition 1 === {{Definition:Integral Element of Algebra/Definition 1}} === Definition 2 === {{Definition:Integral Element of Algebra/Definition 2}} === Definition 3 === {{Definition:Integral Element of Algebra/Definition 3}} === Definition 4 === {{Definition:Integral Element of Algebra/Definition 4}}


Definition:Integral Element of Algebra/Definition 1

The element $b$ is '''integral''' over $A$ {{iff}} it is a root of a monic polynomial in $A[x]$.


Definition:Integral Element of Algebra/Definition 2

The element $b$ is '''integral''' over $A$ {{iff}} the generated subalgebra $A[b]$ is a finitely generated module over $A$.


Definition:Integral Element of Algebra/Definition 3

The element $b$ is '''integral''' over $A$ {{iff}} the generated subalgebra $A[b]$ is contained in a subalgebra $C\leq B$ which is a finitely generated module over $A$.


Definition:Integral Element of Algebra/Definition 4

The element $b$ is '''integral''' over $A$ {{iff}} there exists a [[Definition:Faithful Module|faithful $A[b]$-module]] whose restriction of scalars to $A$ is finitely generated.


Let $T = \left({S, \tau}\right)$ be a topological space. Let $\mathbf C$ be a category. === Definition 1 === {{:Definition:Presheaf on Topological Space/Definition 1}} === Definition 2 === {{:Definition:Presheaf on Topological Space/Definition 2}}


Definition:Sheaf on Topological Space

Let $T = \struct {S, \tau}$ be a topological space. Let $\mathbf C$ be a category. === Definition 1 === {{:Definition:Sheaf on Topological Space/Definition 1}} === Definition 2 === {{:Definition:Sheaf on Topological Space/Definition 2}} === Definition 3 === {{:Definition:Sheaf on Topological Space/Definition 3}}


Definition:Morphism of Presheaves on Topological Space

Let $X$ be a topological space. Let $\mathbf C$ be a category. Let $\mathcal F$ and $\mathcal G$ be $\mathbf C$-valued presheaves on $X$. === Definition 1 === {{definition wanted|using restrictions}} === Definition 2 === A '''morphism of presheaves''' from $\mathcal F$ to $\mathcal G$ is a natural transformation $\mathcal F \to \mathcal G$.


Definition:Abelian Sheaf

Let $X$ be a topological space. === Definition 1 === An '''abelian sheaf''' on $X$ is a sheaf on $X$ with values in the category of abelian groups. === Definition 2 === An '''abelian sheaf''' on $X$ is a sheaf of $\underline{\mathbb{Z}}$-modules on $X$, where $\underline{\mathbb{Z}}$ is the constant sheaf with value the ring of integers $\Z$.


Definition:Stalk of Presheaf

Let $T = \left({S, \tau}\right)$ be a topological space. Let $\mathbf C$ be a category which has all small inductive limits. Let $\mathcal F$ be a $\mathbf C$-valued presheaf on $T$. Let $x \in S$. The '''stalk''' $\mathcal F_x$ of $\mathcal F$ at $x$ is the inductive limit: :$\displaystyle\varinjlim_{U \ni x} \mathcal F \left({U}\right)$ over all open $U \subseteq S$ containing $x$.


Definition:Morphism of Sheaves

Let $X$ be a topological space. Let $\mathbf C$ be a category. Let $\mathcal F$ and $\mathcal G$ be $\mathbf C$-valued sheaves on $X$. A '''sheaf morphism''' from $\mathcal F$ to $\mathcal G$ is a presheaf morphism $\mathcal F \to\mathcal G$.


Definition:Category of Sheaves on Topological Space

Let $X$ be a topological space.  Let $\mathbf C$ be a category. The '''category of $\mathbf C$-valued sheaves on $X$''' is the category with: {{DefineCategory |ob = $\mathbf C$-valued sheaves on $X$ |mor = morphisms of presheaves |id = identity morphisms on presheaves |comp = composition of morphisms of presheaves }}


Definition:Category of Presheaves on Topological Space

Let $X$ be a topological space.  Let $\mathbf C$ be a category. The '''category of $\mathbf C$-valued presheaves on $X$''' is the category with: {{DefineCategory |ob = $\mathbf C$-valued presheaves on $X$ |mor = morphisms of presheaves |id = identity morphisms on presheaves |comp = composition of morphisms of presheaves }}


Definition:Ringed Space

A '''ringed space''' is a pair $\struct {X, \OO_X}$ where: :$X$ is a topological space :$\OO_X$ is a sheaf of rings with unity on $X$.


Definition:Locally Ringed Space

A '''locally ringed space''' is a ringed space $\struct {X, \OO_X}$ such that the stalks of the structure sheaf $\OO_X$ are local rings.


Definition:Natural Transformation/Covariant Functors

Let $F, G : \mathbf C \to \mathbf D$ be covariant functors. A '''natural transformation''' $\eta$ from $F$ to $G$ is a mapping on $\mathbf C$ such that: :$(1): \quad$ For all $x \in \mathbf C$, $\eta_x$ is a morphism from $\map F x$ to $\map G x$. :$(2): \quad$ For all $x, y \in C$ and morphism $f: x \to y$, the following diagram commutes: ::$\xymatrix{ \map F x \ar[d]^{\eta_x} \ar[r]^{\map F f} & \map F y \ar[d]^{\eta_y} \\ \map G x \ar[r]^{\map G f}                 & \map G y}$


Definition:Natural Transformation/Contravariant Functors

Let $F, G: \mathbf C \to \mathbf D$ be contravariant functors. A '''natural transformation''' $\eta$ from $F$ to $G$ is a mapping on $\mathbf C$ such that: :$(1): \quad$ For all $x \in \mathbf C$, $\eta_x$ is a morphism from $\map F x$ to $\map G x$. :$(2): \quad$ For all $x, y \in C$ and morphism $f: x \to y$, the following diagram commutes: ::$\xymatrix{ \map F x \ar[d]^{\eta_x} & \map F y \ar[d]^{\eta_y} \ar[l]^{\map F f}  \\ \map G x                 & \map G y \ar[l]^{\map G f} }$


Definition:Natural Transformation

Let $\mathbf C$ and $\mathbf D$ be categories. === Covariant Functors === {{Definition:Natural Transformation/Covariant Functors}} === Contravariant Functors === {{Definition:Natural Transformation/Contravariant Functors}}


Definition:Natural Isomorphism

Let $\mathbf C$ and $\mathbf D$ be categories. === Covariant Functors === {{Definition:Natural Isomorphism between Covariant Functors}} === Contravariant Functors === {{Definition:Natural Isomorphism between Contravariant Functors}}


Definition:Natural Isomorphism between Covariant Functors

Let $F, G : \mathbf C \to \mathbf D$ be covariant functors. === Definition 1 === A '''natural isomorphism''' from $F$ to $G$ is a natural transformation $\eta : F \to G$ such that for all $x\in \mathbf C$, $\eta_x : F(x) \to G(x)$ is an isomorphism. === Definition 2 === A '''natural isomorphism''' from $F$ to $G$ is an isomorphism in the functor category $\operatorname{Funct}(\mathbf C, \mathbf D)$, that is, a natural transformation $\eta : F \to G$ for which there exists a natural transformation $\xi : G \to F$ such that the compositions $\xi \circ \eta = 1_F$ and $\eta \circ \xi = 1_G$ are identity natural transformations.


Definition:Hom Functor

Let $\mathbf{Set}$ be the category of sets. Let $\mathbf C$ be a locally small category. === Hom Bifunctor === {{Definition:Hom Bifunctor}} === Covariant Hom Functor === {{Definition:Covariant Hom Functor}} === Contravariant Hom Functor === {{Definition:Contravariant Hom Functor}} {{definition wanted|for enriched categories}}


Definition:Covariant Hom Functor

Let $C \in \mathbf C_0$ be an object of $\mathbf C$. The '''covariant hom functor based at $C$''', $\operatorname{Hom}_{\mathbf C} \left({C, \cdot}\right): \mathbf C \to \mathbf{Set}$, is the covariant functor defined by: {{begin-axiom}} {{axiom|lc= Object functor:        |m = \operatorname{Hom}_{\mathbf C} \left({C, D}\right) = \operatorname{Hom}_{\mathbf C} \left({C, D}\right) }} {{axiom|lc= Morphism functor:        |m = \operatorname{Hom}_{\mathbf C} \left({C, f}\right): \operatorname{Hom}_{\mathbf C} \left({C, A}\right) \to \operatorname{Hom}_{\mathbf C} \left({C, B}\right), g \mapsto f \circ g        |rc= for $f: A \to B$ }} {{end-axiom}} where $\operatorname{Hom}_{\mathbf C} \left({C, D}\right)$ denotes a hom set. Thus, the morphism functor is defined to be postcomposition.


Definition:Hom Bifunctor

The '''hom bifunctor''' on $\mathbf C$ is the covariant functor $\map {\operatorname {Hom} } {-, -} : \mathbf C^{\operatorname {op} } \times \mathbf C \to \mathbf {Set}$ from the product with the opposite category to the category of sets such that: :$(1): \quad \map {\operatorname {Hom} } {a, b}$ is the hom class :$(2): \quad$ If $\tuple {f^{\operatorname {op} }, g}: \tuple {a, b} \to \tuple {c, d}$ is a morphism, $\map {\operatorname {Hom} } {f^{\operatorname{op} }, g}: \map {\operatorname {Hom} } {a, b} \to \map {\operatorname {Hom} } {c, d}$ is $f_* \circ g^*$, the postcomposition with $g$ composed with the precomposition with $f$.


Definition:Lehmer's Electromechanical Sieve

rightLehmer's Electromechanical Sieve '''Lehmer's electromechanical sieve''' is a device which was invented for the purpose of solving number theory problems, such as: : determining the prime decomposition of integers : determining whether an integer is prime. It consists of: : a mechanical assemblage of gearwheels : photo detector : an electronic amplifier. It is currently located at the [http://www.computerhistory.org/ Computer History Museum], where its lot number is $\text X 85.82$. {{NamedforDef|Derrick Henry Lehmer|cat = Lehmer D H}}


Definition:Category of Abelian Groups

The '''category of abelian groups''' is the category $\mathbf{Ab}$ with: {{DefineCategory |ob = abelian groups |mor = group homomorphisms |id = identity mappings |comp = composition of mappings }}


Definition:Category of Open Sets

Let $T = \left({S, \tau}\right)$ be a topological space. === Definition 1 === {{:Definition:Category of Open Sets/Definition 1}} === Definition 2 === {{:Definition:Category of Open Sets/Definition 2}}


Definition:Direct Image of Presheaf

Let $\mathbf C$ be a category. Let $T_1 = \left({S_1, \tau_1}\right)$ and $T_2 = \left({S_2, \tau_2}\right)$ be topological spaces. Let $f : T_1 \to T_2$ be continuous. Let $\mathcal F$ be a $\mathbf C$-valued presheaf on $T_1$. === Definition 1 === {{:Definition:Direct Image of Presheaf/Definition 1}} === Definition 2 === {{:Definition:Direct Image of Presheaf/Definition 2}}


Definition:Direct Image of Sheaf

Let $X$ be a topological space. Let $\mathbf C$ be a category. Let $\mathcal F$ be a $\mathbf C$-valued sheaf on $X$. The '''direct image''' of $\mathcal F$ is the direct image of the presheaf $\mathcal F$.


Definition:Étalé Space of Presheaf

Let $X$ be a topological space. Let $\mathbf C$ be a category. Let $\mathcal F$ be a $\mathbf C$-valued presheaf on $X$. The '''étalé space''' of $\mathcal F$ is the pair $\struct {\map {\operatorname {\acute Et} } {\mathcal F}, \pi}$ where: :$\map {\operatorname {\acute Et} } {\mathcal F}$ is the disjoint union $\displaystyle \bigsqcup_{x \mathop \in X} \mathcal F_x$ of stalks of $\mathcal F$ :$\pi: \map {\operatorname {\acute Et} } {\mathcal F} \to X$ is the canonical projection. === Topology on Étalé Space === {{Definition:Topology on Étalé Space of Presheaf}}


Definition:Associated Section of Étalé Space

Let $T = \struct {S, \tau}$ be a topological space. Let $\mathcal F$ be a presheaf of sets. Let $\map {\operatorname {\acute Et} } {\mathcal F}$ be its étalé space. Let $U \subseteq S$ be open in $T$. Let $s \in \map {\mathcal F} U$ be a section. The '''associated section''' of $\map {\operatorname {\acute Et} } {\mathcal F}$ is the mapping: :$\overline s : U \to \map {\operatorname {\acute Et} } {\mathcal F} : x \mapsto \tuple {x, s_x}$ where $s_x$ is the image in the stalk at $x$.


Definition:Topology on Étalé Space of Presheaf

The '''topology on $\map {\operatorname {\acute Et} } {\mathcal F}$''' is the final topology with respect to the sections associated to elements of $\map {\mathcal F} U$ with $U \subseteq S$ open.


Definition:Trapdoor Function

A '''trapdoor function''' is a mapping $f$ which is: :for a given $x$ in the domain of $f$ is easy to calculate  :for a given $y$ in the image of $f$ is hard to calculate unless a specific piece of information $t$ is supplied, in which case it is easy. That piece of information is known as the '''trapdoor'''.


Definition:Section of Étalé Space

Let $T = \left({S, \tau}\right)$ be a topological space. Let $\mathcal F$ be a presheaf of sets on $T$. Let $\left({\operatorname{\acute Et} \left({\mathcal F}\right), \pi}\right)$ be its étalé space. Let $U \subseteq S$ be open. === Definition 1 === {{:Definition:Section of Étalé Space/Definition 1}} === Definition 2 === {{:Definition:Section of Étalé Space/Definition 2}}


Definition:Sheafification

Let $X$ be a topological space.  Let $\mathbf C$ be a category. Let $\mathcal F$ be a $\mathbf C$-valued presheaf on $X$. === Definition 1 === {{definition wanted|as a representation of a functor}} === Definition 2 === Let $\mathcal F$ be a presheaf of sets. The '''sheafification''' of $\mathcal F$ is the sheaf of continuous sections of its étalé space $\map {\operatorname {\acute Et} } {\mathcal F}$.


Definition:Inverse Image Presheaf

Let $T_1 = \left({S_1, \tau_1}\right)$ and $T_2 = \left({S_2, \tau_2}\right)$ be topological spaces. Let $f: T_1 \to T_2$ be continuous. Let $\mathbf C$ be a category which has all small inductive limits. Let $\mathcal F$ be a $\mathbf C$-valued presheaf on $T_2$. The '''inverse image presheaf''' of $\mathcal F$ via $f$ is the presheaf $f^{-1}_{\operatorname{Psh}} \mathcal F$ on $T_1$ with: : $\left({f^{-1}_{\operatorname{Psh}} \mathcal F}\right) \left({U}\right) = \displaystyle \varinjlim_{V \supseteq f \left({U}\right)} \mathcal F \left({V}\right)$ where the inductive limit goes over open $V \subseteq Y$ : $\operatorname{res}^U_W$ is the induced map on the inductive limit of the subset $\left\{ {V: V \supseteq f \left({U}\right)}\right\} \subseteq \left\{ {V : V \supseteq f \left({W}\right)}\right\}$


Definition:Unique up to Unique Isomorphism

Let $\mathbf C$ be a category. Let $S \subseteq \operatorname{Ob}(\mathbf C)$ be a subclass of its objects. The class $S$ is '''unique up to unique isomorphism''' {{iff}} for all objects $s,t \in S$ there is a unique isomorphism from $s$ to $t$.


Definition:Zero Object

Let $\mathbf C$ be a category. A '''zero object''' of $\mathbf C$ is an object which is initial and terminal.


Definition:RSA 129

'''RSA $129$''' is the name given to the semiprime: :$114 \, 381 \, 625 \, 757 \, 888 \, 867 \, 669 \, 235 \, 779 \, 976 \, 146 \, 612 \, 010 \, 218 \, 296 \, 721 \, 242 \, 362 \, 562 \, 561 \, 842 \, 935 \, 706 \, 935 \, 245 \, 733 \, 897 \, 830 \, 597 \, 123 \, 563 \, 958 \, 705 \, 058 \, 989 \, 075 \, 147 \, 599 \, 290 \, 026 \, 879 \, 543 \, 541$ Its factors are: :$3 \, 490 \, 529 \, 510 \, 847 \, 650 \, 949 \, 147 \, 849 \, 619 \, 903 \, 898 \, 133 \, 417 \, 764 \, 638 \, 493 \, 387 \, 843 \, 990 \, 820 \, 577$ and: :$32 \, 769 \, 132 \, 993 \, 266 \, 709 \, 549 \, 961 \, 988 \, 190 \, 834 \, 461 \, 413 \, 177 \, 642 \, 967 \, 992 \, 942 \, 539 \, 798 \, 288 \, 533$


Definition:Tychonoff Topology/Natural Sub-Basis

The '''natural sub-basis on $X$''' is defined as: :$\mathcal S = \set {\pr_i^{-1} \sqbrk U: i \in I, \, U \in \tau_i}$


Definition:Tychonoff Topology/Natural Basis

The '''natural basis on $X$''' is defined as the basis generated by $\SS$.


Definition:RSA 130

'''RSA $130$''' is the name given to the semiprime: :$1 \, 807 \, 082 \, 088 \, 687 \, 404 \, 805 \, 951 \, 656 \, 164 \, 405 \, 905 \, 566 \, 278 \, 102 \, 516 \, 769 \, 401 \, 349 \, 170 \, 127 \, 021 \, 450 \, 056 \, 662 \, 540 \, 244 \, 048 \, 387 \, 341 \, 127 \, 590 \, 812 \, 303 \, 371 \, 781 \, 887 \, 966 \, 563 \, 182 \, 013 \, 214 \, 880 \, 557$ Its factors are: :$39 \, 685 \, 999 \, 459 \, 597 \, 454 \, 290 \, 161 \, 126 \, 162 \, 883 \, 786 \, 067 \, 576 \, 449 \, 112 \, 810 \, 064 \, 832 \, 555 \, 157 \, 243$ and: :$45 \, 534 \, 498 \, 646 \, 735 \, 972 \, 188 \, 403 \, 686 \, 897 \, 274 \, 408 \, 864 \, 356 \, 301 \, 263 \, 205 \, 069 \, 600 \, 999 \, 044 \, 599$


Definition:Mersenne Prime/Index

The '''index''' of the '''Mersenne prime''' $M_p = 2^p - 1$ is the (prime) number $p$.


Definition:Direct Limit

Let $\mathbf C$ be a category. Let $I$ be an upward directed set. Let $\mathbf I$ be its order category. Let $D : \mathbf I \to \mathbf C$ be a diagram. The '''direct limit''' of $D$ is the colimit of the diagram $D$.


Definition:Contravariant Hom Functor

Let $C \in \mathbf C_0$ be an object of $\mathbf C$. The '''contravariant hom functor based at $C$''': : $\operatorname{Hom}_{\mathbf C} \left({\cdot, C}\right): \mathbf C \to \mathbf{Set}$ is the covariant functor defined by: {{begin-axiom}} {{axiom | lc= Object functor:         | m = \operatorname{Hom}_{\mathbf C} \left({B, C}\right) = \operatorname{Hom}_{\mathbf C} \left({B, C}\right) }} {{axiom | lc= Morphism functor:         | m = \operatorname{Hom}_{\mathbf C} \left({f, C}\right): \operatorname{Hom}_{\mathbf C} \left({B, C}\right) \to \operatorname{Hom}_{\mathbf C} \left({A, C}\right), g \mapsto g \circ f         | rc= for $f: A \to B$ }} {{end-axiom}} where $\operatorname{Hom}_{\mathbf C} \left({B, C}\right)$ denotes a hom set. Thus, the morphism functor is defined to be precomposition.


Definition:Representable Functor

Let $\mathbf C$ be a locally small category. Let $\mathbf{Set}$ be the category of sets. Let $F : \mathbf C \to \mathbf{Set}$ be a covariant functor. Then $F$ is '''representable''' if there exists an object $a\in \mathbf C$ such that $F$ is naturally isomorphic to the covariant hom functor $\operatorname{Hom}(a,-)$. {{DefinitionWanted|contravariant}}


Definition:Yoneda Functor/Yoneda Embedding

The '''Yoneda embedding''' of $C$ is the covariant functor $h_- : C \to \left[{C^{\operatorname{op}}, \mathbf{Set} }\right]$ which sends: : an object $X\in C$ to the contravariant hom-functor $h_X = \operatorname{Hom} \left({-, X}\right)$ : a morphism $f : X \to Y$ to the postcomposition natural transformation $h_f : \operatorname{Hom} \left({-, X}\right) \to \operatorname{Hom} \left({-, Y}\right)$


Definition:Yoneda Functor/Contravariant

The '''contravariant Yoneda functor''' of $C$ is the contravariant functor $h^- : C \to \left[{C, \mathbf{Set} }\right]$ which sends : an object $X \in C$ to the covariant hom-functor $h^X = \operatorname{Hom} \left({X, -}\right)$ : a morphism $f : X \to Y$ to the precomposition natural transformation $h^f : \operatorname{Hom} \left({Y, -}\right) \to \operatorname{Hom} \left({X, -}\right) : g \mapsto g \circ f$


Definition:Yoneda Functor

Let $C$ be a locally small category. Let $C^{\operatorname{op}}$ be its opposite category. Let $\mathbf{Set}$ be the category of sets. Let $\left[{C^{\operatorname{op}}, \mathbf{Set} }\right]$ be the functor category between them. === Yoneda Embedding === {{Definition:Yoneda Embedding}} === Contravariant Yoneda Functor === {{Definition:Contravariant Yoneda Functor}}


Definition:Whiskering

Let $B,C,D,E$ be categories. Let $F, G : C \to D$ be covariant functors. Let $\eta : F \to G$ be a natural transformation. === Natural Transformation followed by Functor === Let $H : D \to E$ be a covariant functor. The '''right whiskering''' of $H$ and $\eta$ is the natural transformation $H\eta : H \circ F \to H \circ G$ between compositions of functors defined by $(H\eta)_A = H(\eta_A)$ for $A\in C$. === Functor followed by Natural Transformation === Let $K : B \to C$ be a covariant functor. The '''left whiskering''' of $\eta$ and $K$ is the natural transformation $\eta K : F \circ K \to G \circ K$ between compositions of functors defined by $(\eta K)_A = \eta_{K(A)}$ for $A\in B$.


Definition:Functor Evaluation Bifunctor

Let $C$ and $D$ be categories. Let $\operatorname{Funct}(C, D)$ be their covariant functor category. Let $\operatorname{Funct}(C, D) \times C$ be the product category. The '''evaluation bifunctor''' $\operatorname{ev} : \operatorname{Funct}(C, D) \times C \to D$ is the covariant functor that sends: *an object $(F, a)$ to $F(a)$ *a morphism $(\eta, f) : (F, a) \to (G, b)$ to $G(f) \circ \eta_a = \eta_b \circ F(f)$


Definition:Vertical Composition of Natural Transformations

Let $C$ and $D$ be categories. Let $F,G,H : C \to D$ be covariant functors. Let $\eta : F \to G$ and $\xi : G \to H$ be natural transformations. The '''vertical composition''' of $\eta$ and $\xi$ is the natural transformation $\xi \circ \eta : F \Rightarrow H$ with $(\xi \circ \eta)_A = \xi_A \circ \eta_A$ for $A\in C$.


Definition:Titanic Prime

A '''titanic prime''' is a prime number with $1000$ digits or more.


Definition:Representation of Functor

Let $\mathbf C$ be a locally small category. Let $\mathbf{Set}$ be the category of sets. Let $F : \mathbf C \to \mathbf{Set}$ be a covariant functor. A '''representation''' of $F$ is a pair $(A, \eta)$ where $\eta : \operatorname{Hom}(A, -) \to F$ is a natural isomorphism with the covariant hom functor of $A$.


Definition:Fully Faithful Functor

Let $C$ and $D$ be categories. Let $F : C \to D$ be a covariant functor. The functor $F$ is '''fully faithful''' {{iff}} it is full and faithful. Category:Definitions/Category Theory i96p1iro3gt54jq9tgt3qy5ryxnud99


Definition:Embedding of Categories

Let $C$ and $D$ be categories. Let $F : C \to D$ be a functor. === Definition 1 === The functor $F$ is an '''embedding''' {{iff}} it is: * injective on objects * faithful === Definition 2 === The functor $F$ is an '''embedding''' {{iff}} it is injective on morphisms. === Definition 3 === The functor $F$ is an '''embedding''' {{iff}} it is a monomorphisms in the category of categories.


Definition:Group Category

Let $G$ be a group. The '''group category''' of $G$ is the monoid category of $G$. Category:Definitions/Examples of Categories 4574r9kgtovfd1xa7xh0kz05tjwo9q9


Definition:Primorial Prime

A '''primorial prime''' is a prime number $p$ such that $p + 1$ is a primorial. That is, a prime number which is $1$ less than a primorial.


Definition:Covariant Power Set Functor

The '''(covariant) power set functor''' $\mathcal P : \mathbf{Set} \to \mathbf{Set}$ is the covariant functor which sends: * An object $x$ to its power set $\mathcal P(x)$. * A morphism $f : x \to y$ to the direct image mapping $\mathcal P(f) : \mathcal P(x) \to \mathcal P(y)$.


Definition:Contravariant Power Set Functor

The '''contravariant power set functor''' $\overline{\mathcal P} : \mathbf{Set} \to \mathbf{Set}$ is the contravariant functor which sends: * An object $x$ to its power set $\mathcal P(x)$. * A morphism $f : x \to y$ to the inverse image mapping $\overline{\mathcal P}(f) : \mathcal P(y) \to \mathcal P(x)$.


Definition:Power Set Functor

=== Covariant === {{Definition:Covariant Power Set Functor}} === Contravariant === {{Definition:Contravariant Power Set Functor}} {{definition wanted|Create the analogous definition for relations.}} Category:Definitions/Examples of Functors qzkvcpd1lr99266stgimt8nmreynnz1


Definition:Gigantic Prime

A '''gigantic prime''' is a prime number with $10 \, 000$ digits or more.


Definition:Mapping Induced on Powerset

The '''mapping induced on a powerset''' can be either of two concepts: == Direct Image Mapping == {{:Definition:Direct Image Mapping}} == Inverse Image Mapping == {{:Definition:Inverse Image Mapping}}


Definition:Proth Prime

A '''Proth prime''' is a Proth number which is also a prime.


Definition:Proth Number

A '''Proth number''' is a natural number of the form: :$N = k \times 2^n + 1$ where: :$k$ is an odd integer :$n$ is a positive integer :$2^n > k$ The condition is included, otherwise every odd integer greater than $1$ would be a '''Proth number'''.


Definition:Hyperoperation

=== Hyperoperation Sequence === {{:Definition:Hyperoperation/Sequence}} === $n$th Hyperoperation === {{:Definition:Hyperoperation/Nth Hyperoperation}}


Definition:Hyperoperation/Sequence

The '''hyperoperation sequence''' is the sequence $\left\langle{H_n}\right\rangle$ of binary operations $H_n : \Z_{\ge 0} \times \Z_{\ge 0} \to \Z_{\ge 0}$, defined as: $\forall n, x, y \in \Z_{\ge 0}: H_n \left({x, y}\right) = \begin{cases} y + 1 & : n = 0 \\ x & : n = 1, y = 0 \\ 0 & : n = 2, y = 0 \\ 1 & : n > 2, y = 0 \\ H_{n - 1} \left({x, H_n \left({x, y - 1}\right)}\right) & : n > 0, y > 0 \end{cases}$


Definition:Hyperoperation/Nth Hyperoperation

The $n$th term of $\left\langle{H_n}\right\rangle$, which is the binary operation $H_n : \Z_{\ge 0} \times \Z_{\ge 0} \to \Z_{\ge 0}$, is known as the '''$n$th hyperoperation'''.


Definition:Power (Algebra)/Integer/Knuth Notation

In certain contexts in number theory, the symbol $\uparrow$ is used to denote the (usually) integer power operation: :$x \uparrow y := x^y$ This notation is usually referred to as '''Knuth (uparrow) notation'''.


Definition:Gigaplex

A '''gigaplex''' is defined to be $10$ to the power of a (short scale) billion, that is $10^{1 \, 000 \, 000 \, 000}$.


Definition:Billion

=== Short Scale === {{:Definition:Billion/Short Scale}} === Long Scale === {{:Definition:Billion/Long Scale}}


Definition:Billion/Short Scale

'''Billion''' is a name for $10^9$ in the short scale system: :'''One billion''' $= 10^{3 \times 2 + 3}$


Definition:Billion/Long Scale

'''Billion''' is a name for $10^{12}$ in the long scale system: :'''One billion''' $= 10^{6 \times 2}$


Definition:Skewes' Number

'''Skewes' number''' is: :$10^{10^{10^{34} } }$ In Knuth notation this can be presented as: :$10 \uparrow \paren {10 \uparrow \paren {10 \uparrow 34} }$


Definition:Graham's Number

Let $3 \uparrow 3$ denote Knuth's notation for powers: :$3 \uparrow 3 := 3^3$ Further, let: :$3 \uparrow \uparrow 3 := 3 \uparrow \paren {3 \uparrow 3} = 3^{\paren {3^3} }$ and so define: :$3 \underbrace {\uparrow \uparrow \ldots \uparrow \uparrow}_n 3 := 3 \underbrace {\uparrow \uparrow \ldots \uparrow \uparrow}_{n - 1} \paren {3 \underbrace {\uparrow \uparrow \ldots \uparrow \uparrow}_{n - 1} 3}$ Thus, for example: :$3 \uparrow \uparrow \uparrow \uparrow 3 = 3 \uparrow \uparrow \uparrow \paren {3 \uparrow \uparrow \uparrow 3}$ Let: : $n_1 := 3 \underbrace {\uparrow \uparrow \ldots \uparrow \uparrow}_{3 \uparrow \uparrow \uparrow \uparrow 3} 3$ That is, a total of $3 \uparrow \uparrow \uparrow \uparrow 3$ instances of $\uparrow$. Similarly, let: :$n_2 := 3 \underbrace {\uparrow \uparrow \ldots \uparrow \uparrow}_{n_1} 3$ In general for $m \ge 2$: :$n_m := 3 \underbrace {\uparrow \uparrow \ldots \uparrow \uparrow}_{n_{m - 1} } 3$ and so specifically: :$n_{63} := 3 \underbrace {\uparrow \uparrow \ldots \uparrow \uparrow}_{n_{62} } 3$ It is this $n_{63}$ which is defined as '''Graham's number'''. {{NamedforDef|Ronald Lewis Graham|cat = Graham}}


Definition:Metrizable Topology/Linguistic Note

The UK English spelling of '''metrizable''' is '''metrisable''', but it is rarely found.


Definition:Noetherian Topological Space/Definition 1

A topological space $T = \left({S, \tau}\right)$ is '''Noetherian''' {{iff}} its set of closed sets, ordered by inclusion, satisfies the descending chain condition.


Definition:Noetherian Topological Space/Definition 2

A topological space $T = \left({S, \tau}\right)$ is '''Noetherian''' {{iff}} its set of open sets, ordered by inclusion, satisfies the ascending chain condition.


Definition:Everywhere Dense/Definition 1

The subset $H$ is '''(everywhere) dense in $T$''' {{iff}}: :$H^- = S$ where $H^-$ is the closure of $H$.


Definition:Everywhere Dense/Definition 2

The subset $H$ is '''(everywhere) dense in $T$''' {{iff}} the intersection of $H$ with every open subset of $T$ is non-empty: :$\forall U \in \tau: H \cap U \ne \O$


Definition:Product Space (Topology)/Factor Space

Each of the topological spaces $\struct{S_i, \tau_i}$ are called the '''factors''' of $\struct{S, \tau}$, and can be referred to as '''factor spaces'''.


Definition:Finer Filter on Set

Let $\FF \subseteq \FF'$. Then  $\FF'$ is '''finer''' than $\FF$.


Definition:Coarser Filter on Set

Let $\FF \subseteq \FF'$. Then $\FF$ is '''coarser''' than $\FF$.


Definition:Finer Filter on Set/Strictly Finer

Let $\FF \subset \FF'$, that is, $\FF \subseteq \FF'$ but $\FF \ne \FF'$. Then $\FF'$ is '''strictly finer''' than $\FF$.


Definition:Comparable Filters on Set

Let $S$ be a set. Let $\powerset S$ be the power set of $S$. Let $\FF, \FF' \subset \powerset S$ be two filters on $S$. $\FF$ and $\FF'$ are '''comparable''' {{iff}} one is finer (or coarser) than the other.


Definition:Ultrafilter on Set/Definition 2

Let $S$ be a set. Let $\FF \subseteq \powerset S$ be a filter on $S$. Then $\FF$ is an '''ultrafilter (on $S$)''' {{iff}}: :for every $A \subseteq S$ and $B \subseteq S$ such that $A \cap B = \O$ and $A \cup B \in \FF$, either $A \in \FF$ or $B \in \FF$.


Definition:Ultrafilter on Set/Definition 3

Let $S$ be a set. Let $\FF \subseteq \powerset S$ be a filter on $S$. Then $\FF$ is an '''ultrafilter (on $S$)''' {{iff}}: :for every $A \subseteq S$, either $A \in \FF$ or $\relcomp S A \in \FF$ where $\relcomp S A$ is the relative complement of $A$ in $S$, that is, $S \setminus A$.


Definition:Ring under Ring

Let $A$ be a commutative ring with unity. A '''ring under $A$''' is a pair $(B, f)$ where: * $B$ is a commutative ring with unity * $f : A \to B$ is a unital ring homomorphism


Definition:Principal Ultrafilter/Nonprincipal

Let $\FF \subset \powerset S$ be an ultrafilter on $S$ which does not have a cluster point. Then $\FF$ is a '''nonprincipal ultrafilter'''  on $S$.


Definition:Limit Point/Filter Basis/Definition 1

A point $x \in S$ is called a '''limit point of $\BB$''' {{iff}} $\FF$ converges on $x$. $\BB$ is likewise said to converge on $x$.


Definition:Limit Point/Filter Basis/Definition 2

A point $x \in S$ is called a '''limit point of $\BB$''' {{iff}} every neighborhood of $x$ contains a set of $\BB$.


Definition:Determinant

=== Determinant of Matrix === {{:Definition:Determinant of Matrix}} === Determinant of Linear Operator === {{:Definition:Determinant of Linear Operator}}


Definition:Compact Space/Topology/Definition 1

A topological space $T = \struct {S, \tau}$ is '''compact''' {{iff}} every open cover for $S$ has a finite subcover.


Definition:Compact Space/Topology/Definition 2

A topological space $T = \struct {S, \tau}$ is '''compact''' {{iff}} it satisfies the Finite Intersection Axiom.


Definition:Compact Space/Topology/Definition 3

A topological space $T = \struct {S, \tau}$ is '''compact''' {{iff}} $\tau$ has a sub-basis $\BB$ such that: :from every cover of $S$ by elements of $\BB$, a finite subcover of $S$ can be selected.


Definition:Compact Space/Topology/Definition 4

A topological space $T = \left({S, \tau}\right)$ is '''compact''' {{iff}} every filter on $S$ has a limit point in $S$.


Definition:Countably Compact Space/Definition 1

A topological space $T = \struct {S, \tau}$ is '''countably compact''' {{iff}}: :every countable open cover of $T$ has a finite subcover.


Definition:Countably Compact Space/Definition 2

A topological space $T = \struct {S, \tau}$ is '''countably compact''' {{iff}}: :every countable set of closed sets of $T$ whose intersection is empty has a finite subset whose intersection is empty. That is, $T$ satisfies the countable finite intersection axiom.


Definition:Countably Compact Space/Definition 3

A topological space $T = \struct {S, \tau}$ is '''countably compact''' {{iff}}: :every infinite sequence in $S$ has an accumulation point in $S$.


Definition:Countably Compact Space/Definition 5

A topological space $T = \struct {S, \tau}$ is '''countably compact''' {{iff}}: :every infinite subset of $S$ has an $\omega$-accumulation point in $S$.


Definition:Normed Algebra

Let $R$ be a division ring with norm $\norm {\,\cdot\,}_R$. A '''normed algebra''' over $R$ is a pair $\struct {A, \norm{\,\cdot\,} }$ where: :$A$ is a algebra over $R$ :$ \norm{\,\cdot\,} $ is a norm on $A$.


Definition:Normed Ring

Let $\norm {\, \cdot \,}$ be a norm on $R$. Then $\struct {R, \norm {\, \cdot \,} }$ is a '''normed ring'''.


Definition:Convergent Product/Normed Algebra

Let $\mathbb K$ be a division ring with norm $\norm{\,\cdot\,}_{\mathbb K}$. Let $\struct{ A, \norm{\,\cdot\,} }$ be an associative normed unital algebra over $\mathbb K$. Let $\sequence{a_n}$ be a sequence in $A$. === Definition 1 === The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ is '''convergent''' {{Iff}} there exists $n_0\in\N$ such that: # $a_n$ is invertible for $n \geq n_0$ # the sequence of partial products of $\displaystyle \prod_{n \mathop = n_0}^\infty a_n$ converges to some invertible $b\in A^\times$. === Definition 2: for complete algebras === Let $\struct{ A, \norm{\,\cdot\,} }$ be complete. The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ is '''convergent''' {{Iff}} there exists $n_0\in\N$ such that: : the sequence of partial products of $\displaystyle \prod_{n \mathop = n_0}^\infty a_n$ converges to some invertible $a\in A^\times$.


Definition:Vectorization of Matrix

Let $S$ be a set. Let $m, n \ge 1$ be natural numbers. Let $A = \sqbrk {a_{i j} }$ be a $m \times n$ matrix over $S$. === Definition 1 === The '''vectorization''' of $A$ is the $m n \times 1$ column matrix: :$\map {\operatorname {vec} } A = \sqbrk {a_{11}, \ldots, a_{m1}, a_{12}, \ldots, a_{m2}, \ldots, a_{1n}, \ldots, a_{mn} }^\intercal$ informally obtained by stacking the columns of $A$. That is: :$\map {\operatorname {vec} } A_k = a_{\floor {k/m}, k \bmod m}$ where: :$\floor {\, \cdot \,}$ is the floor function :$\bmod$ is the modulo operation. === Definition 2 === Let $R$ be a ring with unity. Let $A$ be an $m\times n$ matrix over $R$. The '''vectorization''' of $A$ is its coordinate vector with respect to the standard matrix basis.


Definition:Block Matrix

Let $S$ be a set. Let $m, n \ge 1$ be positive integers. Let $A = \sqbrk {A_{i j} }$ be an $m \times n$ matrix of matrices over $S$. Let for every $i \in \set {1, \ldots, m}$, the elements of the $i$th row of $A$ have equal height $m_i$. Let for every $j \in \set {1,\ldots, n}$ the elements of the $j$th column of $A$ have equal width $n_i$. Define $M = \displaystyle \sum_{i \mathop = 1}^m m_i$ and $N = \displaystyle \sum_{i \mathop = 1}^n n_i$ as indexed summations. Let more generally $M_k = \displaystyle \sum_{i \mathop = 1}^k m_i$ and $N_l = \displaystyle \sum_{i \mathop = 1}^l n_i$ for $k \in \set {0, \ldots, m}$ and $l \in \set {0, \ldots, n}$. Then the '''block matrix''' of $A$ is the $M \times N$ matrix $\sqbrk {b_{i j} }$ over $S$ defined as the union of the mappings: :$b_{i j} = \sqbrk {A_{kl} }_{i - M_{k - 1}, j - N_{l - 1} }$ on $\set {M_{k - 1}, \ldots, M_k} \times \set {N_{l - 1}, \ldots, N_l}$ for $k \in \set {1, \ldots, m}$ and $ l \in \set {1, \ldots, n}$.


Definition:Standard Matrix Basis

Let $R$ be a ring with unity. Let $m,n\geq1$ be positive integers. Let $i, j \in \left\{ {1, \ldots, m}\right\} \times \{ 1, \ldots, n\}$. The '''standard matrix basis''' of $m\times n$ matrices over $R$ is the ordered basis of standard basis matrices ordered by the colexicographic order on $\left\{ {1, \ldots, m}\right\} \times \{ 1, \ldots, n\}$.


Definition:Sigma-Locally Compact Space

$T$ is '''$\sigma$-locally compact''' {{iff}}: : $T$ is $\sigma$-compact : $T$ is locally compact That is, $T$ is '''$\sigma$-locally compact''' {{iff}}: : $T$ is the union of countably many compact subspaces : every point of $S$ has a local basis $\mathcal B$ such that all elements of $\mathcal B$ are  compact.


Definition:Normed Unital Algebra

Let $R$ be a division ring with norm $\norm {\,\cdot\,}_R$. A '''normed unital algebra''' over $R$ is a pair $\struct {A, \norm {\,\cdot\,} }$ where: :$A$ is a unital algebra over $R$ :$\norm {\,\cdot\,}$ is a norm on $A$.


Definition:Strongly Locally Compact Space/Definition 1

The space $T$ is '''strongly locally compact''' {{iff}}: :every point of $S$ is contained in an open set whose closure is compact.


Definition:Strongly Locally Compact Space/Definition 2

The space $T$ is '''strongly locally compact''' {{iff}}: :every point has a closed compact neighborhood. That is: :every point of $S$ is contained in an open set which is contained in a closed compact subspace.


Definition:Ringed Space/Structure Sheaf

Let $\left({X, \mathcal O_X}\right)$ be a ringed space. The '''structure sheaf''' of $\left({X, \mathcal O_X}\right)$ is the term $\mathcal O_X$.


Definition:Category of Rings

The '''category of rings''', denoted $\mathbf{Ring}$, is the category with: {{DefineCategory | ob = rings | mor = ring homomorphisms |id = identity mappings |comp = composition of mappings }} === Category of rings with unity === {{Definition:Category of Rings with Unity}}


Definition:Category of Rings with Unity

The '''category of rings with unity''' is the category with: {{DefineCategory | ob = rings with unity | mor = unital ring homomorphisms |id = identity mappings |comp = composition of mappings }}


Definition:Morphism of Ringed Spaces

Let $(X, \mathcal O_X)$ and $(Y, \mathcal O_Y)$ be ringed spaces. === Definition 1 === A '''morphism of ringed spaces''' from $(X, \mathcal O_X)$ to $(Y, \mathcal O_Y)$ is a pair $(f, f^\sharp)$ where: * $f : X \to Y$ is continuous * $f^\sharp : \mathcal O_Y \to f_* \mathcal O_X$ is a morphism of sheaves to the direct image sheaf of $\mathcal O_X$ via $f$ === Definition 2 === A '''morphism of ringed spaces''' from $(X, \mathcal O_X)$ to $(Y, \mathcal O_Y)$ is a pair $(f, f^\sharp)$ where: * $f : X \to Y$ is continuous * $f^\sharp : f^{-1}\mathcal O_Y \to \mathcal O_X$ is a morphism of sheaves from the inverse image sheaf of $\mathcal O_Y$ via $f$ === Definition 3 === {{definition wanted|a description using open subsets, between the two above}}


Definition:Restriction of Presheaf to Open Set

Let $C$ be a category. Let $X$ be a topological space. Let $\mathcal F$ be a $C$-valued presheaf on $X$. Let $U \subset X$ be open. The '''restriction''' of $\mathcal F$ to $U$ is the restriction of the contravariant functor $\mathcal F$ to the subcategory $U$.


Definition:Scheme

A '''scheme''' is a locally ringed space $\struct {X, \OO_X}$ such that every point $x \in X$ has an open neighborhood $U$ such that the restriction of $\struct {X, \OO_X}$ to $U$ is an affine scheme.


Definition:Affine Scheme

An '''affine scheme''' is a ringed space which is isomorphic to the spectrum of a commutative ring with unity.


Definition:Sum of Ideals of Ring

Let $R$ be a ring. === Two ideals === Let $I$ and $J$ be ideals of $R$. Their '''sum''' is the ideal equal to their subset sum: :$I + J  = \set {i + j : i \in I \land j \in J}$ === Multiple Ideals === {{definition wanted|1) explicit description, and 2) using the iterated induced operation on subsets}}


Definition:Unique up to Unique Morphism

Let $\mathbf C$ be a category. Let $S \subseteq \operatorname{Ob}(\mathbf C)$ be a subclass of its objects. === Definition 1 === The class $S$ is '''unique up to unique morphism''' {{iff}} for all objects $s,t \in S$ there is a unique morphism from $s$ to $t$. === Definition 2 === The class $S$ is '''unique up to unique morphism''' {{iff}} for all objects $s,t \in S$ there is a unique morphism from $s$ to $t$, and it is an isomorphism.


Definition:Set of Literals

Let $S$ be a set. A '''set of literals''' on $S$ is a triple $\struct {S^\pm, \iota, \theta}$ where: :$S^\pm$ is a set :$\iota : S \to S^\pm$ is a mapping, the '''canonical injection''' :$\theta : S^\pm \to S^\pm$ is an involution without fixed points, the '''inversion mapping''', and we also denote $\map \theta s = s^{-1}$ such that $S^\pm = \iota \sqbrk S \sqcup \theta \sqbrk {\iota \sqbrk S}$ is the disjoint union of the image of $S$ under $\iota$ and its image under $\theta$. Explicitly, $S^\pm$ can be defined as follows. Let $S^\pm = S \sqcup S = S \times \set 0 \cup S \times \set 1$ be the disjoint union of $S$ with $S$. Let $\iota: S \to S^\pm$ be the canonical mapping: :$s \mapsto \tuple {s, 0}$ Let $\theta : S^\pm \to S^\pm$ be the mapping: :$\tuple {s, i} \mapsto \tuple {s, 1 - i}$ {{tidy|The above phrasing is not up to par}}


Definition:Reduced Group Word on Set

Let $S$ be a set. Let $n \ge 0$ be a natural number. Let $w = w_1 \cdots w_i \cdots w_n$ be a group word on $S$ of length $n$. Then $w$ is '''reduced''' {{iff}} $w_i \ne w_{i + 1}^{-1}$ for all $i \in \set {1, \ldots, n - 1}$


Definition:Elementary Reduction of Group Word on Set

Let $X$ be a set. Let $v$ and $w$ be group words on $X$. Let $n$ be the length of $v$. Then $w$ is an '''elementary reduction''' of $v$ {{iff}}: :$w$ has length $n - 2$ :There exists $k \in \set {1, \ldots, n - 1}$ such that: ::$v_k = v_{k + 1}^{-1}$ ::$w_i = \begin {cases} v_i & : i < k \\ v_{i + 2} & : i > k + 1 \end {cases}$ This is denoted $v \overset 1 \longrightarrow w$.


Definition:Reduction of Group Word on Set

Let $X$ be a set. Let $w$ be a group word on $X$. A '''reduction''' of $w$ is a finite sequence of group words $\tuple {w^{\paren 0}, \ldots, w^{\paren n} }$ such that: :$w^{\paren 0} = w$ :$w_{i + 1}$ is an elementary reduction of $w_i$, for all $i \in \set {0, \ldots, n - 1}$ :$w^{\paren n}$ is reduced


Definition:Reduced Form of Group Word

Let $X$ be a set. Let $w$ be a group word on $X$. The '''reduced form''' $\map {\operatorname {red} } w$ of $w$ is the unique reduced word for which there exists a reduction: :$w = w^{\paren 0} \to w^{\paren 1} \to \cdots \to w^{\paren n} = \map {\operatorname {red} } w$


Definition:Composition of Reduced Group Words

Let $X$ be a set. Let $v$ and $w$ be reduced group words on $X$. The '''composition''' of $v$ and $w$ is the reduced form of their concatenation: :$v \cdot w = \operatorname{red}(vw)$


Definition:Group of Reduced Group Words

Let $S$ be a set. The '''group of reduced group words''' on $S$ is the group $\left({F_S, \circ}\right)$ where: :$F_S$ is the set of reduced group words on $S$ :$\circ$ denotes composition of reduced group words.


Definition:Free Group on Set

Let $X$ be a set. A '''free group''' on $X$ is a certain $X$-pointed group, that is, a pair $(F, \iota)$ where: :$F$ is a group :$\iota : X \to F$ is a mapping that can be defined as follows: === Definition 1: by universal property === A '''free group''' on $X$ is an $X$-pointed group $(F, \iota)$ that satisfies the following universal property: :For every $X$-pointed group $(G, \kappa)$ there exists a unique group homomorphism $\phi : F \to G$ such that $\phi \circ \iota = \kappa$, that is, a morphism of pointed groups $F \to G$. === Definition 2: As the group of reduced group words === The '''free group''' on $X$ is the pair $(F, \iota)$ where: :$F$ is the group of reduced group words on $X$ :$\iota : X \to F$ is the canonical injection


Definition:Group Defined by Presentation

Let $S$ be a set. Let $R$ be a set of group words on $S$. Let $F_S$ be the group of reduced group words on $S$. Let $\operatorname{red} \left({R}\right)$ be the set of reduced forms of elements of $R$. Let $N$ be the normal subgroup generated by $\operatorname{red} \left({R}\right)$ in $F$. The '''group defined by the presentation''' $\langle S \mid R \rangle$ is the quotient group $F_S / N$.


Definition:Group Word on Set

Let $S$ be a set. A '''group word''' on $S$ is an ordered tuple on the set of literals $S^\pm$ of $S$.


Definition:Rank of Free Group

Let $F$ be a free group. The '''rank''' of $F$ is the dimension of its abelianization as a module over $\Z$.


Definition:Commutator Subgroup

Let $G$ be a group. Its '''commutator subgroup''' $\sqbrk {G, G}$ is the subgroup generated by all commutators. === Higher derived subgroups === Let $n \ge 0$ be a natural number. The $n$th '''derived subgroup''' $G^{\paren n}$ is recursively defined as: :$G^{\paren n} = \begin{cases} G & : n = 0 \\ \sqbrk {G^{\paren {n - 1} }, G^{\paren {n - 1} } } & : n \ge 1 \end{cases}$


Definition:Abelianization of Group

Let $G$ be a group. Its '''abelianization''' is the quotient by its commutator subgroup: :$G^{\mathrm {ab} } = G / \sqbrk {G, G}$


Definition:Cartesian Closed Set

Let $S$ be a set. Then $S$ is '''cartesian closed''' {{iff}} the cartesian product $S \times S$ is a subset of $S$.


Definition:Group Relation on Set

Let $X$ be a set. A '''group relation''' on $X$ is a pair $(u, v)$ where $u$ and $v$ are group words on $X$. A '''group relation''' $(u, v)$ is also denoted $u=v$.


Definition:Generator of Algebraic Structure

Let $\left({A, \circ}\right)$ be an algebraic structure. Let $G \subset A$ be a subset. === Definition 1 === The subset $G$ is a '''generator''' of $A$ {{iff}} $A$ is the algebraic substructure generated by $G$. === Definition 2 === The subset $G$ is a '''generator''' of $A$ {{iff}}: * $\forall x, y \in G: x \circ y \in A$; * $\forall z \in A: \exists x, y \in W \left({G}\right): z = x \circ y$ where $W \left({G}\right)$ is the set of words of $G$. That is, every element in $A$ can be formed as the product of a finite number of elements of $G$. If $G$ is such a set, then we can write $A = \left \langle {G}\right \rangle$.


Definition:Cartesian Product/Uncountable

Let $I$ be an indexing set with uncountable cardinality. Let $\family {S_\alpha}_{\alpha \mathop \in I}$ be a family of sets indexed by $I$. The '''cartesian product''' of $\family {S_\alpha}$ is denoted: :$\displaystyle \prod_{\alpha \mathop \in I} S_\alpha$


Definition:Cauchy Sequence in Topological Group

Let $G$ be a topological group with identity element $e$. Let $\sequence {a_n}$ be a sequence in $G$. Then $\sequence {a_n}$ is a '''Cauchy sequence''' {{iff}} for every neighborhood $U$ of $e$ there exists $N \in \N$ such that for all $m, n \ge N : a_n a_m^{-1} \in U$.


Definition:Semiring of Natural Numbers

The '''semiring of natural numbers''' is the semiring $\struct {\N, +, \times}$ where: :$\N$ is the set of natural numbers :$+$ is the addition :$\times$ is the multiplication


Definition:Localization of Ring at Prime Ideal

Let $A$ be a commutative ring with unity. Let $\mathfrak p$ be a prime ideal of $A$. The '''localization of $A$ at $\mathfrak p$''' is the localization of $A$ at the complement $A\setminus\mathfrak p$: :$A_{\mathfrak p} = (A\setminus\mathfrak p)^{-1}A$


Definition:Localization of Ring at Element

Let $A$ be a commutative ring with unity. Let $f\in A$ be an element. The '''localization of $A$ at $f$''' is the localization of $A$ at the set of powers $\{1, f, f^2, \ldots\}$: :$A_f = (\{1, f, f^2, \ldots\})^{-1}A$


Definition:Saturation of Multiplicatively Closed Subset of Ring

Let $A$ be a commutative ring with unity. Let $S \subset A$ be a multiplicatively closed subset. === Definition 1 === The '''saturation''' of $S$ is the smallest saturated multiplicatively closed subset of $A$ containing $S$. That is, it is the intersection of all saturated multiplicatively closed subsets containing $S$. {{circularStructure|this is a general issue with definitions of examples of closure operators}} === Definition 2 === The '''saturation''' of $S$ is the set of divisors of elements of $S$. === Definition 3 === The '''saturation''' of $S$ is the set of elements whose image in the localization $A_S$ is a unit of $A$. === Definition 4 === The '''saturation''' of $S$ is the complement relative to $A$ of the union of prime ideals that are disjoint from $S$: :$\map {\operatorname {Sat} } S = A \setminus \displaystyle \bigcup \set {\mathfrak p \in \operatorname{Spec} A: \mathfrak p \cap S = \O}$


Definition:Closure of Set under Closure Operator

Let $S$ be a set. Let $\operatorname{cl}$ be a closure operator on $S$. Let $T \subseteq S$ be a subset of $S$. The '''closure''' of $T$ is its image $\operatorname{cl}(T)$.


Let $T = \struct {S, \tau}$ be a topological space. Let $x \in S$. === Definition 1 === {{Definition:Weakly Locally Connected at Point/Definition 1}} === Definition 2 === {{Definition:Weakly Locally Connected at Point/Definition 2}}


Definition:Zero Morphism via Zero Object

Let $C$ be a category with has a zero object $0$. Let $a,b\in C$ be objects. The '''zero morphism $0$''' from $a$ to $b$ is the composition of the unique morphism $a \to 0$ and the unique morphism $0 \to b$: :$0 : a \to 0 \to b$


Definition:Pointed Category

Let $C$ be a category. Then $C$ is '''pointed''' {{iff}} $C$ has a zero object. Category:Definitions/Category Theory 8fy4dm6wwomuet3up4qrlrjhr06rgaj


Definition:Universal Object

Let $C$ be a category. A '''universal object''' of $C$ is an object that is initial or terminal.


Definition:Disconnected (Topology)/Topological Space

A topological space $T$ is '''disconnected''' {{iff}} it is not connected.


Definition:Kernel (Category Theory)

Let $C$ be a category. Let $f : A \to B $ be a morphism. === Definition 1: for categories with initial objects === Let $C$ have an initial object $0$. A '''kernel''' of $f$ is a morphism $\map \ker f \to A$ which is a pullback of the unique morphism $0 \to B$ via $f$ to $A$. === Definition 2: for categories with zero objects === Let $C$ have an zero object $0$. A '''kernel''' of $f$ is the equalizer of $f$ and the zero morphism $0 : A \to B$.


Definition:Generated Field Extension

Let $E / F$ be a field extension. Let $S \subset E$ be a subset of $E$. === Definition 1 === {{:Definition:Generated Field Extension/Definition 1}} === Definition 2 === {{:Definition:Generated Field Extension/Definition 2}}


Definition:Non-Associative Ring

A '''non-associative ring''' is a ringoid $\struct {R, +, \times}$ such that $\struct {R, +}$ is an abelian group.


Definition:Nonassociative Ring with Unity

A '''non-associative ring with unity''' is a non-associative ring $(R, +, \times)$ such that $(R, \times)$ is a unitary magma.


Definition:Group Action of GL(2,Q) on Irrational Numbers

Let $\GL {2, \Q}$ be the general linear group on the field of rational numbers. Let $\R \setminus \Q$ be the set of irrational numbers. The '''standard group action''' of $\GL {2, \Q}$ on $\R \setminus \Q$ is the group action: :$\GL {2, \Q} \times \paren {\R \setminus \Q} \to \R \setminus \Q$: :$\paren {\begin {pmatrix} a & b \\ c & d \end {pmatrix}, x} \mapsto \dfrac {a x + b} {c x + d}$


Definition:Disconnected (Topology)/Set

{{:Definition:Disconnected (Topology)/Set/Definition 1}}


Definition:Disconnected (Topology)/Points

Let $T = \left({S, \tau}\right)$ be a topological space. Let $a, b \in S$. Then $a$ and $b$ are '''disconnected (in $T$)''' {{iff}} they are not '''connected (in $T$)'''.


Definition:Local Dimension of Topological Space

Let $X$ be a topological space. Let $x \in X$. The '''local dimension of $X$ at $x$''' is the supremum of lengths of chains of closed irreducible sets of $T$ containing $x$, ordered by inclusion. Thus, the '''Krull dimension''' is $\infty$ {{iff}} there exist arbitrarily long chains containing $x$.


Definition:Krull Dimension of Scheme

Let $\struct {X, \mathcal O_X}$ be a scheme. Its '''Krull dimension''' is the Krull dimension of $X$.


Definition:Disconnected (Topology)/Set/Definition 1

$H$ is a '''disconnected set of $T$''' {{iff}} it is not a '''connected set of $T$'''.


Definition:Disconnected (Topology)/Set/Definition 2

$H$ is a '''disconnected set of $T$''' {{iff}} there exist open sets $U$ and $V$ of $T$ such that: : $H \subseteq U \cup V$ : $H \cap U \cap V = \O$ : $U \cap H \ne \O$ and: : $V \cap H \ne \O$


Definition:Local Krull Dimension of Scheme

Let $\struct {X, \mathcal O_X}$ be a scheme. Let $x \in X$. The '''local Krull dimension at $x$''' is the local dimension of $X$ at $x$.


Definition:Vanishing Set of Subset of Ring

Let $A$ be a commutative ring with unity. Let $S \subseteq A$ be a subset. The '''vanishing set''' of $S$ is the set of prime ideals of $A$ containing $S$: :$V(S) = \{\mathfrak p \in \operatorname{Spec}(A) : \mathfrak p \supseteq S \}$