Definition:Cancellable Operation

Let $\left ({S, \circ}\right)$ be an algebraic structure. === Left Cancellable Operation === {{:Definition:Left Cancellable Operation}} === Right Cancellable Operation === {{:Definition:Right Cancellable Operation}} === Cancellable Operation === The operation $\circ$ in $\struct {S, \circ}$ is '''cancellable''' {{iff}}: * $\forall a, b, c \in S: a \circ b = a \circ c \implies b = c$ * $\forall a, b, c \in S: a \circ c = b \circ c \implies a = b$ That is, {{iff}} it is both a left cancellable operation and a right cancellable operation.


Definition:Algebraic System

An '''algebraic system''' is a mathematical system $\SS = \struct {E, O}$ where: :$E$ is a non-empty set of elements :$O$ is a set of finitary operations on $E$.


Definition:Successor Mapping on Natural Numbers

Let $\N$ be the set of natural numbers. Let $s: \N \to \N$ be the mapping defined as: :$s = \set {\tuple {x, y}: x \in \N, y = x + 1}$ Considering $\N$ defined as a Peano structure, this is seen to be an instance of a successor mapping.


Definition:Formal Grammar/Top-Down

A '''top-down grammar''' for $\mathcal L$ is a formal grammar which allows well-formed formulas to be built from a single metasymbol. Such a grammar can be made explicit by declaring that: * A metasymbol may be replaced by a letter of $\mathcal A$. * A metasymbol may be replaced by certain collations labeled with metasymbols and signs of $\mathcal A$. From the words thus generated, those not containing any metasymbols are the well-formed formulas.


Definition:Formal Grammar/Bottom-Up

Let $\mathcal L$ be a formal language whose alphabet is $\mathcal A$. <section notitle="true" name="tc"> A '''bottom-up grammar''' for $\mathcal L$ is a formal grammar whose rules of formation allow the user to build well-formed formulas from primitive symbols, in the following way: * Letters are well-formed formulas. * A collection of specified collations of well-formed formulas, possibly labeled with additional signs, are also well-formed formulas. In certain use cases, the first clause is adjusted to allow for more complex situations, for example in the bottom-up specification of predicate logic {{transclude:Definition:Formal Grammar/Bottom-Up/Extremal Clause   |section = tc   |title = Extremal Clause   |header = 3   |link = true }} </section>


Definition:Finite Extended Real Number

An extended real number is defined as '''finite''' {{iff}} it is a real number.


Definition:Infinite

=== Infinite Cardinal === {{:Definition:Infinite Cardinal}} === Infinite Set === {{:Definition:Infinite Set}} === Infinity === {{:Definition:Infinity}}


Definition:Cross-Relation

Let $\struct {S, \circ}$ be a commutative semigroup. Let $\struct {S_1, \circ_{\restriction_1} }, \struct {S_2, \circ_{\restriction_2} }$ be subsemigroups of $S$, where $\circ_{\restriction_1}$ and $\circ_{\restriction_2}$ are the restrictions of $\circ$ to $S_1$ and $S_2$ respectively. Let $\struct {S_1 \times S_2, \oplus}$ be the (external) direct product of $\struct {S_1, \circ_{\restriction_1} }$ and $\struct {S_2, \circ_{\restriction_2} }$, where $\oplus$ is the operation on $S_1 \times S_2$ induced by $\circ_{\restriction_1}$ on $S_1$ and $\circ_{\restriction_2}$ on $S_2$. Let $\boxtimes$ be the relation on $S_1 \times S_2$ defined as: :$\tuple {x_1, y_1} \boxtimes \tuple {x_2, y_2} \iff x_1 \circ y_2 = x_2 \circ y_1$ This relation $\boxtimes$ is referred to as the '''cross-relation on $\struct {S_1 \times S_2, \oplus}$'''.


Definition:Additive Semiring

An '''additive semiring''' is a semiring with a commutative distributand. That is, an '''additive semiring''' is a ringoid $\left({S, *, \circ}\right)$ in which: : $(1): \quad \left({S, *}\right)$ forms a commutative semigroup : $(2): \quad \left({S, \circ}\right)$ forms a semigroup. === Additive Semiring Axioms === {{:Definition:Additive Semiring/Axioms}}


Definition:A Fortiori

'''A fortiori''' knowledge arises from stronger facts already established. For example, in: :''I am too old for this game; therefore, my father is also too old.'' the knowledge: :''My father is too old for this game.'' arises '''a fortiori''' from the stronger knowledge that: :''I am too old for this game.'' An '''a fortiori''' argument is most commonly used by applying a general fact in a particular case.


Definition:Rule of Formation

Let $\mathcal F$ be a formal language whose alphabet is $\mathcal A$. The '''rules of formation''' of $\mathcal F$ are the rules which define how to construct collations in $\mathcal A$ which are well-formed. That is, the '''rules of formation''' tell you how to build collations featuring symbols from the alphabet $\mathcal A$ which are part of the formal language $\mathcal F$. The '''rules of formation''' of a formal language together constitute its formal grammar. There are no strict guidelines on how a '''rule of formation''' should look like, since they are employed to ''produce'' such strict guidelines. Thus, these '''rules of formation''' are often phrased in natural language, but their exact form is to some extent arbitrary.


Definition:Cross-Relation on Natural Numbers

Consider the commutative semigroup $\left({\N, +}\right)$ composed of the natural numbers $\N$ and addition $+$. Let $\left({\N \times \N, \oplus}\right)$ be the (external) direct product of $\left({\N, +}\right)$ with itself, where $\oplus$ is the operation on $\N \times \N$ induced by $+$ on $\N$. Let $\boxtimes$ be the relation on $\N \times \N$ defined as: :$\left({x_1, y_1}\right) \boxtimes \left({x_2, y_2}\right) \iff x_1 + y_2 = x_2 + y_1$ This relation $\boxtimes$ is referred to as the '''cross-relation on $\left({\N \times \N, \oplus}\right)$'''.


Definition:Definitional Abbreviation

When discussing a formal language, some particular WFFs may occur very often. If such WFFs are very unwieldy to write and obscure what the author tries to express, it is convenient to introduce a shorthand for them. Such a shorthand is called a '''definitional abbreviation'''. It does ''not'' in any way alter the meaning or formal structure of a sentence, but is purely a method to keep expressions readable to human eyes.


Definition:Function Symbol

Let $\mathcal L$ be a formal language (for example, the language of predicate logic $\mathcal L_1$). A '''function symbol''' is a letter of $\mathcal L$ used to describe a function. The name '''function symbol''' is a gesture to the reader to make clear what such a symbol should (intuitively) represent in the formal language $\mathcal L$.


Definition:Alphabetic Substitution

Consider the (abbreviated) WFF $Q x: \mathbf C$. Let $y$ be another variable such that $y$ does not occur in $\mathbf C$. Let $\mathbf C'$ be the WFF resulting from replacing all free occurrences of $x$ in $\mathbf C$ with $y$. Then to all intents and purposes, the WFFs: :$Q x: \mathbf C$ :$Q y: \mathbf C'$ will have the same interpretation. Thus we may change the free occurrences of any variable for another variable symbol. This change is called '''alphabetic substitution'''.


Definition:Classes of WFFs

The set of all WFFs of $\mathcal L_1$ formed with relation symbols from $\mathcal P$ and function symbols from $\mathcal F$ can be denoted $WFF \left({\mathcal P, \mathcal F}\right)$. If so desired, the parameters can also be emphasized by writing $WFF \left({\mathcal P, \mathcal F, \mathcal K}\right)$ instead. To specify $\mathcal P$, one speaks of '''WFFs with relation symbols from $\mathcal P$'''. To specify $\mathcal F$, one speaks of '''WFFs with function symbols from $\mathcal F$'''. To specify $\mathcal K$, one speaks of '''WFFs with parameters from $\mathcal K$'''. Of course, combinations of these are possible. Several classes of WFFs are often considered and have special names. === Plain WFF === {{:Definition:Classes of WFFs/Plain WFF}} === Sentence === {{:Definition:Classes of WFFs/Sentence}} === Plain Sentence === {{:Definition:Classes of WFFs/Plain Sentence}}


Definition:Classes of WFFs/Sentence

A WFF is said to be a '''sentence''' {{iff}} it contains no free variables. To denote particular classes of '''sentences''', $SENT \left({\mathcal P, \mathcal F, \mathcal K}\right)$ and analogues may be used, similar to the notation for classes of WFFs.


Definition:Classes of WFFs/Plain Sentence

A WFF is said to be a '''plain sentence''' iff it is both plain and a sentence. That is, if it contains free variables nor parameters. Thus, '''plain sentences''' are those WFFs which are in $SENT \left({\mathcal P, \mathcal F, \varnothing}\right)$.


Definition:Compound Statement/Ill-Formed

The substatements in a compound statement, which are joined by a connective, may be compound statements themselves.  It is clearly necessary that the interpretation of such a compound statement is unambiguous.  A compound statement is said to be '''ill-formed''' if it is ambiguous as to how its substatements are grouped by the action of the connectives. For example, in natural language: :''I would like some juice or water with ice.'' can mean either: :''I would like some juice, or water with ice.'' or: :''I would like some juice with ice, or water with ice.''


Definition:Set/Implicit Set Definition/Multipart Infinite Set

Let $S$ be a set. Suppose $S$ is to contain: :$(1): \quad$ a never-ending list of elements and :$(2): \quad$ other elements which are unrelated to that list (perhaps another never-ending list). Then a '''semicolon''' is used to separate the various conceptual parts: :$S = \set {1, 3, 5, \ldots; 2, 4, 6, \ldots; \text{red}, \text{orange}, \text{green} }$ Note that ''without'' the semicolon it would appear as though the first list (of odd numbers) '''continued''' as the second list (of even numbers) which in turn '''continued''' as a list of colours, which is absurd.


Definition:Mathematical Logic

'''Mathematical logic''' is a sub-branch of symbolic logic in which the foundations of the assumptions upon which rest mathematics itself are investigated and made rigorous.


Definition:Multi-Value Logic

'''Multi-value logic''' is a branch of logic in which it is admissible for a statements to have a truth value other than just true or false.


Definition:Modal Logic

'''Modal logic''' is a branch of logic in which truth values are more complex than being merely true or false, and which distinguishes between different "modes" of truth.


Definition:Naive Set Theory

'''Naïve set theory''', in contrast with axiomatic set theory, is an approach to set theory which assumes the existence of a universal set, despite the fact that such an assumption leads to paradoxes. A popular alternative (and inaccurate) definition describes this as a : ''non-formalized definition of set theory which describes sets and the relations between them using natural language.'' However, the discipline is founded upon quite as rigid a set of axioms, namely, those of propositional and predicate logic.


Definition:Pure Set Theory

'''Pure set theory''' is a system of set theory in which all elements of sets are themselves sets.


Definition:Set Intersection/Set of Sets

Let $\Bbb S$ be a set of sets The '''intersection of $\Bbb S$''' is: :$\ds \bigcap \Bbb S := \set {x: \forall S \in \Bbb S: x \in S}$ That is, the set of all objects that are elements of all the elements of $\Bbb S$. Thus: :$\ds \bigcap \set {S, T} := S \cap T$


Definition:Set Intersection/Countable Intersection

Let $\mathbb S$ be a set of sets. Let $\left\langle{S_n}\right\rangle_{n \mathop \in \N}$ be a sequence in $\mathbb S$. Let $S$ be the intersection of $\left\langle{S_n}\right\rangle_{n \mathop \in \N}$: :$\displaystyle S = \bigcap_{n \mathop \in \N} S_n$ Then $S$ is a '''countable intersection''' of sets in $\mathbb S$.


Definition:Set Union/Set of Sets

Let $\mathbb S$ be a set of sets. The '''union of $\mathbb S$''' is: :$\ds \bigcup \mathbb S := \set {x: \exists X \in \mathbb S: x \in X}$ That is, the set of all elements of all elements of $\mathbb S$. Thus the general union of two sets can be defined as: :$\ds \bigcup \set {S, T} = S \cup T$


Definition:Set Union/Countable Union

Let $\mathbb S$ be a set of sets. Let $\left\langle{S_n}\right\rangle_{n \mathop \in \N}$ be a sequence in $\mathbb S$. Let $S$ be the union of $\left\langle{S_n}\right\rangle_{n \mathop \in \N}$: :$\displaystyle S = \bigcup_{n \mathop \in \N} S_n$ Then $S$ is a '''countable union''' of sets in $\mathbb S$.


Definition:Set Union/Finite Union

Let $S = S_1 \cup S_2 \cup \ldots \cup S_n$. Then: :$\displaystyle S = \bigcup_{i \mathop \in \N^*_n} S_i = \set {x: \exists i \in \N^*_n: x \in S_i}$ where $\N^*_n = \set {1, 2, 3, \ldots, n}$. If it is clear from the context that $i \in \N^*_n$, we can also write $\displaystyle \bigcup_{\N^*_n} S_i$.


Definition:Proof System

A '''proof system''' $\mathscr P$ for $\mathcal L$ comprises: * '''Axioms''' and/or '''axiom schemata'''; * '''Rules of inference''' for deriving theorems. It is usual that a '''proof system''' does this by declaring certain arguments concerning $\mathcal L$ to be valid. Informally, a '''proof system''' amounts to a precise account of what constitutes a '''(formal) proof'''.


Definition:Set Intersection/Finite Intersection

Let $S = S_1 \cap S_2 \cap \ldots \cap S_n$. Then: :$\displaystyle S = \bigcap_{i \mathop \in \N^*_n} S_i := \set {x: \forall i \in \N^*_n: x \in S_i}$ where $\N^*_n = \set {1, 2, 3, \ldots, n}$. If it is clear from the context that $i \in \N^*_n$, we can also write $\displaystyle \bigcap_{\N^*_n} S_i$.


Definition:Set Union/Family of Sets

Let $I$ be an indexing set. Let $\family {S_i}_{i \mathop \in I}$ be a family of sets indexed by $I$. Then the '''union''' of $\family {S_i}$ is defined as: :$\ds \bigcup_{i \mathop \in I} S_i := \set {x: \exists i \in I: x \in S_i}$


Definition:Set Intersection/Family of Sets

Let $I$ be an indexing set. Let $\family {S_i}_{i \mathop \in I}$ be a family of sets indexed by $I$. Then the '''intersection''' of $\family {S_i}$ is defined as: :$\displaystyle \bigcap_{i \mathop \in I} S_i := \set {x: \forall i \in I: x \in S_i}$


Definition:Zero (Ordinal)

The '''zero ordinal''', denoted $0$, is the empty set $\O$.


Definition:Generalized Momentum

The '''generalized momentum of analytical (Lagrangian, Hamiltonian) formulations of classical mechanics''' is defined as the partial derivative of the Lagrangian with regards to the time derivative of generalized coordinates: :$p_i = \dfrac {\partial\mathcal L} {\partial \dot q_i}$ where: : $p_i$ is the $i$th coordinate of the generalized momenta : $\mathcal L$ is the Lagrangian : $\dot q_i$ is the time derivative of the generalized coordinates $q_i$. Category:Definitions/Dimensions of Measurement 5trw1vacbts59y03ekptzhiw2a21xus


Definition:Action Applied by System

The '''action applied by a system''' from state $1$ to state $2$ is defined as the definite integral of the Lagrangian over time from state $1$ to state $2$: :$\displaystyle S_{12} = \int_{t_1}^{t_2} \mathcal L \ \mathrm d t$ where: : $S_{12}$ is the '''action''' from $1$ to $2$ : $t$ is time : $\mathcal L$ is the Lagrangian. Category:Definitions/Dimensions of Measurement l2q6txwc7jphfgt8q8sil12fyacu1iy


Definition:Set Product

Let $S$ and $T$ be sets. Let $P$ be a set and let $\phi_1: P \to S$ and $\phi_2: P \to T$ be mappings such that: :For all sets $X$ and all mappings $f_1: X \to S$ and $f_2: X \to T$ there exists a unique mapping $h: X \to P$ such that: ::$\phi_1 \circ h = f_1$ ::$\phi_2 \circ h = f_2$ :that is, such that: ::$\begin{xy}\xymatrix@+1em@L+3px{ &  X   \ar[ld]_*+{f_1}   \ar@{-->}[d]^*+{h}   \ar[rd]^*+{f_2} \\  S &  P   \ar[l]^*+{\phi_1}   \ar[r]_*+{\phi_2} &  T }\end{xy}$ :is a commutative diagram. Then $P$, together with the mappings $\phi_1$ and $\phi_2$, is called '''a product of $S$ and $T$'''.


Definition:Integral of Integrable Function over Measurable Set

Let $\struct {X, \Sigma, \mu}$ be a measure space, and let $E \in \Sigma$. Let $f: X \to \overline \R$ be a $\mu$-integrable function. Then the '''$\mu$-integral of $f$ over $E$''' is defined by: :$\displaystyle \int_E f \rd \mu := \int \chi_E \cdot f \rd \mu$ where: :$\chi_E$ is the characteristic function of $E$ :$\chi_E \cdot f$ is the pointwise product of $\chi_E$ and $f$ :the integral sign on the {{RHS}} denotes $\mu$-integration of the function $\chi_E \cdot f$.


Definition:Measure with Density

Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Let $f: X \to \overline{\R}_{\ge 0}$ be a positive $\mu$-measurable function. Then the '''measure with density $f$ with respect to $\mu$''', denoted $f \mu$, is defined by: :$f \mu \left({E}\right) := \displaystyle \int_E f \, \mathrm d \mu$ where $\displaystyle \int_E f \, \mathrm d \mu$ is the $\mu$-integral of $f$ over $E$.


Definition:Indexing Set/Family of Sets

Let $\mathcal S$ be a set of sets. Let $I$ be an indexing set. Let $\family {S_i}_{i \mathop \in I}$ be a family of elements of $\mathcal S$ indexed by $I$. Then $\family {S_i}_{i \mathop \in I}$ is referred to as an '''indexed family of sets'''.


Definition:Set Partition/Finite Expansion

Let $S$ be a set. Let $\Bbb S = \set {S_1, S_2, \ldots, S_n}$ form a partition of $S$. Then the representation by such a partition $\displaystyle \bigcup_{k \mathop = 1}^n S_k = S$ is also called a '''finite expansion''' of $S$. The notations: :$S = S_1 \mid S_2 \mid \cdots \mid S_n$ or: :$\Bbb S = \set {S_1 \mid S_2 \mid \cdots \mid S_n}$ are sometimes seen.


Definition:Partitioning

Let $S$ be a set. Let $\family {S_i}_{i \mathop \in I}$ be a family of subsets of $S$ such that: :$(1): \quad \forall i \in I: S_i \ne \O$, that is, none of $S_i$ is empty :$(2): \quad \ds S = \bigcup_{i \mathop \in I} S_i$, that is, $S_i$ is the union of $\family {S_i}_{i \mathop \in I}$ :$(3): \quad \forall i, j \in I: i \ne j \implies S_i \cap S_j = \O$, that is, the elements of $\family {S_i}_{i \mathop \in I}$ are pairwise disjoint. Then $\family {S_i}_{i \in I}$ is a '''partitioning''' of $S$. The image of this '''partitioning''' is the set $\set {S_i: i \in I}$ and is called a partition of $S$. Note the difference between: :the '''partitioning''', which is an indexing function (that is a mapping) and :the partition, which is the effect (that is, the image) of that mapping.


Definition:Set Partition/Component

The elements $S_1, S_2, \ldots \in \mathbb S$ are known as the '''components''' of the partition.


Definition:P-Sequence Space

Let $p \in \R$, $p \ge 1$. The '''$p$-sequence space''', denoted $\ell^p$ or $\ell^p \left({\N}\right)$, is defined as: :$\displaystyle \ell^p := \left\{{\left\langle{z_n}\right\rangle_{n \in \N} \in \C^\N: \sum_{n \mathop = 0}^\infty \left\vert{z_n}\right\vert^p < \infty}\right\}$ As such, $\ell^p$ is a subspace of $\C^\N$, the space of all complex sequences. {{stub}}


Definition:Set Product/Family of Sets

Let $\family {S_i}_{i \mathop \in I}$ be an indexed family of sets. Let $P$ be a set. Let $\family {\phi_i}_{i \mathop \in I}$ be an indexed family of mappings $\phi_i: P \to S_i$ for all $i \in I$ such that: :For all sets $X$ and all indexed families $\family {f_i}_{i \mathop \in I}$ of mappings $f_i: X \to S_i$ there exists a unique mapping $h: X \to P$ such that: ::$\forall i \in I: \phi_i \circ h = f_i$ :that is, such that for all $i \in I$: ::$\begin{xy}\xymatrix@+1em@L+3px{  X   \ar@{-->}[d]_*+{h}   \ar[dr]^*+{f_i} \\  P   \ar[r]_*{\phi_i} &  S_i }\end{xy}$ :is a commutative diagram. Then $P$, together with the family of mappings $\family {\phi_i}_{i \mathop \in I}$, is called '''a product of (the family) $\family {S_i}_{i \mathop \in I}$'''. This '''product of $\family {S_i}_{i \mathop \in I}$''' can be denoted $\struct {P, \family {\phi_i}_{i \mathop \in I} }$.


=== Definition 1 === {{:Definition:Cartesian Product/Family of Sets/Definition 1}} === Definition 2 === {{:Definition:Cartesian Product/Family of Sets/Definition 2}} === Uncountable Cartesian Product === Using this notation, it is then possible to define the Cartesian product of an uncountable family: {{:Definition:Cartesian Product/Uncountable}}


Definition:Cartesian Product/Cartesian Space

Let $S$ be a set. The '''cartesian $n$th power of $S$''', or '''$S$ to the power of $n$''', is defined as: :$\displaystyle S^n = \prod_{k \mathop = 1}^n S = \set {\tuple {x_1, x_2, \ldots, x_n}: \forall k \in \N^*_n: x_k \in S}$ Thus $S^n = \underbrace {S \times S \times \cdots \times S}_{\text{$n$ times} }$ Alternatively it can be defined recursively: :$S^n = \begin{cases} S: & n = 1 \\ S \times S^{n - 1} & n > 1 \end{cases}$ The set $S^n$ called a '''cartesian space'''. An element $x_j$ of an ordered tuple $\tuple {x_1, x_2, \ldots, x_n}$ of a '''cartesian space''' $S^n$ is known as a basis element of $S^n$.


Definition:Projection (Mapping Theory)/Family of Sets

Let $\family {S_i}_{i \mathop \in I}$ be a family of sets. Let $\displaystyle \prod_{i \mathop \in I} S_i$ be the Cartesian product of $\family {S_i}_{i \mathop \in I}$. For each $j \in I$, the '''$j$th projection on $\displaystyle S = \prod_{i \mathop \in I} S_i$''' is the mapping $\pr_j: S \to S_j$ defined by: :$\map {\pr_j} {\family {s_i}_{i \mathop \in I} } = s_j$ where $\family {s_i}_{i \mathop \in I}$ is an arbitrary element of $\displaystyle \prod_{i \mathop \in I} S_i$.


Definition:Seminorm

Let $\struct{K, +, \circ}$ be a division ring with norm $\norm{\,\cdot\,}_K$. Let $V$ be a vector space over $\struct{K, \norm{\,\cdot\,}_K}$, with zero $0_V$. A '''seminorm''' on $V$ is a map from $V$ to the positive reals $\norm{\cdot}: V \to \R_{\ge 0}$ satisfying the following properties (for all $x,y \in V$ and $\lambda \in K$): {| | align="left" | '''N2:''' | align="left" | Positive homogeneity: | align="left" | $\norm{\lambda x} = \norm{\lambda}_K \times \norm{x}$ |- | align="left" | '''N3:''' | align="left" | Triangle inequality: | align="left" | $\norm {x + y} \leq \norm{x} + \norm{y}$ |} {{refactor|Use Axiom template}} These may be referred to as the '''seminorm axioms'''. The '''N2''' and '''N3''' markings originate from the fact that these axioms are also used in defining norms.


Definition:P-Seminorm

Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Let $p \in \R$, $p \ge 1$. Let $\mathcal{L}^p \left({\mu}\right)$ be Lebesgue $p$-space for $\mu$. The '''$p$-seminorm''' on $\mathcal{L}^p \left({\mu}\right)$ is the mapping $\left\Vert{\cdot}\right\Vert_p : \mathcal{L}^p \left({\mu}\right) \to \R_{\ge 0}$ defined by: :$\displaystyle \forall f \in \mathcal{L}^p \left({\mu}\right): \left\Vert{f}\right\Vert_p := \left({\int \left\vert{f}\right\vert^p \, \mathrm d \mu}\right)^{1/p}$ That the '''$p$-seminorm''' is in fact a seminorm is proved on $p$-Seminorm is Seminorm.


Definition:Friedman Number

A '''Friedman number (base $n$)''' is a (positive) integer which is the result of an expression in base $n$ arithmetic which contains exactly its digits. The expression is subject to the following constraints: :$(1): \quad$ The arithmetic operators $+$, $-$, $\times$, $\div$ and exponentiation are the only operators which are allowed. :$(2): \quad$ Parentheses are allowed, but only in order to override the default operator precedence, otherwise every number would trivially be Friedman by $n = (n)$. :$(3): \quad$ Leading zeroes are not allowed, otherwise other numbers would trivially be Friedman by, for example, $011 = 10 + 1$.


Definition:Lebesgue Space/L-Infinity

The '''Lebesgue $\infty$-space for $\mu$''', denoted $\mathcal{L}^\infty \left({\mu}\right)$, is defined as: :$\displaystyle \mathcal{L}^\infty \left({\mu}\right) := \left\{{f \in \mathcal M \left({\Sigma}\right): \text{$f$ is a.e. bounded}}\right\}$ and so consists of all $\Sigma$-measurable $f: X \to \R$ that are almost everywhere bounded, that is, subject to: :$\exists c \in \R: \mu \left({\left\{{\left\vert{f}\right\vert > c}\right\}}\right) = 0$ {{refactor|The following can be split off into separate pages, as they need proving in their own right.}} $\mathcal{L}^\infty \left({\mu}\right)$ can be endowed with the supremum seminorm $\left\Vert{\cdot}\right\Vert_\infty$ by: :$\displaystyle \forall f \in \mathcal{L}^\infty \left({\mu}\right): \left\Vert{f}\right\Vert_\infty := \inf \, \left\{{c \ge 0: \mu \left({\left\{{\left\vert{f}\right\vert > c}\right\}}\right) = 0}\right\}$ If, subsequently, we introduce the equivalence $\sim$ by: :$f \sim g \iff \left\Vert{f - g}\right\Vert_\infty = 0$ we obtain the quotient space $L^\infty \left({\mu}\right) := \mathcal{L}^\infty \left({\mu}\right) / \sim$, which is also called '''Lebesgue $\infty$-space for $\mu$'''.


Definition:Product Sigma-Algebra

Let $\struct {X, \Sigma_1}$ and $\struct {Y, \Sigma_2}$ be measurable spaces. The '''product $\sigma$-algebra''' of $\Sigma_1$ and $\Sigma_2$ is denoted $\Sigma_1 \otimes \Sigma_2$, and defined as: :$\Sigma_1 \otimes \Sigma_2 := \map \sigma {\set {S_1 \times S_2: S_1 \in \Sigma_1 \land S_2 \in \Sigma_2} }$ where: :$\sigma$ denotes generated $\sigma$-algebra :$\times$ denotes Cartesian product. This '''product $\sigma$-algebra''' $\Sigma_1 \otimes \Sigma_2$ is a $\sigma$-algebra on $X \times Y$. === Product of Measurable Spaces === {{:Definition:Product of Measurable Spaces}}


Definition:Product of Measurable Spaces

The '''product of $\left({X, \Sigma_1}\right)$ and $\left({Y, \Sigma_2}\right)$''' is the measurable space: :$\left({X \times Y, \Sigma_1 \otimes \Sigma_2}\right)$


Definition:Product Measure

Let $\left({X, \Sigma_1, \mu}\right)$ and $\left({Y, \Sigma_2, \nu}\right)$ be $\sigma$-finite measure spaces. Let $\left({X \times Y, \Sigma_1 \otimes \Sigma_2}\right)$ be the product measurable space of $\left({X, \Sigma_1}\right)$ and $\left({Y, \Sigma_2}\right)$. The '''product measure of $\mu$ and $\nu$''', denoted $\mu \times \nu$, is the measure defined by: :$\forall E_1 \in \Sigma_1, E_2 \in \Sigma_2: \mu \times \nu \left({E_1 \times E_2}\right) = \mu \left({E_1}\right) \nu \left({E_2}\right)$ That this uniquely defines a measure on $\Sigma_1 \otimes \Sigma_2$ is shown on Uniqueness of Product Measures and Existence of Product Measures. === Product Measure Space === {{:Definition:Product Measure Space}}


Definition:Product Measure Space

The '''product of $\left({X, \Sigma_1, \mu}\right)$ and $\left({Y, \Sigma_2, \nu}\right)$''' is the measure space: :$\left({X \times Y, \Sigma_1 \otimes \Sigma_2, \mu \times \nu}\right)$


Definition:Survival Function

Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Let $f: X \to \overline{\R}$ be a positive $\Sigma$-measurable function. The '''survival function of $f$''' is the mapping $F: \R \to \R$ defined by: :$\forall t \in \R: F \left({t}\right) := \mu \left({\left\{{f \ge t}\right\}}\right)$ where $\left\{{f > t}\right\}$ is the set $\left\{{x \in X: f \left({x}\right) \ge t}\right\}$.


Definition:Characteristic Function (Set Theory)/Set

Let $E \subseteq S$. The '''characteristic function of $E$''' is the function $\chi_E: S \to \set {0, 1}$ defined as: :$\map {\chi_E} x = \begin {cases} 1 & : x \in E  \\  0 & : x \notin E \end {cases}$ That is: :$\map {\chi_E} x = \begin {cases} 1 & : x \in E  \\  0 & : x \in \relcomp S E \end {cases}$ where $\relcomp S E$ denotes the complement of $E$ relative to $S$.


Definition:Characteristic Function (Set Theory)/Relation

The concept of a characteristic function of a subset carries over directly to relations. Let $\RR \subseteq S \times T$ be a relation. The '''characteristic function of $\RR$''' is the function $\chi_\RR: S \times T \to \set {0, 1}$ defined as: :$\map {\chi_\RR} {x, y} = \begin {cases} 1 & : \tuple {x, y} \in \RR  \\  0 & : \tuple {x, y} \notin \RR \end{cases}$ It can be expressed in Iverson bracket notation as: :$\map {\chi_\RR} {x, y} = \sqbrk {\tuple {x, y} \in \RR}$ More generally, let $\displaystyle \mathbb S = \prod_{i \mathop = 1}^n S_i = S_1 \times S_2 \times \ldots \times S_n$ be the cartesian product of $n$ sets $S_1, S_2, \ldots, S_n$. Let $\RR \subseteq \mathbb S$ be an $n$-ary relation on $\mathbb S$. The '''characteristic function of $\RR$''' is the function $\chi_\RR: \mathbb S \to \set {0, 1}$ defined as: :$\map {\chi_\RR} {s_1, s_2, \ldots, s_n} = \begin {cases} 1 & : \tuple {s_1, s_2, \ldots, s_n} \in \RR  \\  0 & : \tuple {s_1, s_2, \ldots, s_n} \notin \RR \end {cases}$ It can be expressed in Iverson bracket notation as: :$\map {\chi_\RR} {s_1, s_2, \ldots, s_n} = \sqbrk {\tuple {s_1, s_2, \ldots, s_n} \in \RR}$


Definition:Zero (Cardinal)

The cardinal associated with the empty set $\O$ is called '''zero''', and is denoted $0$. More informally, this means that '''zero''' is defined as being '''the number of elements in the empty set'''.


Definition:One (Cardinal)

The cardinal associated with the singleton $\left\{{\varnothing}\right\}$ is called '''one''', and is denoted $1$. More informally, this means that '''one''' is defined as being '''the number of elements in a singleton set'''.


Definition:Product of Cardinals

Let $A$ and $B$ be sets. Let $\mathbf a$ and $\mathbf b$ be the cardinals associated respectively with $A$ and $B$. Then the '''product''' of $\mathbf a$ and $\mathbf b$ is defined as: :$\mathbf a \mathbf b := \operatorname{Card} \left({A \times B}\right)$ where: :$A \times B$ denotes the Cartesian product of $A$ and $B$ :$\operatorname{Card} \left({A \times B}\right)$ denotes the cardinal associated with $A \times B$.


Definition:Quasigroup/Right Quasigroup

$\left({S, \circ}\right)$ is a '''right quasigroup''' {{iff}}: :for all $a \in S$, the right regular representation $\rho_a$ is a permutations on $S$. That is: :$\forall a, b \in S: \exists ! x: x \circ a = b$


Definition:(1-3) Parastrophe

$\struct {S, *}$ is a '''$(1-3)$ parastrophe of $\struct {S, \circ}$''' {{iff}}: :$\forall x_1, x_2, x_3 \in S: x_1 \circ x_2 = x_3 \iff x_3 * x_2 = x_1$


Definition:Quasigroup/Left Quasigroup

$\left({S, \circ}\right)$ is a '''left quasigroup''' {{iff}}: :for all $a \in S$, the left regular representation $\lambda_a$ is a permutations on $S$. That is: :$\forall a, b \in S: \exists ! x: a \circ x = b$


Definition:(2-3) Parastrophe

$\struct {S, *}$ is a '''$(2-3)$ parastrophe of $\struct {S, \circ}$''' {{iff}}: :$\forall x_1, x_2, x_3 \in S: x_1 \circ x_2 = x_3 \iff x_1 * x_3 = x_2$


Definition:B-Algebra

Let $\struct {X, \circ}$ be an algebraic structure. Then $\struct {X, \circ}$ is a '''$B$-algebra''' {{iff}}: {{begin-axiom}} {{axiom | n = \text {AC}         | q = \forall x, y \in X         | m = x \circ y \in X }} {{axiom | n = \text A 0         | m = \exists 0 \in X }} {{axiom | n = \text A 1         | q = \forall x \in X         | m = x \circ x = 0 }} {{axiom | n = \text A 2         | q = \forall x \in X         | m = x \circ 0 = x }} {{axiom | n = \text A 3         | q = \forall x, y, z \in X         | m = \paren {x \circ y} \circ z = x \circ \paren {z \circ \paren {0 \circ y} } }} {{end-axiom}}


Definition:Power (B-Algebra)

Let $\struct {X, \circ}$ be a $B$-algebra. For any $x \in X$ and $n \in \N$, define the '''$n$th power of $x$''', denoted $x^n$, inductively: :$x^n = \begin{cases} 0 & \text {if $n = 0$} \\ x^{n - 1} \circ \paren {0 \circ x} & \text {if $n \ge 1$} \end{cases}$


Definition:Commutative B-Algebra

Let $\struct {X, \circ}$ be a $B$-algebra. Then $\struct {X, \circ}$ is said to be '''$0$-commutative''' (or just '''commutative''') {{iff}}: :$\forall x, y \in X: x \circ (0 \circ y) = y \circ (0 \circ x)$


Definition:Sum of Cardinals

Let $A$ and $B$ be sets. Let $\mathbf a$ and $\mathbf b$ be the cardinals associated with $A$ and $B$ respectively. Then the '''sum of $\mathbf a$ and $\mathbf b$''' is defined as: :$\mathbf a + \mathbf b = \operatorname{Card} \left({A \sqcup B}\right)$ where: :$A \sqcup B$ denotes the disjoint union of $A$ and $B$ :$\operatorname{Card} \left({A \sqcup B}\right)$ denotes the cardinal associated with $A \cup B$.


Definition:Self-Inverse Element

Let $\struct {S, \circ}$ be a monoid whose identity element is $e$. Let $x \in S$ be an element of $S$. === Definition 1 === {{:Definition:Self-Inverse Element/Definition 1}} === Definition 2 === {{:Definition:Self-Inverse Element/Definition 2}}


Definition:Finite Cardinal

Let $\mathbf a$ be a cardinal. Then $\mathbf a$ is described as '''finite''' {{iff}}: :$\mathbf a < \mathbf a + \mathbf 1$ where $\mathbf 1$ is (cardinal) one. That is, such that $\mathbf a \ne \mathbf a + \mathbf 1$.


Definition:Infinite Cardinal

Let $\mathbf a$ be a cardinal. Then $\mathbf a$ is described as '''infinite''' {{iff}}: :$\mathbf a = \mathbf a + \mathbf 1$ where $\mathbf 1$ is (cardinal) one.


Definition:Convolution of Measurable Functions

Let $f, g: \R^n \to \R$ be $\mathcal B^n$-measurable functions such that for all $x \in \R^n$: :$\displaystyle \int_{\R^n} f \left({x - y}\right) g \left({y}\right) \, \mathrm d \lambda^n \left({y}\right)$ is finite. The '''convolution of $f$ and $g$''', denoted $f * g$, is the mapping defined by: :$\displaystyle f * g: \R^n \to \R, f * g \left({x}\right) := \int_{\R^n} f \left({x - y}\right) g \left({y}\right) \, \mathrm d \lambda^n \left({y}\right)$


Definition:Convolution of Measurable Function and Measure

Let $\mu$ be a measure on the Borel $\sigma$-algebra $\mathcal B^n$ on $\R^n$. Let $f: \R^n \to \R$ be a $\mathcal B^n$-measurable function such that for all $x \in \R^n$: :$\displaystyle \int_{\R^n} f \left({x - y}\right) \, \mathrm d \mu \left({y}\right)$ is finite. The '''convolution of $f$ and $\mu$''', denoted $f * \mu$, is the mapping defined by: :$\displaystyle f * \mu: \R^n \to \R, f * \mu \left({x}\right) := \int_{\R^n} f \left({x - y}\right) \, \mathrm d \mu \left({y}\right)$


Definition:Convolution of Measures

Let $\mu$ and $\nu$ be measures on the Borel $\sigma$-algebra $\mathcal B^n$ on $\R^n$. The '''convolution of $\mu$ and $\nu$''', denoted $\mu * \nu$, is the measure defined by: :$\displaystyle \mu * \nu: \mathcal B^n \to \overline \R, \mu * \nu \left({B}\right) := \int_{\R^n} \chi_B \left({x + y}\right) \, \mathrm d \mu \left({x}\right) \, \mathrm d \nu \left({y}\right)$ where $\chi_B$ is the characteristic function of $B$.


Definition:Group Direct Product/General Definition

Let $\family {\struct {G_i, \circ_i} }_{i \mathop \in I}$ be a family of groups. Let $\displaystyle G = \prod_{i \mathop \in I} G_i$ be their cartesian product. Let $\circ$ be the operation defined on $G$ as: :$\circ := \family {g_i}_{i \mathop \in I} \circ \family {h_i}_{i \mathop \in I} = \family {g_i \circ_i h_i}_{i \mathop \in I}$ for all sequences in $G$. The group $\struct {G, \circ}$  is called the '''(external) direct product''' of $\family {\struct {G_i, \circ_i} }_{i \mathop \in I}$.


Definition:Hölder Continuous

Let $\left({M_1, d_1}\right)$ and $\left({M_2, d_2}\right)$ be metric spaces. Let $\alpha \in \R_{\ge 0}$ be a positive real number. A mapping $f: M_1 \to M_2$ is said to be '''$\alpha$-Hölder continuous''' {{iff}}: :$\exists L \in \R_{\ge 0}: \forall x, y \in M_1: d_2 \left({f \left({x}\right), f \left({y}\right)}\right) \le L \left({d_1 \left({x, y}\right)}\right)^\alpha$ Further, $f$ is said to be '''Hölder continuous''' {{iff}} it is '''$\alpha$-Hölder continuous''' for some $\alpha \in \R_{\ge 0}$.


Definition:Permutation/Ordered Selection

Let $S$ be a set of $n$ elements. Let $r \in \N: r \le n$. An '''$r$-permutation of $S$''' is an ordered selection of $r$ elements of $S$.


Definition:Subsequential Limit

Let $\sequence {x_n}$ be a sequence. Let $\sequence {x_{n_r} }$ be a subsequence of $\sequence {x_n}$. Suppose that $\sequence {x_{n_r} }$ converges to a limit $x$. Then $x$ is called a '''subsequential limit''' of $\sequence {x_n}$.


Definition:Ordinal Function

Let $A$ be an ordinal (we shall allow $A$ to be a proper class). Then the mapping $G : A \to \operatorname{On}$ is called an ''ordinal function''.


Definition:Power of Element/Monoid/Invertible Element

Let $b \in S$ be invertible for $\circ$. Let $n \in \Z$. The definition $b^n = \map {\circ^n} b$ as the $n$th power of $b$ in $\left({S, \circ}\right)$ can be extended to include the inverse of $b$: :$b^{-n} = \paren {b^{-1} }^n$


Definition:Power of Element/Monoid

Let $\struct {S, \circ}$ be a monoid whose identity element is $e$. Let $a \in S$. Let $n \in \N$. The definition $a^n = \map {\circ^n} a$ as the $n$th power of $a$ in a semigroup can be extended to allow an exponent of $0$: :$a^n = \begin {cases} e & : n = 0 \\ a^{n - 1} \circ a & : n > 0 \end{cases}$ or :$n \cdot a = \begin {cases} e & : n = 0 \\ \paren {n - 1} a \circ a & : n > 0 \end{cases}$ The validity of this definition follows from the fact that a monoid has an identity element. === Invertible Element === {{:Definition:Power of Element/Monoid/Invertible Element}}


Definition:Power of Element/Group

Let $\struct {G, \circ}$ be a group whose identity element is $e$. Let $g \in G$. Let $n \in \Z$. The definition $g^n = \map {\circ^n} g$ as the $n$th power of $g$ in a monoid can be extended to allow negative values of $n$: :$g^n = \begin{cases} e & : n = 0 \\ g^{n - 1} \circ g & : n > 0 \\ \paren {g^{-n} }^{-1} & : n < 0 \end{cases}$ or :$n \cdot g = \begin{cases} e & : n = 0 \\ \paren {\paren {n - 1} \cdot g} \circ g & : n > 0 \\ -\paren {-n \cdot g} & : n < 0 \end{cases}$ The validity of this definition follows from the group axioms: $g$ has an inverse element.


Definition:Canonical Order

Let $\operatorname{On}$ be the ordinal class. Let $<$ be the ordinal ordering. Let $\operatorname{Le}$ be the lexicographic ordering on $\operatorname{On} \times \operatorname{On}$. The '''canonical order''' on $\operatorname{On} \times \operatorname{On}$, denoted $R_0$, is defined as follows, for ordinals $\alpha, \beta, \gamma, \delta$: ::$\left({\alpha, \beta}\right) \mathrel{R_0} \left({\gamma, \delta}\right)$ :{{iff}}: ::$\max \left({\alpha, \beta}\right) < \max \left({\gamma, \delta}\right)$ or $\left({\max \left({ \alpha, \beta}\right) = \max \left({\gamma, \delta}\right) \land \left({\alpha, \beta}\right) \mathrel{\operatorname{Le}} \left({\gamma, \delta}\right)}\right)$


Definition:Ordinal Addition

Let $x$ and $y$ be ordinals. The operation of '''ordinal addition''' $x + y$ is defined using transfinite recursion on $y$, as follows. === Base Case === When $y = \varnothing$, define: :$x + \varnothing := x$ === Inductive Case === For a successor ordinal $y^+$, define: :$x + y^+ := \left({x + y}\right)^+$ === Limit Case === Let $y$ be a limit ordinal. Then: :$\displaystyle x + y := \bigcup_{z \mathop \in y} \left({x + z}\right)$


Definition:Relational Loop

Let $\RR$ be a relation on a set $S$. Let $a_1, a_2, \ldots a_n$ be elements of $S$. A '''relational loop on $S$''' takes the form: :$\tuple {a_1 \mathrel \RR a_2 \land a_2 \mathrel \RR a_3 \dots \land a_{n - 1} \mathrel \RR a_n \land a_n \mathrel \RR a_1}$ That is, it is a subset of $\RR$ of the form: :$\set {\tuple {a_1, a_2}, \tuple {a_2, a_3}, \ldots, \tuple {a_{n - 1}, a_n}, \tuple {a_n, a_1} }$


Definition:Real Matrix

A '''real matrix''' is a matrix whose elements are all real numbers.


Definition:Smooth Homotopy

Let $X$ and $Y$ be topological spaces. Let $f: X \to Y$, $g: X \to Y$ be smooth mappings. Then $f$ and $g$ are '''smoothly homotopic''' {{iff}} there exists a smooth mapping: : $H: X \times \left[{0 \,.\,.\, 1}\right] \to Y$ such that: : $H \left({x, 0}\right) = f \left({x}\right)$ and: : $H \left({x, 1}\right) = g \left({x}\right)$ $H$ is called a '''smooth homotopy between $f$ and $g$'''.


Definition:Basic Primitive Recursive Function/Zero Function

The '''zero function''' $\Zero: \N \to \N$ is a basic primitive recursive function, defined as: :$\forall n \in \N: \map \Zero n = 0$


Definition:Basic Primitive Recursive Function/Successor Function

The '''successor function''' $\Succ: \N \to \N$ is a basic primitive recursive function, defined as: :$\forall n \in \N: \map \Succ n = n + 1$


Definition:Basic Primitive Recursive Function/Projection Function

The '''projection functions''' $\pr_j^k: \N^k \to \N$ are basic primitive recursive functions, defined as: :$\forall \tuple {n_1, n_2, \ldots, n_k} \in \N^k: \map {\pr_j^k} {n_1, n_2, \ldots, n_k} = n_j$<ref>The usual notation for the projection function omits the superscript that defines the arity of the particular instance of the projection in question at the time, for example: $\pr_j$. However, in the context of computability theory, it is a ''very good idea'' to be ''completely certain'' of ''exactly'' which projection function is under discussion.</ref> where $j \in \closedint 1 k$.


Definition:Basic Primitive Recursive Function/Identity Function

The identity function $I_\N: \N \to \N$ is a basic primitive recursive function, defined as: :$\forall n \in \N: \map {I_\N} n = n$ Note that this is an implementation of the projection function: :$\pr_1^1: \N \to \N: \map {\pr_1^1} {n_1} = n_1$


Definition:Basic Primitive Recursive Function/URM Computability

The basic primitive recursive functions: * Zero Function * Successor Function * Projection Function * Identity Function are each URM computable by a single-instruction URM program.


Definition:Primitive Recursive/Set

Let $A \subseteq \N$. Then $A$ is a '''primitive recursive set''' iff its characteristic function $\chi_A$ is a primitive recursive function.


Definition:Primitive Recursive/Relation

Let $\mathcal R \subseteq \N^k$ be an $n$-ary relation on $\N^k$. Then $\mathcal R$ is a '''primitive recursive relation''' {{iff}} its characteristic function $\chi_\mathcal R$ is a primitive recursive function.


Definition:Primitive Recursive/Function

A function is '''primitive recursive''' if it can be obtained from basic primitive recursive functions using the operations of substitution and primitive recursion a finite number of times.


Definition:Primitive Recursion/One Variable

Let $a \in \N$ be a natural number. Let $g: \N^2 \to \N$ be a function. Then the function $h: \N \to \N$ is '''obtained from the constant $a$ and $g$ by primitive recursion''' {{iff}}: :$\forall n \in \N: \map h n = \begin {cases} a & : n = 0 \\ \map g {n - 1, \map h {n - 1} } & : n > 0  \end{cases}$


Definition:Primitive Recursion/Partial Function

Let $f: \N^k \to \N$ and $g: \N^{k+2} \to \N$ be partial functions. Let $\tuple {n_1, n_2, \ldots, n_k} \in \N^k$. Then the partial function $h: \N^{k + 1} \to \N$ is '''obtained from $f$ and $g$ by primitive recursion''' {{iff}}: :$\forall n \in \N: \map h {n_1, n_2, \ldots, n_k, n} \approx \begin {cases} \map f {n_1, n_2, \ldots, n_k} & : n = 0 \\ \map g {n_1, n_2, \ldots, n_k, n - 1, \map h {n_1, n_2, \ldots, n_k, n - 1} } & : n > 0  \end{cases}$ where $\approx$ is as defined in Partial Function Equality. Note that $\map h {n_1, n_2, \ldots, n_k, n}$ is defined only when: :$\map h {n_1, n_2, \ldots, n_k, n - 1}$ is defined :$\map g {n_1, n_2, \ldots, n_k, n - 1, \map h {n_1, n_2, \ldots, n_k, n - 1} }$ is defined.


Definition:Primitive Recursion/Several Variables

Let $f: \N^k \to \N$ and $g: \N^{k + 2} \to \N$ be functions. Let $\tuple {n_1, n_2, \ldots, n_k} \in \N^k$. Then the function $h: \N^{k + 1} \to \N$ is '''obtained from $f$ and $g$ by primitive recursion''' {{iff}}: :$\forall n \in \N: \map h {n_1, n_2, \ldots, n_k, n} = \begin {cases} \map f {n_1, n_2, \ldots, n_k} & : n = 0 \\ \map g {n_1, n_2, \ldots, n_k, n - 1, \map h {n_1, n_2, \ldots, n_k, n - 1} } & : n > 0  \end {cases}$


Definition:Minimization/Partial Function

Let $f: \N^{k+1} \to \N$ be a partial function. Let $n = \left({n_1, n_2, \ldots, n_k}\right) \in \N^k$ be fixed. Then the '''minimization operation on $f$''' is written as: :$\mu y \left({f \left({n, y}\right) = 0}\right)$ and is specified as follows: :$\mu y \left({f \left({n, y}\right) = 0}\right) = \begin{cases} z & : f \left({n, z}\right) = 0 \text { and } f \left({n, y}\right) \text{ defined and } \forall y: 0 \le y < z: f \left({n, y}\right) \ne 0 \\ \text{undefined} & : \text{otherwise} \end{cases}$ The partial function: :$g \left({n}\right) \approx \mu y \left({f \left({n, y}\right) = 0}\right)$ obtained in this way (see Partial Function Equality) is said to be obtained from $f$ '''by minimization'''. ==== Note ==== It is not enough for there to exist $z$ such that $f \left({n, z}\right) = 0$. We need to insist that $f \left({n, y}\right)$ is actually defined for all $y$ less than $z$. Otherwise, if we were to try and find $z$ by the recursive technique of trying all $z$ from $0$ up, we would never actually get up as far as $z$ because the undefined value of $f$ for some $y$ is getting in the way. In the context of URM programs, this is significant, as an undefined output from a function is determined by a non-terminating program.


Definition:Recursive/Function

A function is '''recursive''' if it can be obtained from basic primitive recursive functions using the operations of: : substitution : primitive recursion, and  : minimization on a function a finite number of times.


Definition:Recursive/Set

Let $A \subseteq \N$. Then $A$ is a '''recursive set''' iff its characteristic function $\chi_A$ is a recursive function.


Definition:Recursive/Relation

Let $\mathcal R \subseteq \N^k$ be an $n$-ary relation on $\N^k$. Then $\mathcal R$ is a '''recursive relation''' {{iff}} its characteristic function $\chi_\mathcal R$ is a recursive function.


Definition:Special Linear Group

Let $R$ be a commutative ring with unity whose zero is $0$ and unity is $1$. The '''special linear group of order $n$ on $R$''' is the set of square matrices of order $n$ whose determinant is $1$. It is a group under (conventional) matrix multiplication.


Definition:Antiassociative Operation

Let $\circ$ be a binary operation on the set $S$. $\circ$ is an '''antiassociative operation''' on $S$ {{iff}}: :$\forall x, y, z \in S: \left({x \circ y}\right)\circ z \ne x \circ \left({y \circ z}\right)$


Definition:Antiassociative Structure

Let $\left({S, \circ}\right)$ be an algebraic strcuture. Then $\left({S, \circ}\right)$ is an '''antiassociative structure''' {{iff}} $\circ$ is an antiassociative operation. That is, {{iff}}: :$\forall x, y, z \in S: \left({x \circ y}\right)\circ z \ne x \circ \left({y \circ z}\right)$


Definition:Hadamard Product

Let $\struct {S, \cdot}$ be an algebraic structure. Let $\mathbf A = \sqbrk a_{m n}$ be an $m \times n$ matrix over $S$. Let $\mathbf B = \sqbrk b_{m n}$ be an $m \times n$ matrix over $S$. The '''Hadamard product of $\mathbf A$ and $\mathbf B$''' is written $\mathbf A \circ \mathbf B$ and is defined as follows: :$\mathbf A \circ \mathbf B := \mathbf C = \sqbrk c_{m n}$ where: :$\forall i \in \closedint 1 m, j \in \closedint 1 n: c_{i j} = a_{i j} \cdot_R b_{i j}$


Definition:Magma of Sets

Let $X$ be a set. Let $\SS \subseteq \powerset X$ be a set of subsets of $X$. Let $I$ be an index set. For every $i \in I$, let $J_i$ be an index set, and let: :$\phi_i: \powerset X^{J_i} \to \powerset X$ be a partial mapping. Then $\SS$ is a '''magma of sets for $\set {\phi_i: i \in I}$ on $X$''' {{iff}}: :$\forall i \in I: \map {\phi_i} {\family {S_j}_{j \mathop \in J_i} } \in \SS$ for every indexed family $\family {S_j}_{j \mathop \in J_i} \in \SS^{J_i}$ in the domain of $\phi$.


Definition:Magma of Sets Generated by Collection of Subsets

Let $X$ be a set, and let $\Phi := \left\{{\phi_i: i \in I}\right\}$ be a collection of partial mappings with codomain $\mathcal P \left({X}\right)$, the power set of $X$. Let $\mathcal G \subseteq \mathcal P \left({X}\right)$ be a collection of subsets of $X$. Then the '''magma of sets for $\Phi$ generated by $\mathcal G$''' is the unique magma of sets $\mathcal S \subseteq \mathcal P \left({X}\right)$ satisfying: :$(1): \quad \mathcal G \subseteq \mathcal S$ :$(2): \quad \mathcal G \subseteq \mathcal T$ implies that $\mathcal S \subseteq \mathcal T$ for every magma of sets $\mathcal T$ To speak of ''the unique'' '''magma of sets generated by $\mathcal G$''' is justified by Existence and Uniqueness of Magma of Sets Generated by Collection of Subsets.


Definition:Cantor Set/Limit of Decreasing Sequence

Let $\map {I_c} \R$ denote the set of all closed real intervals. Define the mapping $t_1: \map {I_c} \R \to \map {I_c} \R$ by: :$\map {t_1} {\closedint a b} := \closedint a {\dfrac 1 3 \paren {a + b} }$ and similarly $t_3: \map {I_c} \R \to \map {I_c} \R$ by: :$\map {t_3} {\closedint a b} := \closedint {\dfrac 2 3 \paren {a + b} } b$ Note in particular how: :$\map {t_1} {\closedint a b} \subseteq \closedint a b$ :$\map {t_3} {\closedint a b} \subseteq \closedint a b$ Subsequently, define inductively: :$S_0 := \set {\closedint 0 1}$ :$S_{n + 1} := \map {t_1} {C_n} \cup \map {t_3} {C_n}$ and put, for all $n \in \N$: :$C_n := \displaystyle \bigcup S_n$ Note that $C_{n + 1} \subseteq C_n$ for all $n \in \N$, so that this forms a decreasing sequence of sets. Then the '''Cantor set''' $\mathcal C$ is defined as its limit, that is: :$\mathcal C := \displaystyle \bigcap_{n \mathop \in \N} C_n$


Definition:Ordinal Subtraction

Let $x$ and $y$ be ordinals such that $x \le y$. Then the operation of '''ordinal subtraction''' is defined as: :$\displaystyle y - x = \bigcup \left\{{z: x + z = y}\right\}$


Definition:Ordinal Multiplication

Let $x$ and $y$ be ordinals. The operation of '''ordinal multiplication''' $x \times y$ is defined using transfinite recursion as follows: === Base Case === {{begin-eqn}} {{eqn | l = \left({x \times \varnothing}\right)       | r = \varnothing       | c = if $y$ is equal to $\varnothing$ }} {{end-eqn}} === Inductive Case === {{begin-eqn}} {{eqn | l = \left({x \times z^+}\right)       | r = \left({x \times z}\right) + x       | c = if $y$ is the successor of some ordinal $z$ }} {{end-eqn}} === Limit Case === {{begin-eqn}} {{eqn | l = \left({x \times y}\right)       | r = \bigcup_{z \mathop \in y} \left({x \times z}\right)       | c = if $y$ is a limit ordinal }} {{end-eqn}}


Definition:Ordinal Exponentiation

Let $x$ and $y$ be ordinals. '''Ordinal exponentiation''' $x^y$ is defined using Transfinite Recursion: :$\displaystyle x^y = \begin{cases}  0 & : x = 0, \ y \ne 0 \\  & \\ 1 & : x = 0, \ y = 0 \\  & \\ 1 & : x \ne 0, \ y = 0 \\  & \\ \left({x^z \cdot x}\right) & : x \ne 0, \ y = z^+ \\  & \\ \bigcup_{z \mathop \in y} x^z & : x \ne 0, \ y \in K_{II} \\ \end{cases}$ where: * $K_{II}$ is the class of all limit ordinals * $0$ denotes the zero ordinal * $1$ denotes the ordinal $1$, that is: $0^+$, the successor of $0$.


Definition:Ordinal Sum

Let $y$ be an ordinal. Let $\left\langle{A_x}\right\rangle$ be a sequence of ordinals. The '''ordinal sum of $A_x$''' is denoted $\displaystyle \sum_{x \mathop = 1}^y A_x$ and defined using Transfinite Recursion on $y$ as follows: :$\displaystyle \sum_{x \mathop = 1}^\varnothing A_x = \varnothing$ :$\displaystyle \sum_{x \mathop = 1}^{z^+} A_x = \sum_{x \mathop = 1}^z \left({A_x}\right) + A_{z^+}$ :$\displaystyle \sum_{x \mathop = 1}^y A_x = \bigcup_{z \mathop \in y} \left({\sum_{x \mathop = 1}^z A_x}\right)$ for limit ordinals $y$. {{NoSources}} Category:Definitions/Ordinal Arithmetic 8t1l44d9ybtngfihv3ohh31hb978bjt


Definition:Anticommutative/Structure with One Operation

Let $\left({S, \circ}\right)$ be an algebraic structure Then $\circ$ is '''anticommutative on $S$''' {{iff}}: :$\forall x, y \in S: x \circ y = y \circ x \iff x = y$ Equivalently, it can be defined as: :$\forall x, y \in S: x \ne y \iff x \circ y \ne y \circ x$


Definition:Anticommutative/Structure with Two Operations

Let $\left({S, +, \circ}\right)$ be an algebraic structure. Suppose every element $x$ in $\left({S, +}\right)$ has an inverse element $-x$. Then $\circ$ is '''anticommutative on $S$ with respect to $+$''' {{iff}}: :$\forall x, y \in S: x \circ y = -\left({y \circ x}\right)$


Definition:Middle Cancellation Property

Let $\left({S, \circ}\right)$ be a semigroup. Then $\circ$ is said to have the '''middle cancellation property''' iff: :$a \circ x \circ b = c \circ x \circ d \implies a \circ b = c \circ d$ holds for all $a,b,c,d,x \in S$. Category:Definitions/Semigroups rx7p8mvlzf6qo6km7e4d8vpdy9ntb9h


Definition:Cross Cancellation Property

Let $\left({S, \circ}\right)$ be an algebraic structure. Then $\circ$ is said to have the '''cross cancellation property''' iff: :$a \circ b = c \circ a \implies b = c$ holds for all $a, b, c \in S$. Category:Definitions/Abstract Algebra t0eoxvczi15juggfax1qedcpu11zi0p


Definition:Mutatis Mutandis

'''Mutatis mutandis''' is Latin for '''what needs changing having been changed'''. It can be used in place of words like ''analogously'' and ''similarly'', and expresses that an argument can be applied to a new situation in an intuitive fashion. For example the proof of Union is Associative can be made into that for Intersection is Associative by '''mutatis mutandis'''. In this case, this comes down to replacing $\cup$ by $\cap$, and $\lor$ by $\land$. In particular, the essential flow of the argument was not changed in adapting the proof.


Definition:Uniformly Integrable

Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Let $\mathcal F \subseteq \mathcal M \left({\Sigma}\right)$ be a collection of measurable functions. Then $\mathcal F$ is said to be '''uniformly integrable (with respect to $\mu$)''' iff: :$\forall \epsilon > 0: \exists g_\epsilon \in \mathcal{L}^1_+ \left({\mu}\right): \displaystyle \sup_{f \mathop \in \mathcal F} \int_{\left\{{\left\vert{f}\right\vert > g_\epsilon}\right\}} \left\vert{f}\right\vert \, \mathrm d \mu < \epsilon$ where: :$\mathcal{L}^1_+ \left({\mu}\right)$ is the space of positive $\mu$-integrable functions :$\left\{{\left\vert{f}\right\vert > g_\epsilon}\right\}$ is short for $\left\{{x \in X: \left\vert{f \left({x}\right)}\right\vert > g_\epsilon \left({x}\right)}\right\}$


Definition:Structure (Set Theory)

Let $A$ be a class. Let $\mathcal R$ be a relation. The '''relational structure $\left[{A, \mathcal R}\right]$ satisfies well-formed formula $p$''', denoted $\left[{A, \mathcal R}\right] \models p$, shall be defined on the well-formed parts of $p$: {{begin-eqn}} {{eqn | l = \left[{A, \mathcal R}\right] \models x \in y       | o = \iff       | r = \left({x \in A \land y \in A \land x \mathrel {\mathcal R} y}\right) }} {{eqn | l = \left[{A, \mathcal R}\right] \models \neg p       | o = \iff       | r = \neg \left[{A, \mathcal R}\right] \models p }} {{eqn | l = \left[{A, \mathcal R}\right] \models \left({p \land q}\right)       | o = \iff       | r = \left({\left[{A, \mathcal R}\right] \models p \land \left[{A, \mathcal R}\right] \models q}\right) }} {{eqn | l = \left[{A, \mathcal R}\right] \models \forall x: P \left({x}\right)       | o = \iff       | r = \forall x \in A: \left[{A, \mathcal R}\right] \models P \left({x}\right) }} {{end-eqn}}


Definition:Standard Structure

The structure $\left[{A,R}\right]$ is called a ''standard structure'' iff: :$\displaystyle R = \Epsilon \cap \left({ A \times A }\right)$ where $\Epsilon$ denotes the epsilon relation and $\times$ denotes the Cartesian product. With a standard structure, $\left[{A,R}\right] \models p$ shall be abbreviated: :$\displaystyle A \models p \iff \left[{A , E \cap \left({ A \times A }\right)}\right] \models p$


Definition:Relativisation

Let $p$ be a well-formed formula of the language of set theory. Let $A$ be a class. The '''relativisation of $p$ to $A$''' shall be denoted $p^A$ and shall be defined recursively on the symbols in $p$: :$x \in y ^A \iff x \in y$ :$\left({\neg p}\right)^A \iff \neg p^A$ :$\left({p \land q}\right)^A \iff \left({p^A \land q^A}\right)$ :$\left({\forall x: P \left({x}\right)}\right)^A \iff \forall x: \left({ x \in A \implies P \left({x}\right)^A}\right)$ Thus, the relativisation of $p$ is simply the well-formed formula achieved when replacing all instances of $\forall x$ with $\forall x \in A$.


Definition:Restricted Universal Quantifier

Let $A$ be a class in ZF. The '''restricted universal quantifier''' is denoted $\forall x \in A$ and is defined as the following definitional abbreviation: :$\forall x \in A: P \left({x}\right) \quad \text{for} \quad \forall x: \left({x \in A \implies P \left({x}\right)}\right)$ where $P \left({x}\right)$ is any well-formed formula of the language of set theory. Category:Definitions/Predicate Logic Category:Definitions/Zermelo-Fraenkel Class Theory flsam9ljgwf1h557uj8ohqeqyhkdcua


Definition:Restricted Existential Quantifier

Let $A$ be a class in ZF. The '''restricted existential quantifier''' is denoted $\exists x \in A$ and is defined as the following definitional abbreviation: :$\exists x \in A: P \left({x}\right) \quad \text{for} \quad \exists x: \left({x \in A \land P \left({x}\right)}\right)$ where $P \left({x}\right)$ is any well-formed formula of the language of set theory. Category:Definitions/Predicate Logic Category:Definitions/Zermelo-Fraenkel Class Theory re80hztvvvqlai1nkih1dwbwjkeauzu


Definition:Language of Set Theory

{{refactor}} The language of set theory consists of the language of predicate logic with the binary predicate symbol $\in$, denoting membership. === Predicate Symbols === The language of set theory uses only one predicate symbol, $\in$, the membership sign. It is a binary predicate symbol. Using $\in$, other symbols such as $=$ can be defined (see Definition:Set Equality). === Connectives === The language of set theory borrows the connectives from the language of predicate logic.  However, some of the connectives can be considered definitional abbreviations. $\land$ and $\neg$ can be taken as primitive connectives with the following definitional abbreviations: {{begin-eqn}} {{eqn | l = P \implies Q       | o = \operatorname{for}       | r = \neg ( P \land \neg Q )       | c =  }} {{eqn | l = P \lor Q       | o = \operatorname{for}       | r = \neg ( \neg P \land \neg Q )       | c =  }} {{eqn | l = P \iff Q       | o = \operatorname{for}       | r = ( ( P \implies Q ) \land ( Q \implies P ) )       | c = justified because $\implies$ is already a definitional abbreviation }} {{end-eqn}} $\implies$ and $\neg$ can be taken as primitive connectives with the following definitional abbreviations: {{begin-eqn}} {{eqn | l = ( P \land Q )       | o = \operatorname{for}       | r = \neg ( P \implies \neg Q )       | c =  }} {{eqn | l = ( P \lor Q )       | o = \operatorname{for}       | r = ( \neg P \implies Q )       | c =  }} {{eqn | l = ( P \iff Q )       | o = \operatorname{for}       | r = \neg ( ( P \implies Q ) \implies \neg ( Q \implies P ) )       | c =  }} {{end-eqn}} The Sheffer stroke $\mid$ can be taken as a sole primitive connective with the following definitional abbreviations: {{begin-eqn}} {{eqn | l = \neg P       | o = \operatorname{for}       | r = ( P \mid P )       | c =  }} {{eqn | l = ( P \implies Q )       | o = \operatorname{for}       | r = ( P \mid ( Q \mid Q ) )       | c =  }} {{end-eqn}} The other connectives can be defined using $\neg P$ and $( P \implies Q )$ as the "new" primitive connectives. === Quantifiers === The language of set theory adopts the same quantifiers as those in the language of predicate logic. However, only $\forall$ is necessary to adopt as a primitive symbol, and $\exists$ can be defined: :$\exists x: P(x) \operatorname{for} \neg \forall x: \neg P(x)$


Definition:Russell Class

The '''Russell class''', denoted $\operatorname{Ru}$, equals the class of all sets $x$ that are not elements of themselves: :$\operatorname{Ru} = \set {x: x \notin x}$


Definition:Class Equality

Let $A$ and $B$ be classes. === Definition 1 === {{:Definition:Class Equality/Definition 1}} === Definition 2 === {{:Definition:Class Equality/Definition 2}}


Definition:Class/Zermelo-Fraenkel

A '''class''' in $\textrm{ZF}$ is a formal vehicle capturing the intuitive notion of a class, namely a collection of all sets such that a particular condition $P$ holds. In $\textrm{ZF}$, '''classes''' are written using class builder notation: :$\set {x : \map P x}$ {{Questionable|Wouldn't you need the Class Comprehension Axiom for this?}} where $\map P x$ is a statement containing $x$ as a free variable.   More formally, a '''class''' $\set {x: \map P x}$ serves to define the following definitional abbreviations involving the membership symbol: {{begin-eqn}} {{eqn | l = y \in \set {x: \map P x}       | o = \quad \text{for} \quad       | r = \map P y }} {{eqn | l = \set {x: \map P x} \in y       | o = \quad \text{for} \quad       | r = \exists z \in y: \forall x: \paren {x \in z \iff \map P x} }} {{eqn | l = \set {x: \map P x} \in \set {y: \map Q y}       | o = \quad \text{for} \quad       | r = \exists z: \paren {\map Q z \land \forall x: \paren {x \in z \iff \map P x} } }} {{end-eqn}} where: :$x, y ,z$ are variables of $\textrm{ZF}$ :$P, Q$ are propositional functions. Through these "rules", every statement involving $\set {x: \map P x}$ can be reduced to a simpler statement involving only the basic language of set theory.


Definition:Small Class

Let $A$ denote an arbitrary class. Then $A$ is said to be '''small''' {{iff}}: :$\exists x: x = A$ where $=$ denotes class equality and $x$ is a set variable. That is, a class is '''small''' {{iff}} it is equal to some set variable. To denote that a class $A$ is '''small''', the notation $\mathcal M \left({A}\right)$ may be used. Thus: : $\mathcal M \left({A}\right) \iff \exists x: x = A$


Definition:Set Variable

In the language of set theory, '''set variables''' are the variables. '''Set variables''' are sometimes referred to as sets. {{NoSources}} Category:Definitions/Set Theory ppbnk9rv8ysy1yvinsaiakf1mtaqdgs


Definition:Unordered Tuple

An '''unordered tuple''' is a small class in Zermelo-Fraenkel set theory that contains a finite number of elements. It is written using roster notation and is denoted: :$\set {a_1, a_2, \ldots, a_n}$ that is: :$\set {a_1, a_2, \ldots, a_n} = \set {x: x = a_1 \lor x = a_2 \lor \dots \lor x = a_n}$


Definition:Category of Sets

The '''category of sets''', denoted $\mathbf{Set}$ or $\mathbf{Sets}$ is the metacategory with: {{DefineCategory |ob   = sets |mor  = mappings |comp = composition of mappings |id   = identity mappings }}


Definition:Category of Finite Sets

The '''category of finite sets''', denoted $\mathbf{Finset}$ or $\mathbf{Sets}_{\text{fin}}$ is the metacategory with: {{DefineCategory |ob  = All finite sets; |mor = All mappings between finite sets. }}


Definition:Category of Ordered Sets

The '''category of ordered sets''', denoted $\mathbf{OrdSet}$, is the metacategory with: {{DefineCategory | ob = ordered sets | mor = increasing mappings | comp = composition of mappings | id = identity mappings }}


Definition:Category of Relations

The '''category of (binary) relations''', denoted $\mathbf{Rel}$, is the metacategory with: {{DefineCategory |ob   = sets |mor  = binary relations $\mathcal R \subseteq A \times B$. |comp = composition of relations |id   = identity mappings }}


Definition:One (Category)

The '''category one''' $\mathbf 1$, is the category with: {{DefineCategory |ob = One object: $*$ |mor = One morphism: the identity morphism $\operatorname{id}_*$. }}


Definition:Two (Category)

The category $\mathbf 2$, '''two''', is the category: :$\quad * \longrightarrow \star$ with: :Two objects, $*$ and $\star$  :Three morphisms: ::The identity morphism $I_*$ ::The identity morphism $I_\star$ ::One non-identity morphism $* \to \star$


Definition:Three (Category)

The category $\mathbf 3$, '''three''', is the category: ::$\begin{xy} <0em,0em>*+{\circledast} = "x", <5em,0em>*+{\star} = "y", <5em,-5em>*+{\bullet} = "z", "x";"y" **@{-} ?>*@{>}, "y";"z" **@{-} ?>*@{>}, "x";"z" **@{-} ?>*@{>} \end{xy}$ with: :Three objects, $\circledast$, $\star$ and $\bullet$; and  :Six morphisms: :* The three identity morphisms $\operatorname{id}_\circledast$, $\operatorname{id}_\star$, $\operatorname{id}_\bullet$; :* One non-identity morphism $\circledast \to \star$; :* One non-identity morphism $\star \to \bullet$; :* One non-identity morphism $\circledast \to \bullet$.


Definition:Zero (Category)

The category $\mathbf 0$, '''zero''', is the empty category: :$\qquad$ with: :No objects and consequently: :No morphisms.


Definition:Functor/Covariant

Let $\mathbf C$ and $\mathbf D$ be metacategories. A '''covariant functor''' $F: \mathbf C \to \mathbf D$ consists of: * An '''object functor''' $F_0$ that assigns to each object $X$ of $\mathbf C$ an object $FX$ of $\mathbf D$. * An '''arrow functor''' $F_1$ that assigns to each arrow $f: X \to Y$ of $\mathbf C$ an arrow $Ff : FX \to FY$ of $\mathbf D$. These '''functors''' must satisfy, for any morphisms $X \stackrel f \longrightarrow Y \stackrel g \longrightarrow Z$ in $\mathbf C$: :$\map F {g \circ f} = F g \circ F f$ and: :$\map F {\operatorname {id}_X} = \operatorname{id}_{F X}$ where $\operatorname {id}_W$ denotes the identity arrow on an object $W$, and $\circ$ is the composition of morphisms. The behaviour of a '''covariant functor''' can be pictured as follows: ::$\begin{xy} <4em,4em>*{\mathbf C} = "C", <0em,0em>*+{X} = "a", <4em,0em>*+{Y} = "b", <4em,-4em>*+{Z}= "c", "a";"b" **@{-} ?>*@{>} ?<>(.5)*!/_1em/{f}, "b";"c" **@{-} ?>*@{>} ?<>(.5)*!/_1em/{g}, "a";"c" **@{-} ?>*@{>} ?<>(.5)*!/^1em/{g \circ f}, "C"+/r9em/*{\mathbf D}, "C"+/r2em/;"C"+/r6em/ **@{-} ?>*@{>} ?*!/_1em/{F}, "b"+/r2em/+/_2em/;"b"+/r6em/+/_2em/ **@{~} ?>*@2{>} ?<>(.5)*!/_.6em/{F}, "a"+/r13em/*+{FX}="Fa",  "b"+/r13em/*+{FY}="Fb",  "c"+/r13em/*+{FZ}="Fc", "Fa";"Fb" **@{-} ?>*@{>} ?<>(.5)*!/_1em/{Ff}, "Fb";"Fc" **@{-} ?>*@{>} ?<>(.5)*!/_1em/{Fg}, "Fa";"Fc" **@{-} ?>*@{>} ?<>(.7)*!/r3em/{F \left({g \circ f}\right) = \\ Fg \circ Ff}, \end{xy}$


Definition:Functor/Contravariant

{{:Definition:Functor/Contravariant/Definition 1}}


Definition:Composition of Functors

Let $\mathbf C, \mathbf D$ and $\mathbf E$ be metacategories. Let $F: \mathbf C \to \mathbf D$ and $G: \mathbf D \to \mathbf E$ be (covariant) functors. The '''composition of $G$ with $F$''' is the functor $GF: \mathbf C \to \mathbf E$ defined by: :For all objects $C$ of $\mathbf C$: $\hskip{2.9cm} GF \left({C}\right) := G \left({FC}\right)$ :For all morphisms $f: C_1 \to C_2$ of $\mathbf C$: $\quad GF \left({f}\right) := G \left({Ff}\right)$ $GF$ is said to be a '''composite functor'''.


Definition:Identity Functor

Let $\mathbf C$ be a metacategory. The '''identity functor on $\mathbf C$''' is the functor $\operatorname{id}_{\mathbf C}: \mathbf C \to \mathbf C$ defined by: :For all objects $C$ of $\mathbf C$: $\hskip{2.9cm} \operatorname{id}_{\mathbf C} C := C$ :For all morphisms $f: C_1 \to C_2$ of $\mathbf C$: $\quad \operatorname{id}_{\mathbf C} f := f$ That $\operatorname{id}_{\mathbf C}$ constitutes a functor is shown on Identity Functor is Functor.


Definition:Category of Categories

The '''category of categories''', denoted $\mathbf{Cat}$, is the metacategory with: {{DefineCategory |ob = small categories |mor = functors |comp = composition of functors |id = identity functors }}


Definition:Preorder Category

=== Definition 1 === {{:Definition:Preorder Category/Definition 1}} === Definition 2 === {{:Definition:Preorder Category/Definition 2}}


Definition:Order Category

=== Definition 1 === {{:Definition:Order Category/Definition 1}} === Definition 2 === {{:Definition:Order Category/Definition 2}}


Definition:Discrete Category

Let $\CC$ be a metacategory. Then $\CC$ is said to be '''discrete''' {{iff}} it comprises only identity morphisms. If the collection $\CC$ constitutes the objects of $\mathbf C$, then $\mathbf C$ may also be denoted $\map {\mathbf {Dis} } \CC$.


Definition:Monoid Category

Let $\left({S, \circ}\right)$ be a monoid with identity $e_S$. One can interpret $\left({S, \circ}\right)$ as being a category, with: {{DefineCategory | ob = Only one, say $*$ | mor = $a: * \to *$, for all $a \in S$ | comp =  $a \circ b: * \to *$ is defined using the operation $\circ$ of the monoid $S$ | id = $\operatorname{id}_* := e_S: * \to *$ }} The category that so arises is called a '''monoid category'''.


Definition:Category of Monoids

The '''category of monoids''', denoted $\mathbf{Mon}$, is the metacategory with: {{DefineCategory | ob = monoids | mor = monoid homomorphisms }}


Definition:Monoid Homomorphism

Let $\left({S, \circ}\right)$ and $\left({T, *}\right)$ be monoids. Let $\phi: S \to T$ be a mapping such that $\circ$ has the morphism property under $\phi$. That is, $\forall a, b \in S$: :$\phi \left({a \circ b}\right) = \phi \left({a}\right) * \phi \left({b}\right)$ Suppose further that $\phi$ preserves identities, i.e.: :$\phi \left({e_S}\right) = e_T$ Then $\phi: \left({S, \circ}\right) \to \left({T, *}\right)$ is a monoid homomorphism.


Definition:Prime Power

A '''prime power''' $p^n$ is a positive integer which is a prime number $p$ raised to the power $n$. The list of prime powers begins: :$1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, \ldots$ {{OEIS|A000961}} Category:Definitions/Number Theory Category:Definitions/Prime Numbers 1zfofa8rlmrr3xzwh7e6k0lmr49jy8e


Definition:Locally Small Category

Let $\mathbf C$ be a metacategory. Then $\mathbf C$ is said to be '''locally small''' iff all of its hom classes are sets. That is, iff for all objects $X, Y \in \mathbf C_0$ of $\mathbf C$: :$\operatorname{Hom}_{\mathbf C} \left({X, Y}\right) = \left\{{f \in \mathbf C_1 }\ \middle\vert \ {f: X \to Y}\right\}$ is a set.


Definition:Projection Functor

Let $\mathbf C$ and $\mathbf D$ be metacategories, and let $\mathbf C \times \mathbf D$ be their product. The '''first projection functor''' $\operatorname{pr}_1: \mathbf C \times \mathbf D \to \mathbf C$ is defined by: :$\operatorname{pr}_1 \left({C, D}\right) := C$ for all objects $\left({C, D}\right) \in \operatorname{ob} \mathbf C \times \mathbf D$ :$\operatorname{pr}_1 \left({f, g}\right) := f$ for all morphisms $\left({f, g}\right) \in \operatorname{mor} \mathbf C \times \mathbf D$ The '''second projection functor''' $\operatorname{pr}_2: \mathbf C \times \mathbf D \to \mathbf D$ is defined by: :$\operatorname{pr}_2 \left({C, D}\right) := D$ for all objects $\left({C, D}\right) \in \operatorname{ob} \mathbf C \times \mathbf D$ :$\operatorname{pr}_2 \left({f, g}\right) := g$ for all morphisms $\left({f, g}\right) \in \operatorname{mor} \mathbf C \times \mathbf D$ That these constitute functors is shown on Projection Functor is Functor.


Definition:Dual Category

Let $\mathbf C$ be a metacategory. Its '''dual category''', denoted $\mathbf C^{\text{op} }$, is defined as follows: {{DefineCategory |ob = $X^{\text{op} }$, for all $X \in \operatorname{ob}\mathbf C$ |mor = $f^{\text{op} }: D^{\text{op} } \to C^{\text{op} }$ for all $f: C \to D$ in $\mathbf C_1$ |comp = $\left({f^{\text{op} } \circ g^{\text{op} } }\right) := \left({g \circ f}\right)^{\text{op} }$, whenever this is defined |id = $\operatorname{id}_{X^{\text{op} } } := \operatorname{id}_X^{\text{op} }$ }}


Definition:Morphism Category

Let $\mathbf C$ be a metacategory. Its '''morphism category''', denoted $\mathbf C^\to$, is defined as follows: {{DefineCategory |ob = The morphisms $\mathbf C_1$ of $\mathbf C$ |mor = $g: f \to f'$ is a pair $\left({g_1, g_2}\right)$ of morphisms of $\mathbf C$ such that $g_2 \circ f = f' \circ g_1$ |comp = $\left({h_1, h_2}\right) \circ \left({g_1, g_2}\right) := \left({h_1 \circ g_1, h_2 \circ g_2}\right)$, whenever this is defined |id = $\operatorname{id}_{f} := \left({\operatorname{id}_C, \operatorname{id}_D}\right)$ for $f: C \to D$ }}


Definition:Domain Functor

Let $\mathbf C$ be a metacategory. Let $\mathbf C^\to$ be its morphism category. The '''domain functor''' is the functor $\operatorname{\mathbf{dom}}: \mathbf C^\to \to \mathbf C$ defined by: {{begin-axiom}} {{axiom|lc= Object functor:        |m = \operatorname{\mathbf{dom} } f := \operatorname{dom} f }} {{axiom|lc= Morphism functor:        |m = \operatorname{\mathbf{dom} } \left({g_1, g_2}\right) := g_1 }} {{end-axiom}} That it is in fact a functor is shown on Domain Functor is Functor. The functor $\mathbf{dom}$ can be represented as follows: ::$\begin{xy} <0em,2em>*+{A} = "A", <0em,-2em>*+{B} = "B", <4em,2em>*+{A'} = "A2", <4em,-2em>*+{B'} = "B2", "A";"B" **@{-} ?>*@{>} ?*!/^1em/{f}, "A";"A2" **@{-} ?>*@{>} ?*!/_1em/{g_1}, "A2";"B2" **@{-} ?>*@{>} ?*!/_1em/{f'}, "B";"B2" **@{-} ?>*@{>} ?*!/^1em/{g_2}, <6em,0em>;<10em,0em> **@{~} ?>*@2{>} ?*!/_1em/{\mathbf{dom}}, <12em,2em>*+{A} = "AA", <16em,2em>*+{A'} = "AA2", "AA";"AA2" **@{-} ?>*@{>} ?*!/_1em/{g_1}, \end{xy}$ It is thus seen to be an example of a forgetful functor.


Definition:Codomain Functor

Let $\mathbf C$ be a metacategory. Let $\mathbf C^\to$ be its morphism category. The '''codomain functor''' is the functor $\operatorname{\mathbf{cod}}: \mathbf C^\to \to \mathbf C$ defined by: {{begin-axiom}} {{axiom|lc= Object functor:        |m = \operatorname{\mathbf{cod} } f := \operatorname{cod} f }} {{axiom|lc= Morphism functor:        |m = \operatorname{\mathbf{cod} } \left({g_1, g_2}\right) := g_2 }} {{end-axiom}} That it is in fact a functor is shown on Codomain Functor is Functor. The functor $\mathbf{cod}$ can be represented as follows: ::$\begin{xy} <0em,2em>*+{A} = "A", <0em,-2em>*+{B} = "B", <4em,2em>*+{A'} = "A2", <4em,-2em>*+{B'} = "B2", "A";"B" **@{-} ?>*@{>} ?*!/^1em/{f}, "A";"A2" **@{-} ?>*@{>} ?*!/_1em/{g_1}, "A2";"B2" **@{-} ?>*@{>} ?*!/_1em/{f'}, "B";"B2" **@{-} ?>*@{>} ?*!/^1em/{g_2}, <6em,0em>;<10em,0em> **@{~} ?>*@2{>} ?*!/_1em/{\mathbf{cod}}, <12em,-2em>*+{B} = "BB", <16em,-2em>*+{B'} = "BB2", "BB";"BB2" **@{-} ?>*@{>} ?*!/^1em/{g_2}, \end{xy}$ It is thus seen to be an example of a forgetful functor.


Definition:Slice Category

Let $\mathbf C$ be a metacategory. Let $C \in \mathbf C_0$ be an object of $\mathbf C$. The '''slice category''' of $\mathbf C$ over $C$, denoted $\mathbf C / C$, is defined as follows: {{DefineCategory |ob = $f: X \to C$, i.e. the morphisms of $\mathbf C$ with codomain $C$ |mor = $a: f \to f'$, for all morphisms $a \in \mathbf C_1$ with $f' \circ a = f$ |comp = $a \circ b$ is defined precisely as in $\mathbf C$ |id = $\operatorname{id}_f := \operatorname{id}_X$, for $f: X \to C$ }} The morphisms can be displayed using a commutative diagram as follows: ::$\begin{xy} <-3em,0em>*+{X} = "X", <3em,0em>*+{X'} = "X2", <0em,-4em>*+{C} = "C", "X";"X2" **@{-} ?>*@{>} ?*!/_1em/{a}, "X";"C" **@{-} ?>*@{>} ?<>(.3)*!/^1em/{f}, "X2";"C" **@{-} ?>*@{>} ?<>(.3)*!/_1em/{f'}, \end{xy}$


Definition:Permutation on n Letters/Cycle Notation

The $k$-cycle $\rho$ is denoted: :$\begin {pmatrix} i & \map \rho i & \ldots & \map {\rho^{k - 1} } i \end{pmatrix}$ From Existence and Uniqueness of Cycle Decomposition, all permutations can be defined as the product of disjoint cycles. As Disjoint Permutations Commute, the order in which they are performed does not matter. So, for a given permutation $\rho$, the '''cycle notation''' for $\rho$ consists of all the disjoint cycles into which $\rho$ can be decomposed, concatenated as a product. It is conventional to omit $1$-cycles from the expression, and to write those cycles with lowest starting number first.


Definition:Signum Function/Natural Numbers

The '''signum function''' $\sgn: \N \to \set {0, 1}$ is the restriction of the signum function to the natural numbers, defined as: :$\forall n \in \N: \map \sgn n := \begin{cases} 0 & : n = 0 \\ 1 & : n > 0 \end{cases}$


Definition:Signum Function/Signum Complement

Let $\sgn: \N \to \set {0, 1}$ be the signum function on the natural numbers. The '''signum complement function''' $\overline \sgn: \N \to \set {0, 1}$ is defined as: :$\forall n \in \N: \map {\overline \sgn} n := \begin{cases} 1 & : n = 0 \\ 0 & : n > 0 \end{cases}$


Definition:Sign of Permutation

Let $n \in \N$ be a natural number. Let $\N_n$ denote the set of natural numbers $\set {1, 2, \ldots, n}$. Let $\tuple {x_1, x_2, \ldots, x_n}$ be an ordered $n$-tuple of real numbers. Let $\pi$ be a permutation of $\N_n$. Let $\map {\Delta_n} {x_1, x_2, \ldots, x_n}$ be the product of differences of $\tuple {x_1, x_2, \ldots, x_n}$. Let $\pi \cdot \map {\Delta_n} {x_1, x_2, \ldots, x_n}$ be defined as: :$\pi \cdot \map {\Delta_n} {x_1, x_2, \ldots, x_n} := \map {\Delta_n} {x_{\map \pi 1}, x_{\map \pi 2}, \ldots, x_{\map \pi n} }$ The '''sign of $\pi \in S_n$''' is defined as: :$\map \sgn \pi = \begin{cases} \dfrac {\Delta_n} {\pi \cdot \Delta_n} & : \Delta_n \ne 0 \\ 0 & : \Delta_n = 0 \end{cases}$


Definition:Composition Functor on Slice Categories

Let $\mathbf C$ be a metacategory. Let $C$ and $D$ be objects of $\mathbf C$. Let $\mathbf C / C$ and $\mathbf C / D$ be the associated slice categories. Let $g: C \to D$ be a morphism of $\mathbf C$. Then $g$ defines a '''composition functor''' $g_* : \mathbf C / C \to \mathbf C / D$: {{begin-axiom}} {{axiom|lc= Object functor:        |m = g_* f := g \circ f        |rc= The composition $\circ$ is taken in $\mathbf C$ }} {{axiom|lc= Morphism functor:        |m = g_* a := a }} {{end-axiom}} That it is in fact a functor is shown on Composition Functor on Slice Categories is Functor. The effect of $g_*$ is captured in the following commutative diagram: ::$\begin{xy} <-3em,0em>*+{X} = "X", <3em,0em>*+{X'} = "X2", <0em,-4em>*+{C} = "C", <0em,-8em>*+{D} = "D", "X";"X2" **@{-} ?>*@{>} ?*!/_1em/{a}, "X";"C" **@{-} ?>*@{>} ?<>(.4)*!/^.6em/{f}, "X2";"C" **@{-} ?>*@{>} ?<>(.4)*!/_.6em/{f'}, "C";"D" **@{-} ?>*@{>} ?<>(.4)*!/^.6em/{g}, "X";"D" **\crv{<-5em,-4em>} ?>*@{>} ?*!/^1.6em/{g_* f = \\ g \circ f}, "X2";"D" **\crv{<5em,-4em>} ?>*@{>} ?*!/_1.6em/{g_* f' = \\ g \circ f'}, \end{xy}$


Definition:Slice Functor

Let $\mathbf C$ be a metacategory. Let $\mathbf{Cat}$ be the category of categories. The '''slice functor''' is the functor $\mathbf C / \cdot: \mathbf C \to \mathbf{Cat}$ defined by: {{begin-axiom}} {{axiom|lc= Object functor:        |m = \mathbf C / C := \mathbf C / C }} {{axiom|lc= Morphism functor:        |m = \mathbf C / f := f_* }} {{end-axiom}} where $\mathbf C / C$ is a slice category, and $f_*$ is the composition functor defined by $f$. The effect of $\mathbf C / \cdot$ is captured in the following diagram: ::$\begin{xy} <0em,0em>*+{A} = "a", <4em,0em>*+{B} = "b", <4em,-4em>*+{C}= "c", "a";"b" **@{-} ?>*@{>} ?<>(.5)*!/_1em/{f}, "b";"c" **@{-} ?>*@{>} ?<>(.5)*!/_.6em/{g}, "a";"c" **@{-} ?>*@{>} ?<>(.4)*!/^1em/{g \circ f}, "b"+/r4em/+/_3em/;"b"+/r8em/+/_3em/ **@{~} ?>*@2{>} ?*!/_1em/{\mathbf C / \cdot}, "a"+/r13em/*+{\mathbf C / A}="Fa",  "b"+/r14em/*+{\mathbf C / B}="Fb",  "c"+/r14em/+/_1em/*+{\mathbf C / C}="Fc", "Fa";"Fb" **@{-} ?>*@{>} ?<>(.5)*!/_1em/{f_*}, "Fb";"Fc" **@{-} ?>*@{>} ?<>(.5)*!/_1em/{g_*}, "Fa";"Fc" **@{-} ?>*@{>} ?<>(.7)*!/r3em/{\left({g \circ f}\right)_* \\ = g_* f_*}, \end{xy}$ where $g_* f_*$ denotes a composite functor.


Definition:Coslice Category

Let $\mathbf C$ be a metacategory. Let $C \in \mathbf C_0$ be an object of $\mathbf C$. The '''coslice category''' of $\mathbf C$ under $C$, denoted $C / \mathbf C$, is the category with: {{DefineCategory |ob = $f: C \to X$, i.e. the morphisms of $\mathbf C$ with domain $C$ |mor  = $a: f \to f'$, for all morphisms $a \in \mathbf C_1$ with $a \circ f' = f$ |comp = $a \circ b$ is defined precisely as in $\mathbf C$ |id = $\operatorname{id}_f := \operatorname{id}_X$, for $f: C \to X$ }} The morphisms can be displayed using a commutative diagram as follows: ::$\begin{xy} <-3em,0em>*+{X} = "X", <3em,0em>*+{X'} = "X2", <0em,4em>*+{C} = "C", "X";"X2" **@{-} ?>*@{>} ?*!/^1em/{a}, "C";"X" **@{-} ?>*@{>} ?*!/^.6em/{f}, "C";"X2" **@{-} ?>*@{>} ?<>(.6)*!/_1em/{f'}, \end{xy}$


Definition:Category of Pointed Sets

The '''category of pointed sets''', denoted $\mathbf {Set}_*$, is defined as follows: {{DefineCategory | ob = pointed sets $\struct {A, a}$ | mor = pointed mappings | comp = Standard composition of mappings | id = $\map {\mathrm {id}_{\struct {A, a} } } := \mathrm {id}_A$, the identity mapping on $A$ }}


Definition:Kleene Closure

Let $S$ be a set. The '''Kleene closure''' of $S$, denoted $S^*$, is the set of all ordered tuples in $S$.


Definition:Concatenation of Ordered Tuples

Let $S$ be a set. Let $w, w'$ be finite sequences in $S$ of lengths $n$ and $n'$, respectively. Then the '''concatenation of $w$ and $w'$''', denoted $w * w'$ or simply $w w'$, is the sequence of $n + n'$ terms defined by: :$\map {w * w'} i := \begin{cases} \map w i & : \text {if $1 \le i \le n$} \\ \map {w'} {i - n} & : \text {if $n < i \le n + n'$} \end{cases}$ === Algebraic Structure === {{refactor}} Let $S^*$ be the Kleene closure of $S$. Then $\struct {S^*, *}$ is an algebraic structure, and by definition a magma.


Definition:Insertion of Generators

Let $S$ be a set, and let $S^*$ be its Kleene closure. The '''insertion of generators (into $S^*$)''' is the mapping $i: S \to S^*$ defined by: :$i \left({s}\right) := \left\langle{s}\right\rangle$ that is, it sends any element $s$ of $S$ to the one-term sequence containing only $s$.


Definition:Underlying Set Functor

Let $\mathbf{Set}$ be the category of sets. Let $\mathbf C$ be a metacategory with: :A collection of sets as objects; :A collection of mappings as morphisms. The '''underlying set functor''' $\left\vert{\cdot}\right\vert: \mathbf C \to \mathbf{Set}$ is defined in the following cases. {{expand|more to come}} === Category of Monoids === {{:Definition:Underlying Set Functor/Category of Monoids}} Category:Definitions/Category Theory 3ri175rmtt6vzp0uekdndgk3fhoahuu


Definition:Underlying Set Functor/Category of Monoids

Let $\mathbf{Mon}$ be the category of monoids. The '''underlying set functor''' $\left\vert{\cdot}\right\vert : \mathbf{Mon} \to \mathbf{Set}$ is the functor defined by: {{begin-axiom}} {{axiom|lc= Object functor:        |m = \left\vert{\left({M, \circ}\right)}\right\vert := M }} {{axiom|lc= Morphism functor:        |m = \left\vert{f}\right\vert := f }} {{end-axiom}}


Definition:Free Monoid

Let $S$ be a set. A '''free monoid over $S$''' is a monoid $M$ together with a mapping $i: S \to M$, subject to: :For all monoids $N$, for all mappings $f: S \to N$, there is a unique monoid homomorphism $\bar f: M \to N$, such that: ::$\bar f \circ i = f$ This condition is called the '''universal (mapping) property''' or '''UMP''' of the '''free monoid over $S$'''.


Definition:Cantor Normal Form

Let $x$ be an ordinal. The '''Cantor normal form''' of $x$ is an ordinal summation: :$x = \omega^{a_1} n_1 + \dots + \omega^{a_k} n_k$ where: :$k \in \N$ is a natural number :$\omega$ is the minimal infinite successor set :$\langle a_i \rangle$ is a strictly decreasing finite sequence of ordinals. :$\langle n_i \rangle$ is a finite sequence of finite ordinals In summation notation: :$x = \displaystyle \sum_{i \mathop = 1}^k \omega^{a_i} n_i$ {{explain|It still needs to be explained why, when used in pages that link to this, that the summation does not include the object $\omega$ in it, just some ordinal $x$ instead. It is unclear exactly what this definition means, because $\omega$, as currently defined on this website, is the Definition:Minimal Infinite Successor Set. Thus this definition appears to be saying: "Every ordinal (which of course has to include finite ones) can be expressed as finite sums of infinite ordinals." How can a finite number (an ordinal is a number, right?) be expressed as the sum of infinite numbers?}}


Definition:Free Category

Let $G$ be a digraph. The '''free category''' on $G$, denoted $\map {\mathbf C} G$, is the category with: {{DefineCategory | ob = The vertices of $G$ | mor = The walks in $G$ | comp = concatenation of walks | id = $\operatorname{id}_v$ is the empty walk at $v$ }}


Definition:Graph Functor

Let $\mathbf{Set}$ and $\mathbf{Rel}$ be the category of sets and the category of relations, respectively. The '''graph functor''' $G: \mathbf{Set} \to \mathbf{Rel}$ is defined by: {{begin-axiom}} {{axiom|lc= Object functor:        |t = $GX := X$ }} {{axiom|lc= Morphism functor:        |t = $Gf := G_f$, the graph of $f$ }} {{end-axiom}}


Definition:Standard Transitive Model

Let $p$ be a WFF in the language of set theory. Let $A$ be a class. Then $A$ is a '''standard transitive model''' of $p$ {{iff}}: :$A \ne \varnothing$ :$A$ is a transitive class and: :$A \models p$


Definition:Inverse Relation Functor

Let $\mathbf{Rel}$ be the category of relations. The '''inverse relation (contravariant) functor''' $C: \mathbf{Rel} \to \mathbf{Rel}$ is the contravariant functor defined by: {{begin-axiom}} {{axiom|lc= Object functor:        |t = $CX := X$ }} {{axiom|lc= Morphism functor:        |t = $C \mathcal R := \mathcal R^{-1}$, the inverse relation to $\mathcal R$ }} {{end-axiom}} That it is in fact a contravariant functor is shown on Inverse Relation Functor is Contravariant Functor.


=== Definition 1 === {{:Definition:Transitive Closure (Set Theory)/Definition 1}} The following is not equivalent to the above, but they are almost the same. === Definition 2 === {{:Definition:Transitive Closure (Set Theory)/Definition 2}}


Definition:Split Monomorphism

Let $\mathbf C$ be a metacategory. Let $f: C \to D$ be a morphism of $\mathbf C$. Then $f$ is said to be a '''split monomorphism''' iff for some $g: D \to C$, one has: :$g \circ f = \operatorname{id}_C$ where $\operatorname{id}_C$ is the identity morphism of $C$.


Definition:Split Epimorphism

Let $\mathbf C$ be a metacategory. Let $f: C \to D$ be a morphism of $\mathbf C$. Then $f$ is said to be a '''split epimorphism''' {{iff}} for some $g: D \to C$, one has: :$f \circ g = \operatorname{id}_D$ where $\operatorname{id}_D$ is the identity morphism of $D$. That is, {{iff}} $f$ has a section.


Definition:Section (Category Theory)

Let $\mathbf C$ be a metacategory. Let $f: C \to D$ be a morphism of $\mathbf C$. A '''section of $f$''' is a morphism $g: D \to C$ such that: :$f \circ g = \operatorname{id}_D$


Definition:Retraction

Let $\mathbf C$ be a metacategory. Let $f: C \to D$ be a morphism of $\mathbf C$. A '''retraction of $f$''' is a morphism $g: D \to C$ such that: :$g \circ f = \operatorname{id}_C$ === Retract === {{:Definition:Retract}}


Definition:Retract

Let $g: D \to C$ be a retraction of $f$. Then $D$ is said to be a '''retract''' of $C$.


Definition:Projective Object

Let $\mathbf C$ be a metacategory. Let $P \in \mathbf C_0$ be an object of $\mathbf C$. Then $P$ is said to be '''projective''' iff: :For all epimorphisms $e: E \twoheadrightarrow X$ and morphisms $f: P \to X$, there exists $\bar f: P \to E$ such that: ::$\begin{xy} <0em,0em>*+{P} = "P", <4em,0em>*+{X} = "X", <4em,4em>*+{E} = "E", "P";"E" **@{-} ?>*@{>} ?*!/_.6em/{\bar f}, "P";"X" **@{-} ?>*@{>} ?*!/^.6em/{f}, "E";"X" **@{-} ?>*@2{>} ?<>(.7)*{\vee} ?*!/_.6em/{e}, \end{xy}$ :is a commutative diagram, i.e. such that $f = e \circ \bar f$. In this situation, $f$ is said to '''lift across $e$'''.


Definition:Transitive with Respect to a Relation

Let $A$ be a class. Let $\RR$ be a relation on $A$. Let $S$ be a set. Then $S$ is '''transitive with respect to $\RR$''' {{iff}}: :$\forall x \in A: \forall y \in S: \paren {x \mathrel \RR y \implies x \in S}$


Definition:Initial Object

Let $\mathbf C$ be a metacategory. An '''initial object''' of $\mathbf C$ is an object $0 \in \mathbf C_0$ of $\mathbf C$ such that: :For all $C \in \mathbf C_0$, there is a unique morphism $0 \to C$.


Definition:Terminal Object

Let $\mathbf C$ be a metacategory. A '''terminal object''' of $\mathbf C$ is an object $1 \in \mathbf C_0$ of $\mathbf C$ such that: :For all $C \in \mathbf C_0$, there is a unique morphism $C \to 1$.


Definition:Orbit (Group Theory)/Length

The '''length''' of the orbit $\Orb x$ of $x$ is the number of elements of $X$ it contains: :$\size {\Orb x}$


Definition:Orbit (Group Theory)/Set of Orbits

The quotient set $X / \mathcal R_G$ is called the '''set of orbits of $X$ under the action of $G$'''.


Definition:Relational Closure

Let $A$ be a class. Let $\RR \subseteq A \times A$ be a relation. Let $x$ be a small class that is a subset of $A$. The '''relational closure''' of $x$ is the smallest small class containing $x$ that is also $\RR$-transitive.


Definition:Closed Relation

Let $A$ be a class. Let $\mathcal R$ be an arbitrary relation. Then $\mathcal R$ is '''closed''' with respect to $A$ iff $A$ is transitive with respect to $\mathcal R^{-1}$.


Definition:Supertransitive Class

Let $A$ be a transitive class. Then $A$ is said to be a '''supertransitive class''' {{iff}}: :$\forall x \in A: \powerset x \subseteq A$ That is, if $A$ contains the power set of all of its elements. {{explain|This phrasing is a bit awkward; element and subset are bound to be confused}}


Definition:Von Neumann Hierarchy

Let $U$ denote the universal class. The '''von Neumann hierarchy''' is a mapping $V: \On \to U$ on the ordinals, defined via transfinite recursion: :$\map V x = \begin{cases} \O & : x = 0 \\ & \\ \powerset {\map V n} & : x = n^+ \\ & \\ \displaystyle \bigcup_{y \mathop \in x} \map V y & : x \in \operatorname {Lim} \\ \end{cases}$ where: :$\powerset x$ denotes the power set of $x$ :$\operatorname {Lim}$ denotes the set of limit ordinals.


Definition:Boolean Ring

Let $\struct {R, +, \circ}$ be a ring. Then $R$ is called a '''Boolean ring''' {{iff}} $R$ is an idempotent ring with unity.


Definition:Well-Founded Set

Let $S$ be a small class. Let $V \left({ x }\right)$ denote the von Neumann hierarchy. Then $S$ is a '''well-founded set''' {{iff}} there is some ordinal $x$ such that $S \in V\left({x}\right)$.


Definition:Rank (Set Theory)

Let $A$ be a set. Let $V$ denote the von Neumann hierarchy. Then the '''rank''' of $A$ is the smallest ordinal $x$ such that $A \in V \left({x+1}\right)$, given that $x$ exists.


Definition:Prime Number/Negative Prime

A '''negative prime''' is an integer of the form $-p$ where $p$ is a (positive) prime number.


Definition:Composite Number/Negative Composite

A negative integer $n$ is composite {{iff}} $\left|{n}\right|$ is composite.


Definition:Prime Triplet

A '''prime triplet''' is a set of three (positive) prime numbers $\left\{{n, n+2, n+4}\right\}$.


Definition:Power (Algebra)/Integer

Let $x \in \R$ be a real number. Let $n \in \Z$ be an integer. The expression $x^n$ is called '''$x$ to the power of $n$'''. $x^n$ is defined recursively as: :$x^n = \begin{cases} 1 & : n = 0 \\ & \\ x \times x^{n - 1} & : n > 0 \\ & \\ \dfrac {x^{n + 1} } x & : n < 0 \end{cases}$ where $\dfrac{x^{n + 1} } x$ denotes quotient.


Definition:Power (Algebra)/Even Power

Let $x \in \R$ be a real number. Let $n \in \Z$ be an even integer. Then $x^n$ is called an '''even power of $x$'''.


Definition:Power (Algebra)/Odd Power

Let $x \in \R$ be a real number. Let $n \in \Z$ be an odd integer. Then $x^n$ is called an '''odd power of $x$'''


Definition:Power (Algebra)/Rational Number

Let $x \in \R$ be a real number such that $x > 0$. Let $m \in \Z$ be an integer. Let $y = \sqrt [m] x$ be the $m$th root of $x$. Then we can write $y = x^{1/m}$ which means the same thing as $y = \sqrt [m] x$. Thus we can define the power to a positive rational number: Let $r = \dfrac p q \in \Q$ be a positive rational number where $p \in \Z_{\ge 0}, q \in \Z_{> 0}$. Then $x^r$ is defined as: :$x^r = x^{p/q} = \paren {\sqrt [q] x}^p = \sqrt [q] {\paren {x^p} }$. When $r = \dfrac {-p} q \in \Q: r < 0$ we define: :$x^r = x^{-p/q} = \dfrac 1 {x^{p/q}}$ analogously for the negative integer definition.


Definition:Power (Algebra)/Real Number

Let $x \in \R_{>0}$ be a (strictly) positive real number. Let $r \in \R$ be a real number. === Definition 1 === {{:Definition:Power (Algebra)/Real Number/Definition 1}} === Definition 2 === {{:Definition:Power (Algebra)/Real Number/Definition 2}} === Definition 3 === {{:Definition:Power (Algebra)/Real Number/Definition 3}}


Definition:Power (Algebra)/Complex Number

Let $z, k \in \C$ be any complex numbers. '''$z$ to the power of $k$''' is defined as the multifunction: :$z^k := e^{k \ln \paren z}$ where $e^z$ is the exponential function and $\ln$ is the  natural logarithm multifunction.


Definition:Power (Algebra)/Power of Zero

Let $r \in \R$ be a real number. (This includes the situation where $r \in \Z$ or $r \in \Q$.) When $x=0$, $x^r$ is defined as follows: :$0^r = \begin{cases} 1 & : r = 0 \\ 0 & : r > 0 \\ \text{Undefined} & : r < 0 \\ \end{cases}$ This takes account of the awkward case $0^0$: it is "generally accepted" that $0^0 = 1$ as this convention agrees with certain general results which would otherwise need a special case.


Definition:Power (Algebra)/Exponent

In the expression $x^r$, the number $r$ is known as the '''exponent''' of $x$, particularly for $r \in \R$.


Definition:Two (Boolean Algebra)

Denote with $\top$ the canonical tautology. Denote with $\bot$ the canonical contradiction. Define $\mathbf 2 := \set {\bot, \top}$, read '''two'''. When endowed with the logical operations $\lor$, $\land$ and $\neg$, $\mathbf 2$ becomes a Boolean algebra. These operations have the following Cayley tables: :$\begin{array}{c|cc}  \lor & \bot & \top \\ \hline  \bot & \bot & \top \\  \top & \top & \top \end{array} \qquad \begin{array}{c|cc}  \land & \bot & \top \\ \hline  \bot  & \bot & \bot \\  \top  & \bot & \top \end{array} \qquad \begin{array}{c|cc}       & \bot & \top \\ \hline  \neg & \top & \bot \end{array}$ That $\mathbf 2$ thus becomes a Boolean algebra is shown on Two is Boolean Algebra.


Definition:Category of Boolean Algebras

The '''category of Boolean algebras''', denoted $\mathbf{BA}$, is the metacategory with: {{DefineCategory |ob  = Boolean algebras |mor =  Boolean homomorphisms }}


Definition:Operation/Unary Operation

A '''unary operation''' is the special case of an operation where the operation has exactly one operand. Thus, a '''unary operation''' on a set $S$ is a mapping whose domain and codomain are both $S$.


Definition:Constant (Category Theory)

Let $\mathbf C$ be a metacategory, and let $1$ be a terminal object of $\mathbf C$. A '''constant of $\mathbf C$''' is a morphism $f: 1 \to C$ of $\mathbf C$ which has $1$ as its domain.


Definition:Variable Element

Let $\mathbf C$ be a metacategory, and let $C$ be an object of $\mathbf C$. A '''variable element''' of $C$ is a morphism $f: B \to C$ with codomain $C$.


Definition:Enough Constants

Let $\mathbf C$ be a metacategory. Then $\mathbf C$ is said to have '''enough constants''' iff: :For all morphisms $f,g : C \to D$ with $f \ne g$, there is a constant $c: 1 \to C$ such that $f \circ c \ne g \circ c$


Definition:Product (Category Theory)

=== Binary Product === {{:Definition:Product (Category Theory)/Binary Product}} === General Definition === {{:Definition:Product (Category Theory)/General Definition}}


Definition:Cardinal Number

Let $S$ be a set. The '''cardinal number''' of $S$ is defined as follows: :$\card S = \displaystyle \bigcap \set {x \in \On : x \sim S}$ where $\On$ is the class of all ordinals. Compare cardinality.


Definition:Aleph Mapping

Let $\mathcal N'$ denote the class of all infinite cardinal numbers. Then $\aleph$ (that is: '''aleph''') is defined as the unique order isomorphism between the two ordered structures $\left({\operatorname{On}, \in}\right)$ and $\left({\mathcal N', \in}\right)$ where $\operatorname{On}$ denotes the class of ordinal numbers. {{explain|A link is needed to a proof that such a unique order isomorphism exists.}}


Definition:Class of Cardinals

The '''class of cardinals''' is the class consisting of all cardinal numbers: :$\NN = \set {x \in \On: \exists y: x = \size y}$ where $\size y$ denotes the cardinal number corresponding to $y$.


Definition:Infinite Cardinal Class

The '''infinite cardinal class''' is the class of all infinite cardinal numbers. That is: :$\mathcal N’ = \mathcal N \setminus \omega$


Definition:Definiendum

An object being defined in a definition is called a '''definiendum'''.


Definition:Definiens

The part of a definition that explains the meaning of the definiendum is the '''definiens'''.


Definition:Group Axioms/Right

A '''group''' is an algebraic structure $\struct {G, \circ}$ which satisfies the following four conditions: {{begin-axiom}} {{axiom | n = \text G 0         | lc= Closure Axiom         | q = \forall a, b \in G         | m = a \circ b \in G }} {{axiom | n = \text G 1         | lc= Associativity Axiom         | q = \forall a, b, c \in G         | m = a \circ \paren {b \circ c} = \paren {a \circ b} \circ c }} {{axiom | n = \text G_{\text R} 2         | lc= Right Identity Axiom         | q = \exists e \in G: \forall a \in G         | m = a \circ e = a }} {{axiom | n = \text G_{\text R} 3         | lc= Right Inverse Axiom         | q = \forall a \in G: \exists b \in G         | m = a \circ b = e }} {{end-axiom}}


Definition:Group Axioms/Left

A '''group''' is an algebraic structure $\struct {G, \circ}$ which satisfies the following four conditions: {{begin-axiom}} {{axiom | n = \text G 0         | lc= Closure Axiom         | q = \forall a, b \in G         | m = a \circ b \in G }} {{axiom | n = \text G 1         | lc= Associativity Axiom         | q = \forall a, b, c \in G         | m = a \circ \paren {b \circ c} = \paren {a \circ b} \circ c }} {{axiom | n = \text G_{\text L} 2         | lc= Left Identity Axiom         | q = \exists e \in G: \forall a \in G         | m = e \circ a = a }} {{axiom | n = \text G_{\text L} 3         | lc= Left Inverse Axiom         | q = \forall a \in G: \exists b \in G         | m = b \circ a = e }} {{end-axiom}}


Definition:Product of Morphisms

Let $\mathbf C$ be a metacategory. Let $A, A'$ and $B, B'$ be pairs of objects admitting binary products: ::$\begin{xy}\xymatrix@R-1em@C+1em@L+3px{  A &  A \times A'   \ar[l]_*+{p_1}   \ar[r]^*+{p_2} &  A' \\  B &  B \times B'   \ar[l]_*+{q_1}   \ar[r]^*+{q_2} &  B' }\end{xy}$ Let $f: A \to B$ and $f': A' \to B'$ be morphisms. The '''product morphism of $f$ and $f'$''', denoted $f \times f'$, is the unique morphism making the following diagram commute: ::$\begin{xy}\xymatrix@+1em@L+3px{  A   \ar[d]_*+{f} &  A \times A'   \ar[l]_*+{p_1}   \ar[r]^*+{p_2}   \ar@{-->}[d]^*+{\hskip{1.3em} f \times f'} &  A'   \ar[d]^*+{f'} \\  B &  B \times B'   \ar[l]^*+{q_1}   \ar[r]_*+{q_2} &  B' }\end{xy}$ Thus we see that $f \times f'$ is the morphism $\left\langle{fp_1, f'p_2}\right\rangle$.


Definition:Fixed Element of Permutation/Moved

$x$ '''moved by $\pi$''' {{iff}}: :$\map \pi x \ne x$


Definition:Fixed Element of Permutation/Set of Fixed Elements

The set of elements of $S$ which are fixed by $\pi$ can be denoted $\Fix \pi$.


Definition:Permutation on n Letters/Set of Permutations

The '''set of permutations of $\N_n$''' is denoted $S_n$.


Definition:Product Functor

Let $\mathbf C$ be a metacategory with binary products. Let $\mathbf C \times \mathbf C$ be the product category of $\mathbf C$ with itself. The '''product functor''' on $\mathbf C$ is the functor $\times: \mathbf C \times \mathbf C \to \mathbf C$ defined by: {{begin-axiom}} {{axiom|lc= Object functor:        |m = \times \left({C, D}\right) := C \times D }} {{axiom|lc= Morphism functor:        |m = \times \left({f, f'}\right) := f \times f' }} {{end-axiom}} where $C \times D$ is a binary product of $C$ and $D$, and $f \times f'$ is the product of $f$ and $f'$. That it is in fact a functor is shown on Product Functor is Functor.


Definition:One (Ordinal)

The ordinal $1$ is defined as: :$1 = 0^+$ where $0^+$ denotes the successor of the ordinal $0$. Category:Definitions/Ordinals lm8rlyljlz9gn8x9nhximmti7zfq0if


Definition:Two (Ordinal)

The ordinal $2$ is defined as: :$2 = 1^+$ where $1^+$ denotes the successor of the ordinal $1$. Category:Definitions/Ordinals hbior4iy1b4mjhw9j2d374orzwnd996