hcju/proofwikips huggingface dataset:
Definition:Empty Mapping
Let $T$ be a set. Then the mapping whose domain is the empty set and whose codomain is $T$ is called the '''empty mapping''': :$\O \subseteq \O \times T = \O$
Definition:Identity Mapping
The '''identity mapping''' of a set $S$ is the mapping $I_S: S \to S$ defined as: :$I_S = \set {\tuple {x, y} \in S \times S: x = y}$ or alternatively: :$I_S = \set {\tuple {x, x}: x \in S}$ That is: :$I_S: S \to S: \forall x \in S: \map {I_S} x = x$
Definition:Inverse Mapping
Let $S$ and $T$ be sets.
Definition:Injection
=== Definition 1 === {{:Definition:Injection/Definition 1}} === Definition 2 === {{:Definition:Injection/Definition 2}} === Definition 3 === {{:Definition:Injection/Definition 3}} === Definition 4 === {{:Definition:Injection/Definition 4}} === Definition 5 === {{:Definition:Injection/Definition 5}} === Definition 6 === {{:Definition:Injection/Definition 6}}
Definition:Left Cancellable Mapping
A mapping $f: Y \to Z$ is '''left cancellable''' (or '''left-cancellable''') : :$\forall X: \forall \struct {g_1, g_2: X \to Y}: f \circ g_1 = f \circ g_2 \implies g_1 = g_2$ That is, for any set $X$, if $g_1$ and $g_2$ are mappings from $X$ to $Y$: :If $f \circ g_1 = f \circ g_2$ :then $g_1 = g_2$.
Definition:Inclusion Mapping
Let $T$ be a set. Let $S\subseteq T$ be a subset. The '''inclusion mapping''' $i_S: S \to T$ is the mapping defined as: :$i_S: S \to T: \forall x \in S: \map {i_S} x = x$
Definition:Surjection
Let $S$ and $T$ be sets or classes. Let $f: S \to T$ be a mapping from $S$ to $T$.
Definition:Right Cancellable Mapping
A mapping $f: X \to Y$ is '''right cancellable''' (or '''right-cancellable''') : :$\forall Z: \forall \paren {h_1, h_2: Y \to Z}: h_1 \circ f = h_2 \circ f \implies h_1 = h_2$ That is,  for any set $Z$: :If $h_1$ and $h_2$ are mappings from $Y$ to $Z$ :then $h_1 \circ f = h_2 \circ f$ implies $h_1 = h_2$.
Definition:Bijection
=== Definition 1 === {{:Definition:Bijection/Definition 1}} === Definition 2 === {{:Definition:Bijection/Definition 2}} === Definition 3 === {{:Definition:Bijection/Definition 3}} === Definition 4 === {{:Definition:Bijection/Definition 4}} === Definition 5 === {{:Definition:Bijection/Definition 5}}
Definition:Permutation
A bijection $f: S \to S$ from a set $S$ to itself is called a '''permutation on''' (or '''of''') '''$S$'''.
Definition:Inverse Mapping/Definition 2
Let $f: S \to T$ and $g: T \to S$ be mappings. Let: :$g \circ f = I_S$ :$f \circ g = I_T$ where: :$g \circ f$ and $f \circ g$ denotes the composition of $f$ with $g$ in either order :$I_S$ and $I_T$ denote the identity mappings on $S$ and $T$ respectively. That is, $f$ and $g$ are both left inverse mappings and right inverse mappings of each other. Then: :$g$ is '''the inverse (mapping) of $f$''' :$f$ is '''the inverse (mapping) of $g$'''.
Definition:Set Equivalence
Let $S$ and $T$ be sets. Then $S$ and $T$ are '''equivalent''' : :there exists a bijection $f: S \to T$ between the elements of $S$ and those of $T$. That is, if they have the '''same cardinality'''. This can be written $S \sim T$. If $S$ and $T$ are not '''equivalent''' we write $S \nsim T$.
Definition:Injective Restriction
Let $f: S \to T$ be a mapping which is not injective. Then an '''injective restriction''' of $f$ is a restriction $g {\restriction_{S'}}: S' \to T$ of $f$ which is an injection.
Definition:Surjective Restriction
Let $f: S \to T$ be a mapping which is not surjective. Then a '''surjective restriction''' of $f$ is a restriction $g: S' \to T'$ of $f$ such that $g \restriction_{S'}: S' \to T'$ is a surjection.
Definition:Bijective Restriction
Let $f: S \to T$ be a mapping which is not bijective. A '''bijective restriction''' of $f$ is a restriction $f {\restriction_{S' \times T'} }: S' \to T'$ of $f$ such that $f {\restriction_{S' \times T'} }$ is a bijection.
Definition:Set of All Mappings
Let $S$ and $T$ be sets. The '''set of (all) mappings from $S$ to $T$''' is: :$T^S := \set {f \subseteq S \times T: f: S \to T \text { is a mapping} }$
Definition:Projection (Mapping Theory)
Let $S_1, S_2, \ldots, S_j, \ldots, S_n$ be sets. Let $\displaystyle \prod_{i \mathop = 1}^n S_i$ be the Cartesian product of $S_1, S_2, \ldots, S_n$. For each $j \in \set {1, 2, \ldots, n}$, the '''$j$th projection on $\displaystyle S = \prod_{i \mathop = 1}^n S_i$''' is the mapping $\pr_j: S \to S_j$ defined by: :$\map {\pr_j} {s_1, s_2, \ldots, s_j, \ldots, s_n} = s_j$ for all $\tuple {s_1, s_2, \ldots, s_n} \in S$.
Definition:Quotient Mapping
Let $\RR \subseteq S \times S$ be an equivalence on a set $S$. Let $\eqclass s \RR$ be the $\RR$-equivalence class of $s$. Let $S / \RR$ be the quotient set of $S$ determined by $\RR$. Then $q_\RR: S \to S / \RR$ is the '''quotient mapping induced by $\RR$''', and is defined as: :$q_\RR: S \to S / \RR: \map {q_\RR} s = \eqclass s \RR$
Definition:Equivalence Relation Induced by Mapping
Let $f: S \to T$ be a mapping. Let $\RR_f \subseteq S \times S$ be the relation defined as: :$\tuple {s_1, s_2} \in \RR_f \iff \map f {s_1} = \map f {s_2}$ Then $\RR_f$ is known as the '''equivalence (relation) induced by $f$'''.
Definition:Well-Defined
=== Well-Defined Mapping === {{:Definition:Well-Defined/Mapping}} === Well-Defined Relation === The concept can be generalized to include the general relation $\RR: S \to T$. {{:Definition:Well-Defined/Relation}} === Well-Defined Operation === {{:Definition:Well-Defined/Operation}}
Definition:Dominate (Set Theory)
Let $S$ and $T$ be sets. === Definition 1 === {{:Definition:Dominate (Set Theory)/Definition 1}} === Definition 2 === {{:Definition:Dominate (Set Theory)/Definition 2}} === Strictly Dominated === {{:Definition:Dominate (Set Theory)/Strictly Dominate}}
Definition:Ordering
Let $S$ be a set. === Definition 1 === {{:Definition:Ordering/Definition 1}} === Definition 2 === {{:Definition:Ordering/Definition 2}}
Definition:Partially Ordered Set
A '''partially ordered set''' is a relational structure $\left({S, \preceq}\right)$ such that $\preceq$ is a partial ordering. The '''partially ordered set''' $\left({S, \preceq}\right)$ is said to be '''partially ordered by $\preceq$'''.
Definition:Maximal
Let $\struct {S, \preceq}$ be an ordered set. Let $T \subseteq S$ be a subset of $S$. === Ordered Set === {{:Definition:Maximal/Ordered Set}} === Maximal Set === {{:Definition:Maximal/Set}}
Definition:Minimal
=== Relation === {{:Definition:Minimal/Relation}} === Ordered Set === Let $\struct {S, \preceq}$ be an ordered set. Let $T \subseteq S$ be a subset of $S$. {{:Definition:Minimal/Ordered Set}} === Minimal Set === {{:Definition:Minimal/Set}}
Definition:Non-Comparable
Let $\struct {S, \RR}$ be a relational structure. Two elements $x, y \in S, x \ne y$ are '''non-comparable''' if neither $x \mathrel \RR y$ nor $y \mathrel \RR x$. If $x$ and $y$ are not non-comparable then they are '''comparable''', but the latter term is not so frequently encountered.
Definition:Order Isomorphism
=== Definition 1 === {{:Definition:Order Isomorphism/Definition 1}} === Definition 2 === {{:Definition:Order Isomorphism/Definition 2}} Two ordered sets $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ are '''(order) isomorphic''' if there exists such an '''order isomorphism''' between them. $\struct {S, \preceq_1}$ is described as '''(order) isomorphic to''' (or '''with''') $\struct {T, \preceq_2}$, and vice versa. This may be written $\struct {S, \preceq_1} \cong \struct {T, \preceq_2}$. Where no confusion is possible, it may be abbreviated to $S \cong T$. === Well-Ordered Sets === When $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ are well-ordered sets, the condition on the order preservation can be relaxed: {{:Definition:Order Isomorphism/Wosets}}
Definition:Immediate Predecessor Element
Let $\struct {S, \preceq}$ be an ordered set. Let $a, b \in S$. Then $a$ is an '''immediate predecessor (element)''' to $b$ : :$(1): \quad a \prec b$ :$(2): \quad \neg \exists c \in S: a \prec c \prec b$ That is, there exists no element strictly between $a$ and $b$ in the ordering $\preceq$. That is: :$a \prec b$ and $\openint a b = \O$ where $\openint a b$ denotes the open interval from $a$ to $b$. We say that '''$a$ ''immediately'' precedes $b$'''.
Definition:Immediate Successor Element
Let $\struct {S, \preceq}$ be an ordered set. Let $a, b \in S$. Then $a$ is an '''immediate successor (element)''' to $b$  $b$ is an immediate predecessor (element) to $a$. That is, : :$(1): \quad b \prec a$ :$(2): \quad \nexists c \in S: b \prec c \prec a$ That is, there exists no element strictly between $b$ and $a$ in the ordering $\preceq$. That is: :$a \prec b$ and $\openint a b = \O$ where $\openint a b$ denotes the open interval from $a$ to $b$. We say that '''$a$ ''immediately'' succeeds $b$'''.
Definition:Lattice Ordering
Let $\left({S, \preceq}\right)$ be a lattice. Then the ordering $\preceq$ is referred to as a '''lattice ordering'''.
Definition:Complete Lattice
=== Definition 1 === {{:Definition:Complete Lattice/Definition 1}} === Definition 2 === {{:Definition:Complete Lattice/Definition 2}}
Definition:Total Ordering
Let $\mathcal R \subseteq S \times S$ be a relation on a set $S$.
Definition:Totally Ordered Set
Let $\struct {S, \preceq}$ be a relational structure. Then $\struct {S, \preceq}$ is a '''totally ordered set'''  $\preceq$ is a total ordering.
Definition:Chain (Set Theory)
Let $\struct {S, \preceq}$ be an ordered set. A '''chain in $S$''' is a totally ordered subset of $S$. Thus a totally ordered set is itself a '''chain''' in its own right. === Length === {{Definition:Length of Chain}}
Definition:Maximal Chain
Let $\left({S, \preceq}\right)$ be an ordered set. Let $\left({T, \preceq}\right) \subseteq \left({S, \preceq}\right)$ be a chain in $\left({S, \preceq}\right)$ such that there is no other chain in $\left({S, \preceq}\right)$ which has $\left({T, \preceq}\right)$ as a proper subset. Then $\left({T, \preceq}\right)$ is a '''maximal chain in $S$'''.
Definition:Order Embedding
Let $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ be ordered sets. Let $\phi: S \to T$ be a mapping. === Definition 1 === {{:Definition:Order Embedding/Definition 1}} === Definition 2 === {{:Definition:Order Embedding/Definition 2}} === Definition 3 === {{:Definition:Order Embedding/Definition 3}} === Definition 4 === {{:Definition:Order Embedding/Definition 4}}
Definition:Strictly Increasing
=== Ordered Sets === {{:Definition:Strictly Increasing/Mapping}} === Real Functions === This definition continues to hold when $S = T = \R$. {{:Definition:Strictly Increasing/Real Function}} === Sequences === {{:Definition:Strictly Increasing Sequence}}
Definition:Strictly Decreasing
=== Ordered Sets === {{:Definition:Strictly Decreasing/Mapping}} === Real Functions === This definition continues to hold when $S = T = \R$. {{:Definition:Strictly Decreasing/Real Function}} === Sequences === {{:Definition:Strictly Decreasing Sequence}}
Definition:Strictly Monotone
=== Ordered Sets === {{:Definition:Strictly Monotone/Mapping}} === Real Functions === This definition continues to hold when $S = T = \R$: {{:Definition:Strictly Monotone/Real Function}} === Sequences === {{:Definition:Strictly Monotone/Sequence}}
Definition:Increasing
=== Ordered Sets === {{:Definition:Increasing/Mapping}} === Real Functions === This definition continues to hold when $S = T = \R$. {{:Definition:Increasing/Real Function}} === Sequences === {{:Definition:Increasing/Sequence}}
Definition:Decreasing
=== Ordered Sets === {{:Definition:Decreasing/Mapping}} === Real Functions === This definition continues to hold when $S = T = \R$; thus: {{:Definition:Decreasing/Real Function}} === Sequences === {{:Definition:Decreasing Sequence}}
Definition:Monotone (Order Theory)
=== Ordered Sets === {{:Definition:Monotone (Order Theory)/Mapping}} === Real Functions === {{:Definition:Monotone (Order Theory)/Real Function}} === Sequences === {{:Definition:Monotone (Order Theory)/Sequence}}
Definition:Well-Ordered Set
Let $\struct {S, \preceq}$ be an ordered set. Then $\struct {S, \preceq}$ is a '''well-ordered set''' if the ordering $\preceq$ is a well-ordering. That is, if ''every'' non-empty subset of $S$ has a smallest element: :$\forall T \in \powerset S: \exists a \in T: \forall x \in T: a \preceq x$ where $\powerset S$ denotes the power set of $S$. Or, such that $\preceq$ is a well-founded total ordering.
Definition:Entropic
=== Entropic Operation === {{:Definition:Entropic Operation}} === Entropic Structure === {{:Definition:Entropic Structure}}
Definition:Constant Operation
Let $S$ be a set. For a given $c \in S$, a '''constant operation''' on $S$ is defined as: :$\forall x, y \in S: x \left[{c}\right] y = c$
Definition:Left Operation
Let $S$ be a set. For any $x, y \in S$, the '''left operation''' on $S$ is the binary operation defined as: :$\forall x, y \in S: x \leftarrow y = x$
Definition:Right Operation
Let $S$ be a set. For any $x, y \in S$, the '''right operation''' on $S$ is the binary operation defined as: :$\forall x, y \in S: x \rightarrow y = y$
Definition:Max Operation
Let $\struct {S, \preceq}$ be a totally ordered set. The '''max operation''' is the binary operation on $\left({S, \preceq}\right)$ defined as: :$\forall x, y \in S: \map \max {x, y} = \begin{cases} y & : x \preceq y \\ x & : y \preceq x \end{cases}$ === General Definition === {{:Definition:Max Operation/General Definition}}
Definition:Min Operation
Let $\struct {S, \preceq}$ be a totally ordered set. The '''min operation''' is the binary operation on $\struct {S, \preceq}$ defined as: :$\forall x, y \in S: \map \min {x, y} = \begin{cases} x & : x \preceq y \\ y & : y \preceq x \end{cases}$
Definition:Underlying Set
=== Abstract Algebra === {{:Definition:Underlying Set/Abstract Algebra}} === Relational Structure === {{:Definition:Underlying Set/Relational Structure}} === Metric Space === {{:Definition:Underlying Set/Metric Space}} === Topological Space === {{:Definition:Underlying Set/Topological Space}}
Definition:Free Semigroup
A semigroup which has a non-commutative product in which no product can ever be expressed more simply in terms of the other elements is called a '''free semigroup'''. {{wtd|add formal definition}}
Definition:Monoid
A '''monoid''' is a semigroup with an identity element.
Definition:Commutative Monoid
A monoid whose operation is commutative is a '''commutative monoid'''.
Definition:Free Idempotent Monoid
Let $\left({S, \circ}\right)$ be a monoid which is generated by a set of elements, and such that: :$\forall x \in S: x = x \circ x$ Then $\left({S, \circ}\right)$ is a '''free idempotent monoid'''.
Definition:Submonoid
A '''submonoid''' of a monoid $\struct {S, \circ}$ is a monoid $\struct {T, \circ}$ such that $T \subseteq S$.
Definition:Inverse Semigroup
An '''inverse semigroup''' is a semigroup $\struct {S, \circ}$ such that: :$\forall a \in S: \exists! b \in S: a = a \circ b \circ a, b = b \circ a \circ b$
Definition:Invertible Element
Let $\struct {S, \circ}$ be an algebraic structure which has an identity $e_S$. If $x \in S$ has an inverse, then $x$ is said to be '''invertible for $\circ$'''. That is, $x$ is '''invertible''' : :$\exists y \in S: x \circ y = e_S = y \circ x$ === Invertible Operation === {{:Definition:Invertible Operation}}
Definition:Trivial Group
A '''trivial group''' is a group with only one element $e$.
Definition:Quasigroup
A '''quasigroup''' is a magma $\struct {S, \circ}$ which has the Latin square property. That is, such that $\forall a \in S$, the left and right regular representations $\lambda_a$ and $\rho_a$ are permutations on $S$.
Definition:Algebra Loop
An '''algebra loop''' $\struct {S, \circ}$ is a quasigroup with an identity element.
Definition:Relation Compatible with Operation
Let $\struct {S, \circ}$ be a closed algebraic structure. Let $\RR$ be a relation on $S$. Then $\RR$ is '''compatible with $\circ$''' : :$\forall x, y, z \in S: x \mathrel \RR y \implies \paren {x \circ z} \mathrel \RR \paren {y \circ z}$ :$\forall x, y, z \in S: x \mathrel \RR y \implies \paren {z \circ x} \mathrel \RR \paren {z \circ y}$
Definition:Congruence Relation
Let $\struct {S, \circ}$ be an algebraic structure. Let $\mathcal R$ be an equivalence relation on $S$. Then $\mathcal R$ is a '''congruence relation for $\circ$''' : :$\forall x_1, x_2, y_1, y_2 \in S: \paren {x_1 \mathrel {\mathcal R} x_2} \land \paren {y_1 \mathrel {\mathcal R} y_2} \implies \paren {x_1 \circ y_1} \mathrel {\mathcal R} \paren {x_2 \circ y_2}$
Definition:Universally Compatible Relation
A relation $\RR$ is '''universally compatible''' on a set $S$  it is compatible with every closed operation that can be defined on $S$.
Definition:Universally Congruent
A equivalence $\RR$ is '''universally congruent''' on a set $S$  it is a congruence for every closed operation that can be defined on $S$.
Definition:Operation Induced on Quotient Set
Let $\struct {S, \circ}$ be an algebraic structure. Let $\mathcal R$ be a congruence relation on $\struct {S, \circ}$. Let $S / \mathcal R$ be the quotient set of $S$ by $\mathcal R$. The '''operation $\circ_\mathcal R$ induced on $S / \mathcal R$ by $\circ$''' is defined as: :$\eqclass x {\mathcal R} \circ_\mathcal R \eqclass y {\mathcal R} = \eqclass {x \circ y} {\mathcal R}$
Definition:Quotient Structure
Let $\left({S, \circ}\right)$ be an algebraic structure. Let $\mathcal R$ be a congruence relation on $\left({S, \circ}\right)$. Let $S / \mathcal R$ be the quotient set of $S$ by $\mathcal R$. Let $\circ_\mathcal R$ be the operation induced on $S / \mathcal R$ by $\circ$. The '''quotient structure defined by $\mathcal R$''' is the algebraic structure: : $\left({S / \mathcal R, \circ_\mathcal R}\right)$
Definition:External Direct Product
Let $\left({S, \circ_1}\right)$ and $\left({T, \circ_2}\right)$ be algebraic structures. The '''(external) direct product''' $\left({S \times T, \circ}\right)$ of two algebraic structures $\left({S, \circ_1}\right)$ and $\left({T, \circ_2}\right)$ is the set of ordered pairs: :$\left({S \times T, \circ}\right) = \left\{{\left({s, t}\right): s \in S, t \in T}\right\}$ where the operation $\circ$ is defined as: :$\left({s_1, t_1}\right) \circ \left({s_2, t_2}\right) = \left({s_1 \circ_1 s_2, t_1 \circ_2 t_2}\right)$
Definition:Algebraic Substructure
Let $\left({S, \circ}\right)$ be an algebraic structure. Let $T \subseteq S$. Then $\left({T, \circ}\right)$ is an '''algebraic substructure''' of $\left({S, \circ}\right)$.
Definition:Internal Direct Product
Let $\left({S_1, \circ {\restriction_{S_1}} }\right), \left({S_2, \circ {\restriction_{S_2}} }\right)$ be closed algebraic substructures of an algebraic structure $\left({S, \circ}\right)$ where $\circ {\restriction_{S_1}}, \circ {\restriction_{S_2}}$ are the operations induced by the restrictions of $\circ$ to $S_1, S_2$ respectively. The structure $\left({S, \circ}\right)$ is the '''internal direct product of $S_1$ and $S_2$''' if the mapping: :$C: S_1 \times S_2 \to S: C \left({\left({s_1, s_2}\right)}\right) = s_1 \circ s_2$ is an isomorphism from the cartesian product $\left({S_1, \circ {\restriction_{S_1}}}\right) \times \left({S_2, \circ {\restriction_{S_2}}}\right)$ onto $\left({S, \circ}\right)$. The operation $\circ$ on $S$ is the operation induced on $S$ by $\circ {\restriction_{S_1}}$ and $\circ {\restriction_{S_2}}$.
Definition:Homomorphism (Abstract Algebra)
Let $\struct {S, \circ}$ and $\struct {T, *}$ be magmas. Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be a mapping from $\struct {S, \circ}$ to $\struct {T, *}$. Let $\circ$ have the morphism property under $\phi$, that is: :$\forall x, y \in S: \map \phi {x \circ y} = \map \phi x * \map \phi y$ Then $\phi$ is a '''homomorphism'''.
Definition:Morphism Property
Let $\phi: \left({S, \circ}\right) \to \left({T, *}\right)$ be a mapping from one algebraic structure $\left({S, \circ}\right)$ to another $\left({T, *}\right)$. Then $\circ$ has the '''morphism property''' under $\phi$ iff: :$\forall x, y \in S: \phi \left({x \circ y}\right) = \phi \left({x}\right) * \phi \left({y}\right)$
Definition:Epimorphism (Abstract Algebra)
A homomorphism which is a surjection is described as '''epic''', or called an '''epimorphism'''.
Definition:Endomorphism
An '''endomorphism''' is a homomorphism from an algebraic structure into itself.
Definition:Monomorphism (Abstract Algebra)
A homomorphism which is an injection is descibed as '''monic''', or called a '''monomorphism'''.
Definition:Isomorphism (Abstract Algebra)
An '''isomorphism''' is a homomorphism which is a bijection. That is, it is a mapping which is both a monomorphism and an epimorphism.
Definition:Automorphism (Abstract Algebra)
An '''automorphism''' is an isomorphism from an algebraic structure to itself.
Definition:Canonical Injection (Abstract Algebra)
Let $\struct {S_1, \circ_1}$ and $\struct {S_2, \circ_2}$ be algebraic structures with identities $e_1, e_2$ respectively. The following mappings: :$\inj_1: \struct {S_1, \circ_1} \to \struct {S_1, \circ_1} \times \struct {S_2, \circ_2}: \forall x \in S_1: \map {\inj_1} x = \tuple {x, e_2}$ :$\inj_2: \struct {S_2, \circ_2} \to \struct {S_1, \circ_1} \times \struct {S_2, \circ_2}: \forall x \in S_2: \map {\inj_2} x = \tuple {e_1, x}$ are called the '''canonical injections'''. === General Definition === {{:Definition:Canonical Injection (Abstract Algebra)/General Definition}}
Definition:Inverse Completion
Let $\struct {S, \circ}$ be a semigroup. Let $\struct {C, \circ} \subseteq \struct {S, \circ}$ be the subsemigroup of cancellable elements of $\struct {S, \circ}$. Let $\struct {T, \circ'}$ be a semigroup defined such that: :$(1): \quad \struct {S, \circ}$ is a subsemigroup of $\struct {T, \circ'}$ :$(2): \quad$ Every element of $C$ has an inverse in $T$ for $\circ'$ :$(3): \quad \gen {S \cup C^{-1} } = \struct {T, \circ'}$ where: :$\gen {S \cup C^{-1} }$ denotes the semigroup generated by $S \cup C^{-1}$ :$C^{-1}$ denotes the inverse of $C$. Then $\struct {T, \circ'}$ is called an '''inverse completion''' of $\struct {S, \circ}$.
Definition:Inverse of Subset
=== Monoid === {{Definition:Inverse of Subset/Monoid}} === Group === {{Definition:Inverse of Subset/Group}}
Definition:Automorphism Group/Group
Let $\struct {S, *}$ be an algebraic structure. Let $\mathbb S$ be the set of automorphisms of $S$. Then the algebraic structure $\struct {\mathbb S, \circ}$, where $\circ$ denotes composition of mappings, is called the '''automorphism group''' of $S$. The structure $\struct {S, *}$ is usually a group. However, this is not necessary for this definition to be valid. The '''automorphism group of $S$''' is denoted on  as $\Aut S$.
Definition:Inner Automorphism
Let $G$ be a group. Let $x \in G$. Let the mapping $\kappa_x: G \to G$ be defined such that: :$\forall g \in G: \map {\kappa_x} g = x g x^{-1}$ $\kappa_x$ is called the '''inner automorphism of $G$ (given) by $x$'''.
Definition:Proper Subgroup
Let $\struct {G, \circ}$ be a group. Then $\struct {H, \circ}$ is a '''proper subgroup of $\struct {G, \circ}$''' : : $(1): \quad \struct {H, \circ}$ is a subgroup of $\struct {G, \circ}$ : $(2): \quad H \ne G$, i.e. $H \subset G$. The notation $H < G$, or $G > H$, means: : '''$H$ is a proper subgroup of $G$'''. If $H$ is a subgroup of $G$, but it is not specified whether $H = G$ or not, then we write $H \le G$, or $G \ge H$. === Non-Trivial Proper Subgroup === {{:Definition:Proper Subgroup/Non-Trivial}}
Definition:Trivial Subgroup
Let $\struct {G, \circ}$ be a group whose identity is $e$. Then the algebraic structure $\struct {\set e, \circ}$ is called '''the trivial subgroup''' of $\struct {G, \circ}$.
Definition:Conjugate (Group Theory)
Let $\left({G, \circ}\right)$ be a group. === Conjugate of an Element === {{:Definition:Conjugate (Group Theory)/Element}} === Conjugate of a Set === {{:Definition:Conjugate (Group Theory)/Subset}}
Definition:Conjugacy Class
The equivalence classes into which the conjugacy relation divides its group into are called '''conjugacy classes'''. The '''conjugacy class''' of an element $x \in G$ can be denoted $\conjclass x$.