Definition:Computational Method

A '''computational method''' is an ordered quadruple $\left({Q, I, \Omega, f}\right)$ in which: : $Q$ is a set representing the '''states of the computation''' : $I$ is a set representing the '''input to the computation''' : $\Omega$ is a set representing the '''output from the computation''' : $f: Q \to Q$ is a mapping representing the '''computational rule''' subject to the following constraints: : $I \subseteq Q$ and $\Omega \subseteq Q$ : $\forall x \in \Omega: f \left({x}\right) = x$ === Computational Sequence === {{:Definition:Computational Method/Computational Sequence}}


Definition:Set of All Linear Transformations

Let $G$ and $H$ be $R$-modules. Then $\map {\mathrm {Hom}_R} {G, H}$ is the '''set of all linear transformations''' from $G$ to $H$: :$\map {\mathrm {Hom}_R} {G, H} := \set {\phi: G \to H: \phi \mbox{ is a linear transformation} }$ If it is clear (and therefore does not need to be stated) that the scalar ring is $R$, then this can be written $\map {\mathrm {Hom} } {G, H}$. Similarly, $\map {\mathrm {Hom}_R} G$ is the set of all linear operators on $G$: :$\map {\mathrm {Hom}_R} G := \set {\phi: G \to G: \phi \text{ is a linear operator} }$ Again, this can also be written $\map {\mathrm {Hom} } G$.


Definition:Inverse Matrix

Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $\mathbf A$ be a square matrix of order $n$. Let there exist a square matrix $\mathbf B$ of order $n$ such that: :$\mathbf A \mathbf B = \mathbf I_n = \mathbf B \mathbf A$ where $\mathbf I_n$ denotes the unit matrix of order $n$. Then $\mathbf B$ is called the '''inverse of $\mathbf A$''' and is usually denoted $\mathbf A^{-1}$.


Definition:Stirling's Triangle of the Second Kind

$\begin{array}{r|rrrrrrrrrr} n & {n \brace 0} & {n \brace 1} & {n \brace 2} & {n \brace 3} & {n \brace 4} & {n \brace 5} & {n \brace 6} & {n \brace 7} & {n \brace 8} & {n \brace 9} \\ \hline 0 & 1 & 0 &   0 &    0 &    0 &    0 &    0 &   0 &  0  & 0 \\ 1 & 0 & 1 &   0 &    0 &    0 &    0 &    0 &   0 &  0  & 0 \\ 2 & 0 & 1 &   1 &    0 &    0 &    0 &    0 &   0 &  0  & 0 \\ 3 & 0 & 1 &   3 &    1 &    0 &    0 &    0 &   0 &  0  & 0 \\ 4 & 0 & 1 &   7 &    6 &    1 &    0 &    0 &   0 &  0  & 0 \\ 5 & 0 & 1 &  15 &   25 &   10 &    1 &    0 &   0 &  0  & 0 \\ 6 & 0 & 1 &  31 &   90 &   65 &   15 &    1 &   0 &  0  & 0 \\ 7 & 0 & 1 &  63 &  301 &  350 &  140 &   21 &   1 &  0  & 0 \\ 8 & 0 & 1 & 127 &  966 & 1701 & 1050 &  266 &  28 &  1  & 0 \\ 9 & 0 & 1 & 255 & 3025 & 7770 & 6951 & 2646 & 462 & 36  & 1 \\ \end{array}$


Definition:Stirling's Triangle of the First Kind (Signed)

$\begin{array}{r|rrrrrrrrrr} n & \map s {n, 0} & \map s {n, 1} & \map s {n, 2} & \map s {n, 3} & \map s {n, 4} & \map s {n, 5} & \map s {n, 6} & \map s {n, 7} & \map s {n, 8} & \map s {n, 9} \\ \hline 0 & 1 &     0 &       0 &      0 &      0 &     0 &     0 &   0 &   0  & 0 \\ 1 & 0 &     1 &       0 &      0 &      0 &     0 &     0 &   0 &   0  & 0 \\ 2 & 0 &    -1 &       1 &      0 &      0 &     0 &     0 &   0 &   0  & 0 \\ 3 & 0 &     2 &      -3 &      1 &      0 &     0 &     0 &   0 &   0  & 0 \\ 4 & 0 &    -6 &      11 &     -6 &      1 &     0 &     0 &   0 &   0  & 0 \\ 5 & 0 &    24 &     -50 &     35 &    -10 &     1 &     0 &   0 &   0  & 0 \\ 6 & 0 &  -120 &     274 &   -225 &     85 &   -15 &     1 &   0 &   0  & 0 \\ 7 & 0 &   720 &   -1764 &   1624 &   -735 &   175 &   -21 &   1 &   0  & 0 \\ 8 & 0 & -5040 &   13068 & -13132 &   6769 & -1960 &   322 & -28 &   1  & 0 \\ 9 & 0 & 40320 & −109584 & 118124 & −67284 & 22449 & −4536 & 546 & −36  & 1 \\ \end{array}$


Definition:Root of Unity

Let $n \in \Z_{> 0}$ be a strictly positive integer. Let $F$ be a field. The '''$n$th roots of unity of $F$''' are defined as: :$U_n = \left\{{z \in F: z^n = 1}\right\}$


Let $\struct {H_1, \circ {\restriction_{H_1} } }, \struct {H_2, \circ {\restriction_{H_2} } }$ be subgroups of a group $\struct {G, \circ}$ where $\circ {\restriction_{H_1} }, \circ {\restriction_{H_2} }$ are the restrictions of $\circ$ to $H_1, H_2$ respectively. === Definition 1 === {{:Definition:Internal Group Direct Product/Definition 1}} === Definition 2 === {{:Definition:Internal Group Direct Product/Definition 2}} === Definition 3 === {{:Definition:Internal Group Direct Product/Definition 3}}


Definition:Permutable Subgroups

Let $\struct {G, \circ}$ be a group. Let $H$ and $K$ be subgroups of $G$. Let $H \circ K$ denote the subset product of $H$ and $K$. Then $H$ and $K$ are '''permutable''' {{iff}}: :$H \circ K = K \circ H$


Definition:Cycle Type

Let $S_n$ denote the symmetric group on $n$ letters. Let $\rho \in S_n$. From Existence and Uniqueness of Cycle Decomposition, every $\rho$ may be uniquely expressed as a product of disjoint cycles: :$\rho = \tau_1, \tau_2, \ldots, \tau_r$ up to the order of factors. Let $\tau_1, \tau_2, \ldots, \tau_r$ be arranged in increasing order of cycle length. Let the length of the cycle $\tau_i$ be $k_i$. The resulting ordered tuple of cycle lengths $\tuple {k_1, k_2, \ldots, k_r}$ is called the '''cycle type''' of $\rho$. Thus $\sigma$ and $\rho$ have the same '''cycle type''' if they have the same number of cycles of equal length.


Definition:Permutation on Polynomial

Let $\map f {x_1, x_2, \ldots, x_n}$ denote a polynomial in $n$ variables $x_1, x_2, \ldots, x_n$. Let $S_n$ denote the symmetric group on $n$ letters. Let $\pi, \rho \in S_n$. Then $\pi * f$ is the polynomial obtained by applying the permutation $\pi$ to the subscripts on the variables of $f$. This is called the '''permutation on the polynomial $f$ by $\pi$''', or the $f$-permutation by $\pi$.


Definition:Pullback of Quotient Group Isomorphism

Let $G, H$ be groups. Let $N \lhd G, K \lhd H$ be normal subgroups of $G$ and $H$ respectively. Let: :$G / N \cong H / K$ where: :$G / N$ denotes the quotient of $G$ by $N$ :$\cong$ denotes group isomorphism. Let $\theta: G / N \to H / K$ be such a group isomorphism. The '''pullback $G \times^\theta H$ of $G$ and $H$ via $\theta$''' is the subset of $G \times H$ of elements of the form $\tuple {g, h}$ where $\map \theta {g N} = h K$.


Definition:Permutation Group

A '''permutation group''' on a set $S$ is a subgroup of the symmetric group $\struct {\map \Gamma S, \circ}$ on $S$.


Definition:Commutative and Unitary Ring

A '''commutative and unitary ring''' $\struct {R, +, \circ}$ is a ring with unity which is also commutative. That is, it is a ring such that the ring product $\struct {R, \circ}$ is commutative and has an identity element. That is, such that the multiplicative semigroup $\struct {R, \circ}$ is a commutative monoid.


Definition:K-Algebra

Let $k$ be a field. A '''$k$-algebra''' is a unital associative commutative algebra over $k$. Category:Definitions/Commutative Algebra Category:Definitions/Algebras jn4ahs1333akdt43rqk74peimcev5md


Definition:Entropic Operation

Let $\circ$ be an operation on a set $S$ such that: :$\forall a, b, c, d \in S: \left({a \circ b}\right) \circ \left({c \circ d}\right) = \left({a \circ c}\right) \circ \left({b \circ d}\right)$ Then $\circ$ is '''entropic'''.


Definition:Finite Ring Homomorphism

Let $A$ and $B$ be commutative rings with unity. Let $\phi: A \to B$ be a ring homomorphism. === Definition 1 === $\phi$ is '''finite''' {{iff}} $B$ is finite as an algebra over $A$ via $\phi$. === Definition 2 === $\phi$ is '''finite''' {{iff}} there exists a finite number of $b_1, \ldots, b_n$ such that every $b \in B$ can be written as: :$\displaystyle b = \sum_{i \mathop = 1}^n \phi \left({a_i}\right) b_i$ where $a_i \in A$.


Definition:Composition (Combinatorics)

A $k$-composition of a (strictly) positive integer $n \in \Z_{> 0}$ is an ordered $k$-tuple: :$c = \left({c_1, c_2, \ldots, c_k}\right)$ such that: : $(1): \quad c_1 + c_2 + \cdots + c_k = n$ : $(2): \quad \forall i \in \left[{1 \,.\,.\, k}\right]: c_i \in \Z_{> 0}$, that is, all the $c_i$ are strictly positive integers. Category:Definitions/Combinatorics 2iv1b2c8grfnqiu47q6bt64v06zxjsv


Definition:Ideal of Ring/Proper Ideal

A '''proper ideal''' $J$ of $\left({R, +, \circ}\right)$ is an ideal of $R$ such that $J$ is a proper subset of $R$. That is, such that $J \subseteq R$ and $J \ne R$.


Definition:Generated Ideal of Ring

Let $\struct {R, +, \circ}$ be a ring. Let $S \subseteq R$ be a subset. === Definition 1 === {{Definition:Generated Ideal of Ring/Definition 1}} === Definition 2: for commutative rings with unity === {{Definition:Generated Ideal of Ring/Definition 2}} === Definition 3: for rings with unity === {{Definition:Generated Ideal of Ring/Definition 3}}


Definition:Internal Direct Sum of Rings

Let $\left({R, +, \circ}\right)$ be a ring. Let $S_1, S_2, \ldots, S_n$ be a finite sequence of subrings of $R$. Let $\displaystyle S = \prod_{j \mathop = 1}^n S_j$ be the cartesian product of $S_1$ to $S_n$. Then $S$ is the '''(ring) direct sum''' of $S_1, S_2, \ldots, S_n$ {{iff}} the mapping $\phi: S \to R$ defined as: :$\phi\left({\left({x_1, x_2, \ldots, x_n}\right)}\right) = x_1 + x_2 + \cdots x_n$ is an isomorphism from $S$ to $R$.


Definition:Prime Ideal of Ring

Let $R$ be a ring. A '''prime ideal''' of $R$ is a proper ideal $P$ such that: : $I \circ J \subseteq P \implies I \subseteq P \text { or } J \subseteq P$ for any ideals $I$ and $J$ of $R$.


Definition:Generated Division Subring

Let $\left({D, +, \circ}\right)$ be a division ring. Let $S \subseteq D$. The '''division subring generated by $S$''' is the smallest division subring of $D$ containing $S$.


Definition:Isomorphism (Abstract Algebra)/Field Isomorphism

Let $\struct {F, +, \circ}$ and $\struct {K, \oplus, *}$ be fields. Let $\phi: F \to K$ be a (field) homomorphism. Then $\phi$ is a field isomorphism {{iff}} $\phi$ is a bijection.


Definition:Cossist

The '''cossist''' tradition was a school of abacist mathematics developed in Germany in the 15th century. Notable cossists include {{AuthorRef|Johann Faulhaber}}.


Definition:Scalar Multiplication

=== $R$-Algebraic Structure === {{:Definition:Scalar Multiplication/R-Algebraic Structure}} === Module === {{:Definition:Scalar Multiplication/Module}} === Vector Space === {{:Definition:Scalar Multiplication/Vector Space}}


Definition:R-Algebraic Structure Automorphism

Let $\left({S, \ast_1, \ast_2, \ldots, \ast_n, \circ}\right)_R$ be an $R$-algebraic structure. Let $\phi: S \to S$ be an $R$-algebraic structure isomorphism from $S$ to itself. Then $\phi$ is an $R$-algebraic structure automorphism. This definition continues to apply when $S$ is a module, and also when it is a vector space.


Definition:Generator of Module

Let $R$ be a ring. Let $M$ be an $R$-module. Let $S \subseteq M$ be a subset. === Definition 1 === {{:Definition:Generator of Module/Definition 1}} === Definition 2 === {{:Definition:Generator of Module/Definition 2}} === Definition 3 === {{:Definition:Generator of Module/Definition 3}}


Let $\phi: G \to H$ be a linear transformation where $G$ and $H$ are $R$-modules. Let $e_H$ be the identity of $H$. The '''kernel''' of $\phi$ is defined as: :$\map \ker \phi := \phi^{-1} \sqbrk {\set {e_H} }$. where $\phi^{-1} \sqbrk S$ denotes the preimage of $S$ under $\phi$. === In Vector Space === {{:Definition:Kernel of Linear Transformation/Vector Space}}


Definition:R-Algebraic Structure Monomorphism

Let $\left({S, \ast_1, \ast_2, \ldots, \ast_n, \circ}\right)_R$ and $\left({T, \odot_1, \odot_2, \ldots, \odot_n, \otimes}\right)_R$ be $R$-algebraic structures. Then $\phi: S \to T$ is an '''$R$-algebraic structure monomorphism''' {{iff}}: : $(1): \quad \phi$ is an injection : $(2): \quad \forall k: k \in \left[{1 \,.\,.\, n}\right]: \forall x, y \in S: \phi \left({x \ast_k y}\right) = \phi \left({x}\right) \odot_k \phi \left({y}\right)$ : $(3): \quad \forall x \in S: \forall \lambda \in R: \phi \left({\lambda \circ x}\right) = \lambda \otimes \phi \left({x}\right)$. That is, {{iff}}: : $(1): \quad \phi$ is an injection : $(2): \quad \phi$ is an $R$-algebraic structure homomorphism. This definition continues to apply when $S$ and $T$ are modules, and also when they are vector spaces. === Vector Space Monomorphism === {{:Definition:Vector Space Monomorphism}}


Definition:Vector Space Monomorphism

Let $V$ and $W$ be $K$-vector spaces. Then $\phi: V \to W$ is a '''vector space monomorphism''' {{iff}}: : $(1): \quad \phi$ is an injection : $(2): \quad \forall \mathbf x, \mathbf y \in V: \phi \left({\mathbf x + \mathbf y}\right) = \phi \left({\mathbf x}\right) + \phi \left({\mathbf y}\right)$ : $(3): \quad \forall \mathbf x \in V: \forall \lambda \in K: \phi \left({\lambda \mathbf x}\right) = \lambda \phi \left({\mathbf x}\right)$


Definition:Isomorphism (Abstract Algebra)/R-Algebraic Structure Isomorphism/Vector Space Isomorphism

Let $\struct {V, +, \circ }$ and $\struct {W, +', \circ'}$ be $K$-vector spaces. Then $\phi: V \to W$ is a '''vector space isomorphism''' {{iff}}: : $(1): \quad \phi$ is a bijection : $(2): \quad \forall \mathbf x, \mathbf y \in V: \map \phi {\mathbf x + \mathbf y} = \map \phi {\mathbf x} +' \map \phi {\mathbf y}$ : $(3): \quad \forall \mathbf x \in V: \forall \lambda \in K: \map \phi {\lambda \mathbf x} = \lambda \map \phi {\mathbf x}$


Definition:Isomorphism (Abstract Algebra)/R-Algebraic Structure Isomorphism/Module Isomorphism

Let $\struct {G, +_G, \circ}_R$ and $\struct {H, +_H, \circ}_R$ be $R$-modules. Let $\phi: G \to H$ be a module homomorphism. Then $\phi$ is a '''module isomorphism''' {{iff}} $\phi$ is a bijection.


Definition:Quaternion

A '''quaternion''' is a number in the form: : $a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ where: : $a, b, c, d$ are real numbers : $\mathbf 1, \mathbf i, \mathbf j, \mathbf k$ are entities related to each other in the following way: {{begin-eqn}}      {{eqn | l = \mathbf i \mathbf j = -\mathbf j \mathbf i       | r = \mathbf k       | c =   }} {{eqn | l = \mathbf j \mathbf k = -\mathbf k \mathbf j       | r = \mathbf i       | c =  }} {{eqn | l = \mathbf k \mathbf i = -\mathbf i \mathbf k       | r = \mathbf j       | c =  }} {{eqn | l = \mathbf i^2 = \mathbf j^2 = \mathbf k^2 = \mathbf i \mathbf j \mathbf k       | r = -\mathbf 1       | c =  }} {{end-eqn}}


Definition:Conjugate Quaternion

Let $\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ be a quaternion. The conjugate quaternion of $\mathbf x$ is defined as: :$\overline {\mathbf x} = a \mathbf 1 - b \mathbf i - c \mathbf j - d \mathbf k$. === Matrix Form === {{:Definition:Conjugate Quaternion/Matrix Form}} === Ordered Pair of Complex Numbers === {{:Definition:Conjugate Quaternion/Ordered Pair}}


Definition:Subsemiring

Let $\left({S, *, \circ}\right)$ be an algebraic structure with two operations. A '''subsemiring of $\left({S, *, \circ}\right)$''' is a subset $T$ of $S$ such that $\left({T, *_T, \circ_T}\right)$ is a semiring. {{SUBPAGENAME}} 6corsnq5jtxaxkicgktu8idhrahdz68


Definition:Additive Group of Ring

The group $\struct {R, +}$ is known as the '''additive group of $R$'''.


Definition:Additive Subgroup

Let $\struct {R, +, \times}$ be a ring. Let $\struct {R, +}$ be the additive group of $\struct {R, +, \times}$. Let $\struct {S, +}$ be a subgroup of $\struct {R, +}$ Then $\struct {S, +}$ is an '''additive subgroup of ''' $\struct {R, +, \times}$.


Definition:Generated Subring

Let $\struct {R, +, \circ}$ be a ring. Let $S \subseteq R$ be a subset. The '''subring generated by $S$''' is the smallest subring of $R$ containing $S$; that is, it is the intersection of all subrings of $R$ containing $S$.


Definition:Proper Zero Divisor

Let $\struct {R, +, \circ}$ be a ring. A proper zero divisor of $R$ is an element $x \in R^*$ such that: :$\exists y \in R^*: x \circ y = 0_R$ where $R^*$ is defined as $R \setminus \set {0_R}$. That is, it is a zero divisor of $R$ which is '''specifically not $0_R$'''.


Definition:Perfect Graph

A graph is '''perfect''' if no two vertices have the same degree.


Definition:Term of Sequence

The elements of a sequence are known as its '''terms'''. Let $\sequence {x_n}$ be a sequence. Then the '''$k$th term''' of $\sequence {x_n}$ is the ordered pair $\tuple {k, x_k}$.


Definition:Totally Ordered Group

A '''totally ordered group''' is a totally ordered structure $\left({G, \circ, \preceq}\right)$ such that $\left({G, \circ}\right)$ is a group.


Definition:Jacobson Radical

Let $R$ be a commutative ring with unity. Let $\operatorname{maxspec} \left({R}\right)$ be the set of maximal ideals of $R$. Then the '''Jacobson radical''' of $R$ is: :$\displaystyle \operatorname{Jac} \left({R}\right) = \bigcap_{m \mathop \in \operatorname{maxspec} \left({R}\right)} m$ That is, it is the intersection of all maximal ideals of $R$.


Definition:Jacobson Ring

Let $\left({R, +, \circ}\right)$ be a commutative ring with unity. Then $\left({R, +, \circ}\right)$ is a '''Jacobson ring''' {{iff}}: : every prime ideal of $\left({R, +, \circ}\right)$ is an intersection of  maximal ideals.


Definition:Adjugate Matrix

Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$. Let $\mathbf C$ be its cofactor matrix. The '''adjugate matrix''' of $\mathbf A$ is the transpose of $\mathbf C$: :$\adj {\mathbf A} = \mathbf C^\intercal$


Definition:Radical of Ideal of Ring

Let $A$ be a commutative ring with unity. Let $I$ be an ideal of $A$. === Definition 1 === {{Definition:Radical of Ideal of Ring/Definition 1}} === Definition 2 === {{Definition:Radical of Ideal of Ring/Definition 2}}


Definition:Radical Ideal of Ring

Let $A$ be a commutative ring with unity. Let $I$ be an ideal of $A$. Then $I$ is a '''radical ideal''' {{iff}} it is equal to its radical.


Definition:Nilpotent Ring Element

Let $R$ be a ring with zero $0_R$. An element $x \in R$ is '''nilpotent''' {{iff}}: :$\exists n \in \Z_{>0}: x^n = 0_R$


Definition:Reduced Ring

Let $\struct {R, +, \circ}$ be a ring. Then $R$ is a '''reduced ring''' {{iff}} it contains no nilpotent elements except the zero.


Definition:Nilradical of Ring

Let $A$ be a commutative ring with unity. === Definition 1 === {{:Definition:Nilradical of Ring/Definition 1}} === Definition 2 === {{:Definition:Nilradical of Ring/Definition 2}}


Definition:Polynomial Ring/Monoid Ring on Free Monoid on Set

Let $R \sqbrk {\family {X_i: i \in I} }$ be the ring of polynomial forms in $\family {X_i: i \in I}$. The '''polynomial ring in $I$ indeterminates over $R$''' is the ordered triple $\struct {\struct {A, +, \circ}, \iota, \family {X_i: i \in I} }$ {{DefinitionWanted|define the inclusion and indeterminates in this case}}


Definition:Monomial of Free Commutative Monoid

A '''monomial''' in the indexed set $\family {X_j: j \in J}$ is a possibly infinite product: :$\ds \prod_{j \mathop \in J} X_j^{k_j}$ with integer exponents $k_j \ge 0$ such that $k_j = 0$ for all but finitely many $j$. Let $\mathbf X = \family {X_j}_{j \mathop \in J}$ and for a multiindex $k = \paren {k_j}_{j \mathop \in J}$ over $J$ define: :$\ds \mathbf X^k = \prod_{j \mathop \in J} X_j^{k_j}$ Then a '''monomial''' is an object of the form $\mathbf X^k$, where $k$ is a multiindex.


Definition:Free Commutative Monoid

The '''free commutative monoid''' on an indexed set $X = \family {X_j: j \in J}$ is the set $M$ of all monomials under the standard multiplication. That is, it is the set $M$ of all finite sequences of $X$.


Definition:Multiindex

=== Definition 1 === Let $\ds m = \prod_{j \mathop \in J} X_j^{k_j}$ be a monomial in the indexed set $\set {X_j: j \mathop \in J}$. Such a monomial can be expressed implicitly and more compactly by referring only to the sequence of indices: :$k = \sequence {k_j}_{j \mathop \in J}$ and write $m = \mathbf X^k$ without explicit reference to the indexing set. Such an expression is called a '''multiindex''' (or '''multi-index'''). === Definition 2 === Let $J$ be a set. A '''$J$-multiindex''' is a sequence of natural numbers indexed by $J$: :$\displaystyle k = \sequence {k_j}_{j \mathop \in J}$ with only finitely many of the $k_j$ non-zero. === Definition 3 === A '''multiindex''' is an element of $\Z^J$, the free $\Z$-module on $J$, an abelian group of rank over $\Z$ equal to the cardinality of $J$. === Modulus === {{:Definition:Multiindex/Modulus}}


Definition:Pointwise Addition of Mappings

Let $S$ be a non-empty set. Let $\struct {G, \circ}$ be a commutative semigroup. Let $G^S$ be the set of all mappings from $S$ to $G$. Then '''pointwise addition''' on $G^S$ is the binary operation $\circ: G^S \times G^S \to G^S$ (the $\circ$ is the same as for $G$) defined by: :$\forall f, g \in G^S: \forall s \in S: \map {\paren {f \circ g} } s := \map f s \circ \map g s$ The double use of $\circ$ is justified as $\struct {G^S, \circ}$ inherits all abstract-algebraic properties $\struct {G, \circ}$ might have. This is rigorously formulated and proved on Mappings to Algebraic Structure form Similar Algebraic Structure. === Pointwise Multiplication === Let $\circ$ be used with multiplicative notation. Then the operation defined above is called '''pointwise multiplication''' instead.


Definition:Ring of Polynomial Functions

Let $\struct {R, +, \circ}$ be a commutative ring with unity. Let $R \sqbrk {\set {X_j: j \in J} }$ be the ring of polynomial forms over $R$ in the indeterminates $\set {X_j: j \in J}$. Let $R^J$ be the free module on $J$. Let $A$ be the set of all polynomial functions $R^J \to R$. Then the operations $+$ and $\circ$ on $R$ induce pointwise operations on $A$. We denote these operations by the same symbols: :$\forall x \in R^J: \map {\paren {f + g} } x = \map f x + \map g x$ :$\forall x \in R^J: \map {\paren {f \circ g} } x = \map f x \circ \map g x$ The '''ring of polynomial functions''' is the resulting algebraic structure.


Definition:Arithmetic Function

An '''arithmetic function''' or '''arithmetical function''' is a complex valued function defined on the strictly positive integers. That is, a mapping: :$f: \Z_{>0} \to \C$


Definition:Category

A '''category''' is an interpretation of the metacategory axioms within set theory. Because a metacategory is a metagraph, this means that a '''category''' is a graph. Let $\mathfrak U$ be a class of sets. A metacategory $\mathbf C$ is a '''category''' {{iff}}: :$(1): \quad$ The objects form a subset $\mathbf C_0$ or $\operatorname {ob} \ \CC \subseteq \mathfrak U$ :$(2): \quad$ The morphisms form a subset $\mathbf C_1$ or $\operatorname{mor} \ \mathbf C$ or $\operatorname{Hom} \ \mathbf C \subseteq \mathfrak U$


Definition:Abuse of Notation

Mathematical notation can be considered in two ways: * As an aid to mathematical understanding, no more and no less than a useful convention to encapsulate more or less complicated ideas in a completely unambiguous format; * As the reason for mathematical effort, so as to encapsulate a truth as a documented piece of aesthetic beauty in its own right. '''Abuse of notation''' is a technique of using a system of symbology in a way different from that for which it was originally defined. Such abuse may make a train of thought more streamlined, as it is often possible to save considerable redefinition of one's terms. However, such abuse is frequently considered to be incorrect, improper and (in the eyes of many mathematicians) illegal. Philosophers of the various schools practising pragmatism tend to consider that if a notation has been adequately explained, then one should be allowed to use it in whatever way is most useful to communicate one's ideas. On the other hand, such an attitude causes indignation, rage and fury among philosophers whose attraction to mathematics is purely aesthetic.


Definition:Completion (Metric Space)

Let $M_1 = \struct {A, d}$ and $M_2 = \struct {\tilde A, \tilde d}$ be metric spaces. Then $M_2$ is a '''completion''' of $M_1$, or $M_2$ '''completes''' $M_1$, {{iff}}: :$(1): \quad M_2$ is a complete metric space :$(2): \quad A \subseteq \tilde A$ :$(3): \quad A$ is dense in $M_2$ :$(4): \quad \forall x, y \in A : \map {\tilde d} {x, y} = \map d {x, y}$. In terms of restriction of functions, this says that $\map {\tilde d {\restriction_A} } = d$. It is immediate from this definition that a '''completion''' of a metric space $M_1$ consists of: :A complete metric space $M_2$ :An isometry $\phi : A \to \tilde A$ such that $\map \phi A = \set {\map \phi x: x \in A}$ is dense in $M_2$. An isometry is often required to be bijective, so here one should consider $\phi$ as a mapping from $A$ to the image of $\phi$. Therefore to insist that $\phi$ be an isometry, ''in this context'', is to say that $\phi$ must be an injection that preserves the metric of $M_1$.


Definition:Metacategory

A '''metacategory''' is a metagraph subject to extra restrictions. As such, a '''metacategory''' $\mathbf C$ consists of: * A collection $\mathbf C_0$ of objects $X, Y, Z, \ldots$ * A collection $\mathbf C_1$ of morphisms $f, g, h, \ldots$ between its objects The morphisms of $\mathbf C$ are subjected to: {{begin-axiom}} {{axiom | n = C1         | lc= ''Composition''         | t = For objects $X, Y, Z$ and morphisms $X \stackrel {f} {\longrightarrow} Y \stackrel {g} {\longrightarrow} Z$ with $\operatorname{cod} f = \operatorname{dom} \, g$, there exists a morphism: :$g \circ f : X \to Z$ called the composition of $f$ and $g$. }} {{axiom | n = C2         | lc= ''Identity''          | t = For every object $X$, there is a morphism ${\operatorname{id}_X}: X \to X$, called the identity morphism, such that: :$f \circ {\operatorname{id}_X} = f$, and ${\operatorname{id}_X} \circ g = g$ for any object $Y$ and morphisms $f : X \to Y$ and $g : Y \to X$ }} {{axiom | n = C3         | lc= ''Associativity''         | t = For any three morphisms $f, g, h$: :$f \circ \left({g \circ h}\right) = \left({f \circ g}\right) \circ h$ whenever these compositions are defined (according to $(C1)$). }} {{end-axiom}} To describe a '''metacategory''' it is necessary to specify: :* The collection $\mathbf C_0$ of objects :* The collection $\mathbf C_1$ of morphisms :* For each object $X$, an identity morphism ${\operatorname{id}_X}: X \to X$ :* For every appropriate pair of morphisms $f, g$, the composite morphism $g \circ f$ However, the last two are often taken to be implicit when the objects and morphisms are familiar. Of course, after defining these, it is still to be shown that $(C1)$ up to $(C3)$ are satisfied. A '''metacategory''' is purely axiomatic, and does not use set theory. For example, the objects are not "elements of the set of objects", because these axioms are (without further interpretation) unfounded in set theory. For some purposes, it is convenient to have a different description of a '''metacategory'''. Two such descriptions are found on: * Characterization of Metacategory via Equations * Morphisms-Only Metacategory


Definition:Non-Archimedean

=== Non-Archimedean Norm (Vector Space) === {{:Definition:Non-Archimedean/Norm (Vector Space)}} === Non-Archimedean Norm (Division Ring) === {{:Definition:Non-Archimedean/Norm (Division Ring)}} === Non-Archimedean Metric === {{:Definition:Non-Archimedean/Metric}} Category:Definitions/Norm Theory dicdyvqyec5b4uu6ym7vxwb2mloc6j5


Definition:Orthogonal (Linear Algebra)

Let $\struct {V, \innerprod \cdot \cdot}$ be an inner product space. Let $u, v \in V$. Then $u$ and $v$ are '''orthogonal''' {{iff}}: :$\innerprod u v = 0$ === Orthogonal Set === {{:Definition:Orthogonal (Linear Algebra)/Set}} === Vectors in $\R^n$ === {{:Definition:Orthogonal (Linear Algebra)/Real Vector Space}}


Definition:Valued Field

Let $\struct {K, +, \circ}$ be a field. Let $\norm {\,\cdot\,}$ be a norm on $K$. Then $\struct {K, \norm{\,\cdot\,} }$ is a '''valued field'''.


Definition:Minkowski Functional

Let $E$ be a vector space over $\R$. A functional $p: E \to \R$ is called a '''Minkowski functional''' if it satisfies: {{begin-axiom}} {{axiom | n = 1         | q = \forall x \in E, \forall \lambda \in \R_{>0}         | ml= \map p {\lambda x}         | mo= =         | mr= \lambda \map p x         | rc= that is, $p$ is positive homogeneous }} {{axiom | n = 2         | q = \forall x, y \in E         | ml= \map p {x + y}         | mo= \le         | mr= \map p x + \map p y         | rc= that is, $p$ is sub-additive }} {{end-axiom}} {{NamedforDef|Hermann Minkowski|cat = Minkowski}} Category:Definitions/Functional Analysis Category:Definitions/Topology 4kxgn0joo47z28y6odqekq891lzuff1


Definition:Linear Functional

Let $E$ be a vector space over a field $\Bbb F$. Let $D$ be a linear subspace of $E$. A mapping $f : D \to \Bbb F$ is called a '''linear functional''' iff: :$ f \left({\alpha x + \beta y}\right) = \alpha f(x) + \beta f(y)$ holds for all $x, y$ in $L$ and for all $\alpha, \beta$ in $\Bbb F$.


Definition:Euclidean Domain

Let $\struct {R, +, \circ}$ be an integral domain with zero $0_R$. Let there exist a mapping $\nu: R \setminus \set {0_R} \to \N$ with the properties: :$(1): \quad$ For all $a, b \in R, b \ne 0_R$, there exist $q, r \in R$ with $\map \nu r < \map \nu b$, or $r = 0_R$ such that: ::::$a = q \circ b + r$ :$(2): \quad$ For all $a, b \in R, b \ne 0_R$: ::::$\map \nu a \le \map \nu {a \circ b}$ Then $\nu$ is called a '''Euclidean valuation''' and $R$ is called a '''Euclidean domain'''.


Definition:Farey Sequence

The '''Farey sequence''' is a chain of subsets of the reduced rational numbers lying in $\Q \cap \closedint 0 1$. For $Q \in \Z_{>0}$, the '''Farey set''' $F_Q$ is the set of all reduced rational numbers with denominators not larger than $Q$: :$F_Q = \set {\dfrac p q: p = 0, \ldots, Q,\ q = 1, \ldots, Q,\ p \perp q}$ where $p \perp q$ denotes that $p$ and $q$ are coprime.


Definition:Linear Representation

=== Groups === {{:Definition:Linear Representation/Group}} === Algebras === {{:Definition:Linear Representation/Algebra}}


Definition:Dimension (Representation Theory)

Let $\struct {k, +, \circ}$ be a field. Let $V$ be a vector space over $k$ of finite dimension. Let $\GL V$ be the general linear group of $V$. Let $\struct {G, \cdot}$ be a finite group. Let $\rho: G \to \GL V$ be a linear representation of $G$ on $V$. The '''dimension''' or '''degree''' of $\rho$, written $\map \deg \rho$ is the dimension of the vector space $V$. Category:Definitions/Representation Theory c66h1ktiokjtucgu3oi4bqj4i1jxurb


Definition:Faithful Linear Representation of Group

Let $ \left({k, +, \circ}\right)$ be a field. Let $V$ be a vector space over $k$ of finite dimension. Let $\operatorname{GL} \left({V}\right)$ be the general linear group of $V$. Let $\left({G, \cdot}\right)$ be a finite group. Let $\rho : G \to \operatorname{GL} \left({V}\right)$ be a linear representation of $G$ on $V$. Then $\rho$ is '''faithful''' if the kernel of $\rho$ is trivial. {{SUBPAGENAME}} 2gwgnr1htwebwxkr89qk9ty3psbwpvz


Definition:Linear Group Action

Let $\left({V, +, \cdot}\right)$ be a vector space over a field $\left({k, \oplus, \circ}\right)$. Let $G$ be a group. Let $\phi : G \times V \to V$ be an action of $G$ on $V$. Then $\phi$ is a '''(left) linear group action''' {{iff}} it is compatible with the linear structure of $V$ in the following sense: : $(1): \quad \forall v_1, v_2 \in V: g \in G: \phi \left({g, v_1 + v_2}\right) = \phi \left({g, v_1}\right) + \phi \left({g, v_2}\right)$ : $(2): \quad \forall \lambda \in k, g \in G, v \in V: \phi \left({g, \lambda \cdot v}\right) = \lambda \cdot \phi \left({g, v}\right)$ === Right Linear Group Action === {{:Definition:Linear Group Action/Right Linear Group Action}}


Definition:Group Algebra

Let $\left({k, + ,\circ}\right)$ be a field. Let $\left({G, *}\right)$ be a finite group. Then the '''group algebra''' $k G$ or $k \left[{G}\right]$ is the set of all formal sums: :$\displaystyle \sum_{g \in G} \alpha_g g\ :\ \alpha_g \in k$ That is, $k \left[{G}\right]$ is the free vector space over $k$ with basis $G$. {{SUBPAGENAME}} nne3ovn9w8raya0d71gehml4l2xsazi


Definition:Krull Dimension of Ring

Let $\struct {R, +, \circ}$ be a commutative ring with unity. The '''Krull dimension''' of $R$ is the supremum of lengths of chains of prime ideals, ordered by inclusion: :$\map {\operatorname {dim_{Krull} } } R = \sup \set {n \in \N: \exists p_0, \ldots, p_n \in \Spec R: \mathfrak p_0 \subsetneqq \mathfrak p_1 \subsetneqq \cdots \subsetneqq \mathfrak p_n}$ where $\Spec R$ is the prime spectrum of $R$. That is, the '''Krull dimension''' is $\infty$ if there exist arbitrarily long chains.


Definition:Exact Sequence of Modules

Let $\left({R, +, \cdot}\right)$ be a ring. Let: :$(1): \quad \cdots \longrightarrow M_i \stackrel{d_i} {\longrightarrow} M_{i + 1} \stackrel {d_{i + 1}} {\longrightarrow} M_{i + 2} \stackrel {d_{i + 2}} {\longrightarrow} \cdots$ be a sequence of $R$-modules $M_i$ and $R$-module homomorphisms $d_i$. Then the sequence $(1)$ is '''exact''' {{iff}}: : $\forall i: \operatorname{Im} d_i = \ker d_{i + 1}$ where $\operatorname{Im}$ and $\ker$ denote the image and kernel of mappings respectively.


Definition:Differential Module

Let $R$ be a commutative ring with unity. Let $M$ be an $R$-module. A '''differential''' on $M$ is a homomorphism $d:M\to M$ of $R$-modules such that $d^2 = d\circ d = 0$. The pair $(M,d)$ is called a '''differential module'''. {{SUBPAGENAME}} 3bgo6tem59m2851njehp4d3uddkt2p1


Definition:Graded Ring

Let $M$ be a semigroup. An '''$M$-graded ring''' is a pair $\struct {R, f}$ where: :$R$ is a ring :$f$ is a gradation of type $M$ on the additive group of $R$ which is compatible with the ring structure of $R$.


Definition:Graded Module

Let $R$ be a graded commutative ring with unity. Let $M$ be an $R$-module. We say $M$ is $\N$-'''graded''' if it has a decomposition as a direct sum of abelian groups :$\displaystyle R = \bigoplus_{n \in \N} R_n $ such that for all $m, n \in \N$ we have $R_n M_m \subseteq M_{n+m}$. Similarly, we say $M$ is $\Z$-'''graded''' if it has a decomposition as a direct sum of abelian groups :$\displaystyle R = \bigoplus_{n \in \Z} R_n $ such that for all $m,n \in \Z$ we have $R_nM_m \subseteq M_{n+m}$. {{SUBPAGENAME}} q984qkm4593t5sqg0wbyu6xtemvda2q


Definition:Differential Complex

Let $R$ be a commutative ring with unity. Let $\displaystyle M = \bigoplus_{n \in \Z} M^n$ be a $\Z$-graded $R$-module that is also a differential module with differential $d$. Then $M$ is a '''differential complex''' if the differential $d$ satisfies: :$ d \left({M^n}\right) \subseteq M^{n+1}$ for all $n \in \Z$. The notation $d_n := d \restriction_{M_n}$ is often seen.


Definition:Cohomology Groups

{{refactor|various definitions to separate}} Let $\left({M, d}\right)$ be a differential complex with grading: :$\displaystyle M = \bigoplus_{n \in \Z} M^n$ Let $d_n := d \restriction_{M_n}$. Elements of the module $M$ are called '''cochains'''. Elements of the submodule $Z^n \left({M}\right) = \ker \left({d_n}\right)$ are called '''cocycles'''. Elements of the submodule $B^n \left({M}\right) = \operatorname{Im} \left({d_{n-1}}\right)$ are called '''coboundaries'''. The modules (and hence groups) $H^n \left({M}\right) = Z^n \left({M}\right)/B^n \left({M}\right)$ are called the '''cohomology groups''' of the differential complex $\left({M, d}\right)$. Category:Definitions/Homological Algebra 8zswo67pms9v3ccmsdcahy2jkut4mak


Definition:Algebraic Number Field

An '''algebraic number field''' is a finite extension of the field of rational numbers $\Q$.


Definition:Algebraic Integer

Let $K / \Q$ be a number field. Then $\alpha \in K$ is an '''algebraic integer''' if it satisfies a monic polynomial $f \in \Z \left[{X}\right]$. The set of all '''algebraic integers''' in $K$ is denoted $\mathfrak o_K$ or $\mathcal O_K$. By Ring of Algebraic Integers it is a ring, hence usually referred to as the '''ring of algebraic integers''' of $K$. === Quadratic Integer === {{:Definition:Algebraic Integer/Quadratic}}


Definition:Field Norm

Let $K$ be a field and $L / K$ a finite field extension of $K$. Let $\alpha\in L$. === Definition 1 === {{Definition:Field Norm/Definition 1}} === Definition 2: for Galois extensions === {{Definition:Field Norm/Definition 2}}


Definition:Trace (Field Theory)

Let $K$ be a field and $L / K$ a finite field extension of $K$. Then by Vector Space on Field Extension is Vector Space, $L$ is naturally a vector space over $K$. Let $\alpha \in L$, and $\theta_\alpha$ be the linear operator: :$\theta_\alpha: L \to L: \beta \mapsto \alpha\beta$ The '''trace''' $\map {\operatorname {Tr}_{L / K} } \alpha$ of $\alpha$ is the trace of this linear operator.


Definition:Content of Polynomial

=== Integer Polynomial === {{:Definition:Content of Polynomial/Integer}} === Rational Polynomial === {{:Definition:Content of Polynomial/Rational}} === Polynomial in GCD Domain === {{:Definition:Content of Polynomial/GCD Domain}} == Commutative Ring with Unity == {{:Definition:Content of Polynomial/Commutative and Unitary Ring}}


Definition:Ordered Integral Domain

=== Definition 1 === {{:Definition:Ordered Integral Domain/Definition 1}} === Definition 2 === {{:Definition:Ordered Integral Domain/Definition 2}} An ordered integral domain can be denoted: :$\struct {D, +, \times \le}$ where $\le$ is the total ordering induced by the strict positivity property.


Definition:Strict Positivity Property

{{begin-axiom}} {{axiom | n = \text P 1         | lc= Closure under Ring Addition:         | q = \forall a, b \in D         | m = \map P a \land \map P b \implies \map P {a + b} }} {{axiom | n = \text P 2         | lc= Closure under Ring Product:         | q = \forall a, b \in D         | m = \map P a \land \map P b \implies \map P {a \times b} }} {{axiom | n = \text P 3         | lc= Trichotomy Law:         | q = \forall a \in D         | m = \paren {\map P a} \lor \paren {\map P {-a} } \lor \paren {a = 0_D} }} {{axiom | lc= For $\text P 3$, '''exactly one''' condition applies for all $a \in D$. }} {{end-axiom}}


Definition:Strict Negativity Property

Let $\struct {D, +, \times}$ be an ordered integral domain, whose (strict) positivity property is denoted $P$. The '''strict negativity property''' $N$ is defined as: :$\forall a \in D: \map N a \iff \map P {-a}$ This is compatible with the trichotomy law: :$\forall a \in D: \map P a \lor \map P {-a} \lor a = 0_D$ which can therefore be rewritten: :$\forall a \in D: \map P a \lor \map N a \lor a = 0_D$ or even: :$\forall a \in D: \map N a \lor \map N {-a} \lor a = 0_D$


Definition:Repdigit Number

Let $b \in \Z_{>1}$ be an integer greater than $1$. Let a (positive) integer $n$, greater than $b$, be expressed in base $b$. $n$ is a '''repdigit base $b$''' {{iff}} all of the digits of $n$ are the same digit. For example: :$55555$ or: :$3333_4$ that second one being in base 4.


Definition:Octal Notation

'''Octal''' is another word for base $8$. That is, every number $x \in \R$ is expressed in the form: :$\displaystyle x = \sum_{j \mathop \in \Z} r_j 8^j$ where: :$\forall j \in \Z: r_j \in \set {0, 1, 2, 3, 4, 5, 6, 7}$


Definition:Well-Ordered Integral Domain

Let $\struct {D, +, \times \le}$ be an ordered integral domain whose zero is $0_D$. === Definition 1 === {{:Definition:Well-Ordered Integral Domain/Definition 1}} === Definition 2 === {{:Definition:Well-Ordered Integral Domain/Definition 2}}


Definition:Gaussian Integral

=== Gaussian Integral of Two Variables === {{:Definition:Gaussian Integral/Two Variables}} === Gaussian Integral of One Variable === {{:Definition:Gaussian Integral/One Variable}}


Definition:Completed Riemann Zeta Function

The '''completed Riemann zeta function''' $\xi: \C \to \C$ is defined on the complex plane $\C$ as: :$\displaystyle \forall s \in \C: \map \xi x := \begin{cases} \dfrac 1 2 s \paren {s - 1} \pi^{-s/2} \map \Gamma {\dfrac s 2} \map \zeta s & : \map \Re s > 0 \\ \map \xi {1 - s} & : \map \Re s \le 0 \end{cases}$ where $\map \zeta s$ is the Riemann zeta function.


Definition:Order of Entire Function

Let $f: \C \to \C$ be an entire function. Let $\ln$ denote the natural logarithm. === Definition 1 === {{:Definition:Order of Entire Function/Definition 1}} === Definition 2 === {{:Definition:Order of Entire Function/Definition 2}} === Definition 3 === {{:Definition:Order of Entire Function/Definition 3}}


Definition:Proper Subring

A subring $S$ of $R$ is a '''proper subring of $R$''' {{iff}} $S$ is neither the null ring nor $R$ itself.


Definition:Schwarz Function

Let $f: \R \to \C$ be a function. $f$ is a '''Schwarz function''' {{iff}}: :$\forall c \in \R, n \in \N_0: \left\vert{f^{\left({n}\right)} \left({x}\right)}\right\vert = o \left({\left\vert{x}\right\vert^c}\right)$ where: : $f^{\left({n}\right)}$ denotes the $n$th derivative : $o$ is the little-o notation. {{NamedforDef|Karl Hermann Amandus Schwarz|cat = Schwarz}} Category:Definitions/Harmonic Analysis 88ei8d6x22yif1z50vprz1dhfiunrq5


Definition:Common Multiple

Let $S$ be a finite set of non-zero integers, that is: :$S = \set {x_1, x_2, \ldots, x_n: \forall k \in \N^*_n: x_k \in \Z, x_k \ne 0}$ Let $m \in \Z$ such that all the elements of $S$ divide $m$, that is: :$\forall x \in S: x \divides m$ Then $m$ is a '''common multiple''' of all the elements in $S$. {{expand|Expand the concept to general algebraic expressions, for example that $\paren {2 x + 1} \paren {x - 2}$ is a '''common multiple''' of $\paren {2 x + 1}$ and $\paren {x - 2}$}}


Definition:Ring Automorphism

Let $\struct {R, +, \circ}$ be a ring. Let $\phi: R \to R$ be a (ring) isomorphism. Then $\phi$ is a ring automorphism.


Definition:Even Function

Let $X \subset \R$ be a symmetric set of real numbers: :$\forall x \in X: -x \in X$ A real function $f: X \to \R$ is an '''even function''' {{iff}}: :$\forall x \in X: \map f {-x} = \map f x$


Definition:Odd Function

Let $X \subset \R$ be a symmetric set of real numbers: :$\forall x \in X: -x \in X$ A real function $f: X \to \R$ is an '''odd function''' {{iff}}: :$\forall x \in X: \map f {-x} = -\map f x$


Definition:Symmetric Set

Let $G$ be a group. Let $S \subseteq G$ such that: :$\forall x \in S: x^{-1} \in S$ That is, for every element in $S$, its inverse is also in $S$. Then $S$ is a '''symmetric subset of $G$''', or (if $G$ is implicit) $S$ is a '''symmetric set'''. Equivalently, $S \subseteq G$ is a '''symmetric set''' {{iff}}: :$S = S^{-1}$ where $S^{-1}$ is the inverse of $S$. === Symmetric Set of Real Numbers === {{:Definition:Symmetric Set/Real Numbers}} Category:Definitions/Group Theory l1ii5pww7bkn0tufbxo1uud41x0sdy1


Definition:Proper Divisor

Let $\struct {D, +, \circ}$ be an integral domain whose zero is $0_D$ and whose unity is $1_D$. Let $U$ be the group of units of $D$. Let $x, y \in D$. Then $x$ is a '''proper divisor''' of $y$ {{iff}}: :$(1): \quad x \divides y$ :$(2): \quad y \nmid x$ :$(3): \quad x \notin U$ That is: :$(1): \quad x$ is a divisor of $y$ :$(2): \quad x$ is not an associate of $y$ :$(3): \quad x$ is not a unit of $D$ === Integers === As the set of integers form an integral domain, the concept of a '''proper divisor''' is fully applicable to the integers. {{:Definition:Proper Divisor/Integer}}


Definition:Order of Zero

Let $f: \C \to \C$ be a complex function. Let $U \subset \C$ be such that $f$ is analytic in $U$. Let $x \in U$ be a zero of $f$. That is, let $x$ be such that $f \left({x}\right) = 0$. Let $n \in \Z_{\ge 0}$ be the least positive integer such that: : $f^{(n)} \left({x}\right) \ne 0$ where $f^{(n)}$ denotes the $n$th derivative of $f$. Then $n$ is the '''order of the zero''' at $x$. === Simple Zero === {{:Definition:Order of Zero/Simple Zero}} Category:Definitions/Complex Analysis 0p3aj8fbmmcjakfupmr6rqrd6f1we0x


Definition:Order of Pole

Let $f: \C \to \C$ be a complex function. Let $x \in U \subset \C$ be such that $f$ is analytic in $U \setminus \set x$, with a pole at $x$. By Existence of Laurent Series there is a series: :$\displaystyle \map f z = \sum_{n \mathop \ge n_0}^\infty a_j \paren {z - x}^n$ The '''order''' of the pole at $x$ is defined to be $\size {n_0} > 0$. === Simple Pole === {{:Definition:Order of Pole/Simple Pole}} Category:Definitions/Complex Analysis eget6rdxjtf764d0in5c9ejer94a5m8


Definition:Locally Uniform Convergence/Complex Functions

Let $U \subseteq \C$ be an open set. Let $\sequence {f_n}$ be a be a sequence of functions $f_n : U \to \C$. For $z \in U$, let $\map {D_r} z$ be the disk of radius $r$ about $z$. {{explain|Given that "disk" is just another term for "ball", it remains to be specified whether that ball is open or closed.}} Then $f_n$ '''converges to $f$ locally uniformly''' {{iff}}: :for each $z \in U$, there is an $r > 0$ such that $f_n$ converges uniformly to $f$ on $\map {D_r} z$ and: :$\map {D_r} z \subseteq U$


Definition:Primitive Polynomial (Ring Theory)

Let $\Q \sqbrk X$ be the ring of polynomial forms over the field of rational numbers in the indeterminate $X$. Let $f \in \Q \sqbrk X$ be such that: :$\cont f = 1$ where $\cont f$ is the content of $f$. That is: :The greatest common divisor of the coefficients of $f$ is equal to $1$. Then $f$ is described as '''primitive'''.


Definition:Vector Sum

Let $\mathbf u$ and $\mathbf v$ be vector quantities of the same physical property. === Component Definition === {{:Definition:Vector Sum/Component Definition}} === Triangle Law === {{:Definition:Vector Sum/Triangle Law}}


Definition:Linearly Dependent

Let $G$ be an abelian group whose identity is $e$. Let $R$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $\left({G, +_G, \circ}\right)_R$ be a unitary $R$-module. === Sequence === {{:Definition:Linearly Dependent/Sequence}} === Set === {{:Definition:Linearly Dependent/Set}}


Definition:Valuation Ring

Let $\struct {D, +, \circ}$ be a integral domain. Let $K$ be the field of quotients of $D$. Let $K$ be such that: :for all $x \in K$, either $x \in D$ or $x^{-1} \in D$. Then $D$ is a '''valuation ring''' of $K$.


Definition:Tychonoff Topology

Let $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set. Let $X$ be the cartesian product of $\family {X_i}_{i \mathop \in I}$: :$\displaystyle X := \prod_{i \mathop \in I} X_i$ For each $i \in I$, let $\pr_i: X \to X_i$ denote the $i$th projection on $X$: :$\forall \family {x_j}_{j \mathop \in I} \in X: \map {\pr_i} {\family {x_j}_{j \mathop \in I} } = x_i$ The '''Tychonoff topology''' on $X$ is defined as the initial topology $\tau$ on $X$ with respect to $\family {\pr_i}_{i \mathop \in I}$.


Definition:Fréchet Space (Topology)

{{:Definition:Fréchet Space (Topology)/Definition 1}}


Definition:Regular Space

$\struct {S, \tau}$ is a '''regular space''' {{iff}}: :$\struct {S, \tau}$ is a $T_3$ space :$\struct {S, \tau}$ is a $T_0$ (Kolmogorov) space.


Definition:Normal Space

$\struct {S, \tau}$ is a '''normal space''' {{iff}}: :$\struct {S, \tau}$ is a $T_4$ space :$\struct {S, \tau}$ is a $T_1$ (Fréchet) space.


Definition:T3 Space

{{:Definition:T3 Space/Definition 1}}


Definition:T4 Space

{{:Definition:T4 Space/Definition 1}}


Let $T = \left({X, \tau}\right)$ be a topological space. Let $x, y \in X$. Then $x$ and $y$ are '''topologically distinguishable''' {{iff}} they do not have exactly the same neighborhoods. That is, either: :$\exists U \in \tau: x \in U \subseteq N_x \subseteq X: y \notin N_x$ or: :$\exists V \in \tau: y \in V \subseteq N_y \subseteq X: x \notin N_y$ or both. That is, at least one of the points $x$ and $y$ has a neighborhood that is not a neighborhood of the other. === Topologically Indistinguishable === {{:Definition:Topologically Distinguishable/Indistinguishable}} Category:Definitions/Separation Axioms Category:Definitions/Distinct r06vk8aij0zkgl3mnwiy71x8lu4q3ez


Definition:T5 Space

{{:Definition:T5 Space/Definition 1}}


Definition:Completely Normal Space

$\struct {S, \tau}$ is a '''completely normal space''' {{iff}}: :$\struct {S, \tau}$ is a $T_5$ space :$\struct {S, \tau}$ is a $T_1$ (Fréchet) space.


Definition:Separated by Neighborhoods

Let $T = \left({S, \tau}\right)$ be a topological space. === Sets === {{:Definition:Separated by Neighborhoods/Sets}} === Points === {{:Definition:Separated by Neighborhoods/Points}} Category:Definitions/Separation Axioms Category:Definitions/Separated by Neighborhoods kd1ha0mqgbk3v94ybukvzye15f6cscn


Definition:Urysohn Space

$\struct {S, \tau}$ is an '''Urysohn space''' {{iff}}: :For any distinct elements $x, y \in S$ (that is, $x \ne y$), there exists an Urysohn function for $\set x$ and $\set y$.


Let $T = \left({S, \tau}\right)$ be a topological space. === Sets === {{:Definition:Separated by Closed Neighborhoods/Sets}} === Points === {{:Definition:Separated by Closed Neighborhoods/Points}}


Definition:Completely Hausdorff Space

{{:Definition:Completely Hausdorff Space/Definition 1}}


Definition:Urysohn Function

Let $T = \struct {S, \tau}$ be a topological space. Let $A, B \subseteq S$ such that $A \cap B = \O$. Let $f: S \to \closedint 0 1$ be a continuous mapping where $\closedint 0 1$ is the closed unit interval. Then $f$ is an '''Urysohn function for $A$ and $B$''' {{iff}}: :$f {\restriction_A} = 0, f {\restriction_B} = 1$ that is: :$\forall a \in A: \map f a = 0$ :$\forall b \in B: \map f b = 1$


Definition:Dyadic Rational

A '''dyadic rational''' is a rational number whose denominator, when expressed in canonical form, is an integral power of $2$. That is, $a \in \Q$ is '''dyadic''' iff: :$\exists p \in \Z, q \in \Z_{>0}: a = \dfrac p {2^q}$ Category:Definitions/Rational Numbers gq5146yf091y149gjw1gjapawaz37on


Definition:Tychonoff Space

$\struct {S, \tau}$ is a '''Tychonoff Space''' or '''completely regular space''' {{iff}}: :$\struct {S, \tau}$ is a $T_{3 \frac 1 2}$ space :$\struct {S, \tau}$ is a $T_0$ (Kolmogorov) space.


Definition:Limit Point/Topology/Point

Let $a \in S$. A point $x \in S, x \ne a$ is a '''limit point of $a$''' {{iff}} every open neighborhood of $x$ contains $a$. That is, it is a limit point of the singleton $\set a$.


Definition:Expansion of Topology

Let $S$ be a set. Let $\tau_1$ and $\tau_2$ be topologies on $S$ such that $\tau_1 \subseteq \tau_2$. Then $\tau_2$ is an '''expansion''' of $\tau_1$.


Definition:T3 1/2 Space

$\struct {S, \tau}$ is a '''$T_{3 \frac 1 2}$ space''' {{iff}}: :For any closed set $F \subseteq S$ and any point $y \in S$ such that $y \notin F$, there exists an Urysohn function for $F$ and $\set y$.


Definition:Separated by Function

Let $\left({X, \vartheta}\right)$ be a topological space. Let $A, B \subseteq X$. Then $A$ and $B$ are '''separated by function''' iff there exists an Urysohn function for $A$ and $B$. $A$ and $B$ may well be singleton sets $A = \left\{{a}\right\}, B = \left\{{b}\right\}$. In this case $a$ and $b$ are '''separated by function''' iff $A$ and $B$ are '''separated by function'''. {{SUBPAGENAME}} o11tt4onkwpoxop44h4ziff3vtb20rt


Definition:Perfectly T4 Space

$T$ is a '''perfectly $T_4$ space''' {{iff}}: :$(1): \quad T$ is a $T_4$ space :$(2): \quad$ Every closed set in $T$ is a $G_\delta$ set. That is: :Every closed set in $T$ can be written as a countable intersection of open sets of $T$.


Definition:Perfectly Normal Space

$\struct {S, \tau}$ is a '''perfectly normal space''' {{iff}}: :$\struct {S, \tau}$ is a perfectly $T_4$ space :$\struct {S, \tau}$ is a $T_1$ (Fréchet) space.


Definition:Semiregular Space

$\struct {S, \tau}$ is a '''semiregular space''' {{iff}}: :$\struct {S, \tau}$ is a Hausdorff ($T_2$) space :The regular open sets of $T$ form a basis for $T$.


Definition:Almost All

=== Measure Space === {{:Definition:Almost Everywhere}} === Set Theory === {{:Definition:Almost All/Set Theory}}


Definition:Finite Intersection Axiom

Let $T = \struct {S, \tau}$ be a topological space. Let $T$ be such that: :Every set $V_\alpha$ of closed sets of $T$ such that $\displaystyle \bigcap V_\alpha = \O$ contains a finite subset $V_\beta \subseteq V_\alpha$ such that $\displaystyle \bigcap V_\beta = \O$. Then $T$ satisfies the '''finite intersection axiom'''.


Definition:Sigma-Compact Space

Let $T = \struct {S, \tau}$ be a topological space. $T$ is '''$\sigma$-compact''' {{iff}} $S$ is the union of the underlying sets of countably many compact subspaces of $T$.


Definition:Countably Compact Space

=== Definition 1 === {{:Definition:Countably Compact Space/Definition 1}} === Definition 2 === {{:Definition:Countably Compact Space/Definition 2}} === Definition 3 === {{:Definition:Countably Compact Space/Definition 3}} === Definition 4 === {{:Definition:Countably Compact Space/Definition 4}} === Definition 5 === {{:Definition:Countably Compact Space/Definition 5}}


Definition:Countable Finite Intersection Axiom

Let $T = \struct {S, \tau}$ be a topological space. Let $T$ be such that: :Every countable set $V_\alpha$ of closed sets of $T$ such that $\displaystyle \bigcap V_\alpha = \O$ contains a finite subset $V_\beta \subseteq V_\alpha$ such that $\displaystyle \bigcap V_\beta = \O$. Then $T$ satisfies the '''countable finite intersection axiom'''.


Definition:Weakly Countably Compact Space

Let $T = \struct {S, \tau}$ be a topological space. $T$ is '''weakly countably compact''' {{iff}} every infinite subset of $S$ has a limit point in $S$.


Definition:Pseudocompact Space

Let $T = \struct {S, \tau}$ be a topological space. $T$ is '''pseudocompact''' {{iff}} every continuous real-valued function on $S$ is bounded.


Definition:Locally Compact Space

Let $T = \left({S, \tau}\right)$ be a topological space. Then $T$ is '''locally compact''' {{iff}}: : every point of $S$ has a neighborhood basis $\mathcal B$ such that all elements of $\mathcal B$ are compact.


Let $T = \left({S, \tau}\right)$ be a topological space. === Definition 1 === {{Definition:Strongly Locally Compact Space/Definition 1}} === Definition 2 === {{Definition:Strongly Locally Compact Space/Definition 2}}


Definition:Convergent Sequence/Topology

{{:Definition:Convergent Sequence/Topology/Definition 1}}


Definition:Convergent Sequence/Analysis

=== Real Numbers === {{:Definition:Convergent Sequence/Real Numbers}} === Rational Numbers === {{:Definition:Convergent Sequence/Rational Numbers}} === Complex Numbers === {{:Definition:Convergent Sequence/Complex Numbers/Definition 1}}


Definition:Weakly Sigma-Locally Compact Space

Let $T = \struct {S, \tau}$ be a topological space. Then $T$ is '''weakly $\sigma$-locally compact''' {{iff}}: :$T$ is $\sigma$-compact :$T$ is weakly locally compact.


Definition:Countable Chain Condition

Let $T = \struct {S, \tau}$ be a topological space. $T$ satisfies the '''countable chain condition''' {{iff}} every disjoint set of open sets of $T$ is countable.


Definition:Refinement of Cover

Let $S$ be a set. Let $\UU = \set {U_\alpha}$ and $\VV = \set {V_\beta}$ be covers of $S$. Then $\VV$ is a '''refinement''' of $\UU$ {{iff}}: :$\forall V_\beta \in \VV: \exists U_\alpha \in \UU: V_\beta \subseteq U_\alpha$ That is, {{iff}} every element of $\VV$ is the subset of some element of $\UU$. === Finer Cover === {{:Definition:Refinement of Cover/Finer Cover}} === Coarser Cover === {{:Definition:Refinement of Cover/Coarser Cover}}


Definition:Point Finite

Let $S$ be a set. Let $\CC$ be a set of subsets of $S$. Then $\CC$ is '''point finite''' {{iff}} each element of $S$ is an element of finitely many sets in $\CC$: :$\forall s \in S: \card {\set {C \in \CC: s \in C} } < \infty$


Definition:Locally Finite Cover

Let $T = \struct {S, \tau}$ be a topological space. Let $\CC$ be a cover of $S$. Then $\CC$ is '''locally finite''' {{iff}} each element of $S$ has a neighborhood which intersects a finite number of sets in $\CC$.


Definition:Star (Topology)

Let $S$ be a set. Let $\CC$ be a cover for $S$. Let $x \in S$. The '''star of $x$ (with respect to $\CC$)''' is defined as: :$\displaystyle x^* := \bigcup \set {U \in \CC: x \in U}$ That is, the union of all sets in $\CC$ which contain $x$.


Definition:Star Refinement

Let $S$ be a set. Let $\CC$ be a cover for $S$. Let $\VV$ be a cover for $S$ such that: :$ \forall x \in S: \exists U \in \CC: x^* \subseteq U$ where $x^*$ is the star of $x$ with respect to $\VV$. That is: :$\displaystyle x^* := \bigcup \set {V \in \VV: x \in V}$ the union of all sets in $\VV$ which contain $x$. Then $\VV$ is a '''star refinement''' of $\CC$.


Definition:Multiplicity (Complex Analysis)

Let $f: \C \to \C$ be a function. Suppose there is $a \in \C$ such that $\map f a = 0$. Then $a$ is said to be a '''zero of multiplicity $k$''' if there exists non-zero $L \in \R$ such that: :$\displaystyle \lim_{z \mathop \to a} \dfrac {\cmod {\map f z} } {\cmod {z - a}^k} = L$ {{definition wanted|Needs to be expanded so as to be relevant to other fields, not just $\C$.}} Category:Definitions/Complex Analysis muauvhfddwgkk8yblkjl4b26455wrai


Definition:Dirichlet Convolution

Let $f, g$ be arithmetic functions. === Definition 1 === {{:Definition:Dirichlet Convolution/Definition 1}} === Definition 2 === {{:Definition:Dirichlet Convolution/Definition 2}}


Definition:Unit Arithmetic Function

The '''unit arithmetic function''' $u: S \to \Z$ is defined by $u \left({n}\right) = 1$ for all $n$: :$\forall n \in S: u \left({n}\right) = 1$ where $S$ is (in theory) any set, but in this context is usually one of the standard number sets $\Z, \Q, \R, \C$.


Definition:Identity Arithmetic Function

The '''identity arithmetic function''' $\iota: S \to \Z$ is defined for $n \geq 1$ by: :$\forall n \in S: \map \iota n = \delta_{n 1}$ where: :$S$ is (in theory) any set, but in this context is usually one of the standard number sets $\Z, \Q, \R, \C$. :$\delta$ is the Kronecker delta. That is: :$\forall n \in S: \map \iota n = \begin {cases} 1 & : n = 1\\ 0 & : n \ne 1 \end {cases}$


Definition:Metacompact Space

Let $T = \struct {S, \tau}$ be a topological space. $T$ is '''metacompact''' {{iff}} every open cover of $S$ has an open refinement which is point finite.


Definition:Paracompact Space

Let $T = \struct {S, \tau}$ be a topological space. $T$ is '''paracompact''' {{iff}} every open cover of $S$ has an open refinement which is locally finite.


Definition:Fully T4 Space

Let $T = \struct {S, \tau}$ be a topological space. $T$ is '''fully $T_4$''' {{iff}} every open cover of $S$ has a star refinement.


Definition:Fully Normal Space

Let $T = \struct {S, \tau}$ be a topological space. $T$ is '''fully normal''' {{iff}}: :Every open cover of $S$ has a star refinement :All points of $T$ are closed.


Definition:Countably Metacompact Space

Let $T = \struct {S, \tau}$ be a topological space. $T$ is '''countably metacompact''' {{iff}} every countable open cover of $S$ has an open refinement which is point finite.


Definition:Countably Paracompact Space

Let $T = \struct {S, \tau}$ be a topological space. $T$ is '''countably paracompact''' {{iff}} every countable open cover of $S$ has an open refinement which is locally finite.


Definition:Open Refinement

Let $T = \struct {X, \tau}$ be a topological space. Let $\UU$ and $\VV$ be covers of $X$. Then $\VV$ is an '''open refinement''' of $\UU$ {{iff}}: :$(1): \quad \forall V \in \VV: \exists U \in \UU: V \subseteq U$ :$(2): \quad \VV \subseteq \tau$ That is: :$(1): \quad \VV$ is a refinement of $\UU$ :$(2): \quad$ All elements of $\VV$ are open in $T$ Category:Definitions/Covers c777nta7lbjehhhn3u0t0lntbpkebx7


Definition:Little-Omega

Let $f$ and $g$ be real functions. Then: :$f \left({n}\right) \in \omega \left({g \left({n}\right)}\right)$ is equivalent to: :$\displaystyle \lim_{n \to \infty} {\frac{f \left({n}\right)} {g \left({n}\right)}} = \infty$ {{refactor|Extract this into a separate page with proof}} A function $f$ is $\omega \left({g}\right)$ iff $f$ is not $\mathcal O \left({g}\right)$ where $\mathcal O$ is the big-O notation.


Definition:Dirichlet Character

Let $q \in \Z_{>1}$. Let $\paren {\Z / q \Z}$ denote the ring of integers modulo $q$. Let $G = \paren {\Z / q \Z}^\times$ be the group of units of $\paren {\Z / q \Z}$. Let $\C^\times$ be the group of units of $\C$. A '''Dirichlet character modulo q''' is a group homomorphism: :$\chi: G \to \C^\times$ {{refactor|Presumably the following statement is part of the justification for the generalised definition, which needs to be extracted and put into its own page.}} By Reduced Residue System under Multiplication forms Abelian Group, $a + q \Z \in G$ {{iff}} $\map \gcd {a, q} = 1$. It is standard practice to extend $\chi$ to a function on $\Z$ by setting: :$\map \chi A = \begin{cases} \map \chi {a + q \Z} & : \map \gcd {a, q} = 1 \\ 0 & : \text{otherwise} \end{cases}$ {{explain|The nature of $\chi$ when invoked is not apparent from its notation.  How does one understand, when encountering an instance of $\chi$ whether it means the mapping from $G$ (for some $q$) of just $\Z$?}} === Primitive Character === Let $\chi_0$ be the trivial character modulo $q$. Let $q^*$ be the least divisor of $q$ such that: : $\chi = \chi_0 \chi^*$ where $\chi^*$ is some character modulo $q^*$. If $q = q^*$ then $\chi$ is called '''primitive''', otherwise $\chi$ is '''imprimitive'''.


Definition:Dirichlet L-function

Let $\chi : \left({\Z / q \Z}\right)^\times \to \C^\times$ be a Dirichlet character. {{explain|the notation $\left({\Z / q \Z}\right)^\times$}} A '''Dirichlet $L$-function (associated to $\chi$)''' is a Dirichlet series: :$\displaystyle L \left({s, \chi}\right) = \sum_{n \mathop \ge 1} \chi \left({n}\right) n^{-s}$ for all $s \in \C$ such that the sum converges.


Definition:Dirichlet Density

Let $\mathcal P$ be a set of prime numbers. For $s \in \C$, let $\displaystyle f \left({s}\right) = \sum_{p \mathop \in \mathcal P}\: p^{-s}$. $S$ has '''Dirichlet density''' $\alpha$ {{iff}}: :$\displaystyle \lim_{s \mathop \to 1^+} \left\{{\frac {f \left({s}\right)} {\ln \left({s - 1}\right)}}\right\} = - \alpha$ where $1^+$ indicates a limit from above along the real line. {{NamedforDef|Johann Peter Gustav Lejeune Dirichlet|cat = Dirichlet}} Category:Definitions/Analytic Number Theory otagch3ceq5s8l44g2zapaawo3cer6m


Definition:Character (Number Theory)

Let $\left({G, +}\right)$ be a finite abelian group. Let $\left({\C_{\ne 0}, \times}\right)$ be the multiplicative group of complex numbers. A '''character''' of $G$ is a group homomorphism: :$\chi: G \to \C_{\ne 0}$


Definition:Completed Dirichlet L-Function

Let $\chi$ be a primitive Dirichlet character to the modulus $q \geq 1$. Let $\displaystyle \kappa = \frac12\left(1 - \chi(-1)\right)$. Let $\delta = 1$ if $\chi$ is the principal character, and $0$ otherwise. The '''completed Dirichlet $L$-function''' for $\chi$ is defined to be :$\displaystyle \Lambda(s,\chi) = \frac{1 + \kappa}{2}\left( s(1-s) \right)^\delta \Gamma\left( \frac{s+\kappa}2 \right)L(s,\chi)$ where $L(s,\chi)$ is the Dirichlet $L$-function for $\chi$. {{SUBPAGENAME}} sg0f5771dwwj52bt574bbvohaja0j5n


Definition:Limit Point/Filter

Let $T = \struct {S, \tau}$ be a topological space. Let $\FF$ be a filter on $S$. A point $x \in S$ is called a '''limit point of $\FF$''' {{iff}} $\FF$ is finer than the neighborhood filter of $x$.


Definition:Limit Point/Filter Basis

Let $T = \struct {S, \tau}$ be a topological space. Let $\FF$ be a filter on the underlying set $S$ of $T$. Let $\BB$ be a filter basis of $\FF$. === Definition 1 === {{:Definition:Limit Point/Filter Basis/Definition 1}} === Definition 2 === {{:Definition:Limit Point/Filter Basis/Definition 2}}


Definition:Convergent Filter Basis

Let $\struct {S, \tau}$ be a topological space. Let $\BB$ be a filter basis of a filter $\FF$ on $S$. Then $\BB$ '''converges''' to a point $x \in S$ {{iff}}: :$\forall N_x \subseteq S: \exists B \in \BB: B \subseteq N_x$ where $N_x$ is a neighborhood of $x$. That is, a filter basis is '''convergent''' to a point $x$ if every neighborhood of $x$ contains some set of that filter basis.


Definition:Integral Element of Ring Extension

Let $A$ be a commutative ring with unity. Let $R \subseteq A$ be a subring. Then $a \in A$ is said to be '''integral''' over $R$ {{iff}} is is a root of a monic nonzero polynomial over $R$.


Definition:Ring Extension

Let $R$ and $S$ be commutative rings with unity. Let $\phi : R \to S$ be a ring monomorphism. Then $\phi : R \to S$ is a '''ring extension''' of $R$.


Definition:Faithful Module

Let $R$ be a commutative ring with unity. Let $M$ be an $R$-module. Then $M$ is '''faithful''' {{iff}} its annihilator is zero.


Definition:Integral Closure

Let $A$ be an extension of a commutative ring with unity $R$. Let $C$ be the set of all elements of $A$ that are integral over $R$. Then $C$ is called the '''integral closure''' of $R$ in $A$.


Definition:Integrally Closed

=== Ring Extension === {{Definition:Integrally Closed in Ring Extension}} === Integral Domain === {{Definition:Integrally Closed Integral Domain}}


Definition:Multiplicatively Closed Subset of Ring

Let $\left({A, +, \circ}\right)$ be a ring with unity $1_A$ and zero $0_A$. Let $S \subseteq A$ be a subset. Then $S$ is '''multiplicatively closed''' {{iff}}: :$(1): \quad 1_A \in S$ :$(2): \quad x, y \in S \implies x \circ y \in S$


Definition:Localization of Ring

Let $A$ be a commutative ring with unity. Let $S \subseteq A$ be a multiplicatively closed subset of $A$. A '''localization of $A$ at $S$''' is a pair $\struct {A_S, \iota}$ where: :$A_S$ is a commutative ring with unity, the actual '''localization''' :$\iota: A \to A_S$ is a ring homomorphism, the '''localization homomorphism''' such that: :$(1): \quad \map \iota S \subseteq A_S^\times$, where $A_S^\times$ is the group of units of $A_S$ :$(2): \quad$ For every pair $\tuple {B, g}$ where: ::::$B$ is any ring with unity ::::$g: A \to B$ is a ring homomorphism such that $\map g S \subseteq B^\times$ :::there exists a unique ring homomorphism $h: A_S \to B$ such that: ::::$g = h \circ \iota$ That is, the following diagram commutes:  :325px


Definition:Prime Spectrum of Ring

Let $A$ be a commutative ring with unity. The '''prime spectrum''' or '''spectrum''' of $A$ is the set of prime ideals $\mathfrak p$ of $A$: :$\Spec A = \set {\mathfrak p \lhd A: \mathfrak p \text{ is prime} }$


Definition:Maximal Spectrum of Ring

Let $A$ be a commutative ring with unity. The '''maximal spectrum''' of $A$ is the set of maximal ideals of $A$: :$\operatorname{Max} \: \Spec A = \set {\mathfrak m \lhd A : \mathfrak m \text { is maximal} }$ where $I \lhd A$ indicates that $I$ is an ideal of $A$. The notation $\operatorname {Max} \: \Spec A$ is also a shorthand for the locally ringed space :$\struct {\operatorname {Max} \: \Spec A, \tau, \mathcal O_{\map {\operatorname {Max Spec} } A} }$ where: :$\tau$ is the Zariski topology on $\map {\operatorname {Max Spec} } A$ :$\mathcal O_{\map {\operatorname {Max Spec} } A}$ is the structure sheaf of $\map {\operatorname {Max Spec} } A$


Definition:Affine Algebraic Set

Let $k$ be a field. Let $A = k[X_1,\ldots,X_n]$ be the ring of polynomial functions in $n$ variables over $k$. Then a subset $X \subseteq k^n$ is an '''affine algebraic set''' if it is the zero locus of some set $T \subseteq A$.


Definition:Irreducible Space

=== Definition 1 === {{Definition:Irreducible Space/Definition 1}} === Definition 2 === {{Definition:Irreducible Space/Definition 2}} === Definition 3 === {{Definition:Irreducible Space/Definition 3}} === Definition 4 === {{Definition:Irreducible Space/Definition 4}} === Definition 5 === {{Definition:Irreducible Space/Definition 5}} === Definition 6 === {{Definition:Irreducible Space/Definition 6}} === Definition 7 === {{Definition:Irreducible Space/Definition 7}}


Definition:Zero Locus of Set of Polynomials

Let $I \subseteq A$ be a set. Then the '''zero locus''' of $I$ is the set: :$\map V I = \set {x \in k^n : \forall f \in I: \map f x = 0}$


Definition:Locally Euclidean Space

Let $M$ be a topological space. Let $d \in \N$ be a natural number. Then $M$ is a '''locally Euclidean space''' of dimension $d$ if each point in $M$ has an open neighbourhood homeomorphic to an open subset of Euclidean space $\R^d$. === Complex Locally Euclidean Space === {{:Definition:Locally Euclidean Space/Complex}}


Definition:Chart

Let $M$ be a locally Euclidean space, or complex locally Euclidean space of dimension $d$. A '''$d$-dimensional chart''' of $M$ is an ordered pair $\struct {U, \phi}$, where: :$U$ is an open subset of $M$ :$\phi: U \to D$ is a homeomorphism of $U$ onto an open subset $D$ of Euclidean space $\R^d$.


Definition:Differentiable Structure

Let $M$ be a topological space. Let $d$ be a natural number. Let $k \ge 1$ be a natural number. A '''$d$-dimensional differentiable structure of class $\mathcal C^k$''' on $M$ is a non-empty equivalence class of the set of $d$-dimensional $\mathcal C^k$-atlases on $M$ under the equivalence relation of compatibility.


Definition:Atlas

Let $M$ be a topological space. An '''atlas of class $C^k$ and dimension $d$''' on $M$ is a set of $d$-dimensional charts $\mathscr F = \family {\struct {U_\alpha, \phi_\alpha}: \alpha \in A}$ indexed by some set $A$ such that: :$(1): \quad \displaystyle \bigcup_{\alpha \mathop \in A} U_\alpha = M$ :$(2): \quad$ Every two charts $\struct {U, \phi}$ and $\struct {V, \psi}$ are $C^k$-compatible. === Smooth Atlas === {{:Definition:Atlas/Smooth Atlas}}


Definition:Connected Between Two Points

Let $T = \struct {S, \tau}$ be a topological space. Let $a, b \in S$. $T$ is '''connected between (the) two points $a$ and $b$''' {{iff}} each separation of $T$ includes a single open set $U \in \tau$ which contains both $a$ and $b$.


Definition:Quasicomponent

Let $T = \struct {S, \tau}$ be a topological space. Let the relation $\sim$ be defined on $T$ as follows: :$x \sim y \iff T$ is connected between the two points $x$ and $y$ That is, {{iff}} each separation of $T$ includes a single open set $U \in \tau$ which contains both $x$ and $y$. We have that $\sim$ is an equivalence relation, so from the Fundamental Theorem on Equivalence Relations, the points in $T$ can be partitioned into equivalence classes. These equivalence classes are called the '''quasicomponents''' of $T$.


Definition:Arc (Topology)

Let $T$ be a topological space. Let $\mathbb I \subset \R$ be the closed unit interval $\closedint 0 1$. Let $a, b \in T$. An '''arc from $a$ to $b$''' is a path $f: \mathbb I \to T$ such that $f$ is injective. That is, an '''arc from $a$ to $b$''' is a continuous injection $f: \mathbb I \to T$ such that $\map f 0 = a$ and $\map f 1 = b$. The mapping $f$ can be described as an '''arc (in $T$) joining $a$ and $b$'''.


Definition:Path Component

Let $T$ be a topological space.


Definition:Arc-Connected

=== Points in Topological Space === {{:Definition:Arc-Connected/Points}} === Subset of Topological Space === {{:Definition:Arc-Connected/Subset}} === Topological Space === {{:Definition:Arc-Connected/Topological Space}}


Definition:Arc Component

Let $T$ be a topological space. Let us define the relation $\sim$ on $T$ as follows: :$x \sim y \iff x$ and $y$ are arc-connected. We have that $\sim $ is an equivalence relation, so from the Fundamental Theorem on Equivalence Relations, the points in $T$ can be partitioned into equivalence classes. These equivalence classes are called the '''arc components of $T$'''.


Definition:Ring of Integers of Number Field

Let $K$ be a number field. The '''ring of integers''' of $K$, usually denoted $\mathcal O_K$ or $\mathfrak o_K$, is the integral closure of $\Z$ in $K$.


Definition:GCD Domain

A '''GCD domain''' is an integral domain in which any two non-zero elements have a greatest common divisor.


Definition:Ascending Chain Condition/Module

Let $R$ be a commutative ring with unity. Let $M$ be an $R$-module. Let $\left({D, \subseteq}\right)$ be a set of submodules of $M$ ordered by inclusion. Then $M$ is said to have the '''ascending chain condition on submodules''' {{iff}}: :''Every increasing sequence $N_1 \subseteq N_2 \subseteq N_3 \subseteq \cdots$ with $N_i \in D$ eventually stabilizes: $\exists k \in \N: \forall n \in \N, n \ge k: N_n = N_{n+1}$''


Definition:Descending Chain Condition

Let $A$ be a commutative ring with unity. Let $M$ be an $A$-module. Let $(D,\supseteq)$ be a set of submodules of $M$ ordered by inclusion. Then the hypothesis :''Every increasing sequence $N_1 \supseteq N_2 \supseteq N_3 \supseteq \cdots$ with $N_i \in D$ eventually terminates: there is $k \in \N$ such that $N_k = N_{k+1} = \cdots$'' is called the '''descending chain condition''' on the submodules in $D$.


Definition:Maximal Condition

Let $A$ be a commutative ring with unity. Let $M$ be an $A$-module. Let $(D,\subseteq)$ be the set of submodules of $M$ ordered by inclusion. Then the hypothesis :''Every non-empty subset of $D$ has a maximal element'' is called the '''maximal condition''' on submodules.


Definition:Minimal Condition

=== Ordered  set === {{Definition:Minimal Condition/Ordered Set}} === Minimal condition on subsets === Let $S$ be a set. Let $F$ be a set of subsets of $S$. Then $S$ '''satisfies the minimal condition on $F$''' {{iff}} $F$, ordered by inclusion satisfies the minimal condition. ==== Minimal condition on submodules ==== Let $A$ be a commutative ring with unity. Let $M$ be an $A$-module. Let $(D,\supseteq)$ be the set of submodules of $M$ ordered by inclusion. Then the hypothesis :''Every non-empty subset of $D$ has a minimal element'' is called the '''minimal condition''' on submodules.


Definition:Noetherian Module

Let $A$ be a commutative ring with unity. Let $M$ be an $A$-module. === Definition 1 === {{:Definition:Noetherian Module/Definition 1}} === Definition 2 === {{:Definition:Noetherian Module/Definition 2}} === Definition 3 === {{:Definition:Noetherian Module/Definition 3}}


Definition:Artinian Module

Let $A$ be a commutative ring with unity. Let $M$ be an $A$-module Then $M$ is a '''Artinian module''' if either of the following conditions hold: :$(1): \quad$ $M$ satisfies the descending chain condition on submodules :$(2): \quad$ $M$ satisfies the minimal condition on submodules.


Definition:Artinian Ring

Let $A$ be a commutative ring with unity. Then $A$ is '''Artinian''' if it is Artinian as an $A$-module {{NamedforDef|Emil Artin|cat=Artin}} Category:Definitions/Ring Theory 1816gmd3wdbxexj3pmb5o0i5lc7svrd


Definition:Prime Element of Ring

Let $R$ be a commutative ring. Let $p \in R \setminus \set 0$ be any non-zero element of $R$. Then $p$ is a '''prime element of $R$''' {{iff}}: :$(1): \quad p$ is not a unit of $R$  :$(2): \quad$ whenever $a, b \in R$ such that $p$ divides $a b$, then either $p$ divides $a$ or $p$ divides $b$.


Definition:Factorization Domain

A '''factorization domain''' is an integral domain in which every element can be written as a finite product of irreducible elements.


Definition:Primal

Let $A$ be an integral domain. An element $p \in A$ is called '''primal''' if whenever $a, b \in A$ are such that $p$ divides $a b$, then we can write $p = q_1 q_2$, with $q_1$ divides $a$ and $q_2$ divides $B$. Category:Definitions/Ring Theory Category:Definitions/Factorization pgmsxqkkoncitqb13emce2k8nz7syok


Definition:Schreier Domain

A ring $A$ is a '''Schreier domain''' if it is an integrally closed integral domain in which every non-zero element is primal. {{NamedforDef|Otto Schreier|cat = Schreier}} Category:Definitions/Ring Theory Category:Definitions/Factorization br9ia0erqzair0nwjvtxcqj2o30eg8h


Definition:Nagata Criterion

Let $A$ be a ring. The '''Nagata criterion''' reads as follows: : ''Let $S \subseteq A$ be a multiplicatively closed subset of $A$ generated by prime elements. If the localization $A_S$ is a UFD, then so is $A$''. By Localization of UFD is UFD, this is equivalent to: : ''Let $S \subseteq A$ be a multiplicatively closed subset of $A$ generated by prime elements. Then the localization $A_S$ is a UFD {{iff}} $A$ is a UFD''. {{NamedforDef|Masayoshi Nagata|cat = Nagata M}} Category:Definitions/Factorization efl4gt8fnhjlcqc6mzvqnlt3upkzmrm


Definition:Product of Ideals of Ring

Let $\left({R, +, \circ}\right)$ be a commutative ring. Let $I,J$ be ideals of $R$. === Definition 1 === The '''product of $I$ and $J$''' is the set of all finite sums: :$IJ = \{a_1 b_1 + \cdots + a_r b_r : a_i \in I, b_i \in J, r \in \N \}$ === Definition 2 === The '''product of $I$ and $J$''' is the ideal generated by their product as subsets.


Definition:Bézout Domain

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


Definition:Direct Sum of Modules

Let $A$ be a commutative ring with unity. {{explain|For this concept to be more easily understood, it is suggested that the ring be defined using its full specification, i.e. complete with operators, $\left({A, +, \circ}\right)$ and so on, and similarly that a symbol be used to make the module scalar product equally explicit.}} Let $\left\{ {M_i}\right\}_{i \in I}$ be a family of $A$-modules indexed by $I$. {{explain|Worth specifically stating exactly what $I$ is.}} Let $M = \displaystyle \prod_{i \mathop \in I} M_i$ be their direct product. The '''direct sum''' $\displaystyle \bigoplus_{i \mathop \in I} M_i$ is the submodule of $M$ the consisting of the elements of finite support.


Definition:Projective Module

Let $A$ be a ring with unity. Let $M$ be an $A$-module. $M$ is '''projective''' {{iff}} there exists an $A$-module $N$ such that the direct sum $M \oplus N$ is a free module.


Definition:Hereditary Ring

Let $A$ be a commutative ring. We say that $A$ is '''hereditary''' if for every module $M$ over $A$, if $M$ is projective, then so is each submodule of $M$.


Definition:Semihereditary Ring

Let $A$ be a commutative ring. We say that $A$ is '''semihereditary''' if for every module $M$ over $A$, if $M$ is projective, then so is each finitely generated submodule of $M$.


Definition:Prüfer Domain

Let $A$ be an integral domain. Then $A$ is a '''Prüfer domain''' if every finitely generated ideal of $A$ is projective. {{NamedforDef|Heinz Prüfer|cat = Prüfer}} Category:Definitions/Integral Domains c7x5em8c9sysqq9sni7q5de9tjhs3d9


Definition:Height (Ring Theory)

Let $A$ be a commutative ring. Let $\mathfrak p$ be a prime ideal in $A$. The '''height''' of $\mathfrak p$ is the supremum over all $n$ such that there exists a chain of prime ideals: :$\mathfrak p_0 \subsetneqq \mathfrak p_1 \subsetneqq \cdots \subsetneqq \mathfrak p$ {{explain|make this precise, as in Definition:Krull Dimension of Ring}} Category:Definitions/Ring Theory jmeuk9fqjcj18r00abrfbfsh5hs9p1t


Definition:Ultraconnected Space

=== Definition 1 === {{:Definition:Ultraconnected Space/Definition 1}} === Definition 2 === {{:Definition:Ultraconnected Space/Definition 2}} === Definition 3 === {{:Definition:Ultraconnected Space/Definition 3}}


Definition:Addition of Polynomials/Polynomial Forms

Let: : $\displaystyle f = \sum_{k \mathop \in Z} a_k \mathbf X^k$ : $\displaystyle g = \sum_{k \mathop \in Z} b_k \mathbf X^k$ be polynomials in the indeterminates $\left\{{X_j: j \in J}\right\}$ over $R$. {{explain|What is $Z$ in the above? Presumably the integers, in which case they need to be denoted $\Z$ and limited in domain to non-negative? However, because $Z$ is used elsewhere in the exposition of polynomials to mean something else (I will need to hunt around to find out exactly what), I can not take this assumption for granted.}} The operation '''polynomial addition''' is defined as: :$\displaystyle f + g := \sum_{k \mathop \in Z} \left({a_k + b_k}\right) \mathbf X^k$ The expression $f + g$ is known as the '''sum''' of $f$ and $g$.


Definition:Multiplication of Polynomials/Polynomial Forms

Let $\displaystyle f = \sum_{k \mathop \in Z} a_k \mathbf X^k$, $\displaystyle g = \sum_{k \mathop \in Z} b_k \mathbf X^k$ be polynomial forms in the indeterminates $\left\{{X_j: j \in J}\right\}$ over $R$. The '''product of $f$ and $g$''' is defined as: :$\displaystyle f \circ g := \sum_{k \mathop \in Z} c_k \mathbf X^k$  where: :$\displaystyle c_k = \sum_{\substack{p + q \mathop = k \\ p, q \mathop \in Z}} a_p b_q$


Definition:Trivial Topological Space

A '''trivial topological space''' is a topological space with only one element. The open sets of a '''trivial topological space''' $T = \left({\left\{{s}\right\}, \tau}\right)$ are $\varnothing$ and $\left\{{s}\right\}$.


Definition:Locally Connected Space

=== Definition 1: Using Local Bases === {{:Definition:Locally Connected Space/Definition 1}} === Definition 2: Using Neighborhood Bases === {{:Definition:Locally Connected Space/Definition 2}} === Definition 3: Using (Global) Basis === {{:Definition:Locally Connected Space/Definition 3}} === Definition 4: Using Open Components === {{:Definition:Locally Connected Space/Definition 4}}


Definition:Locally Path-Connected Space

=== Definition 1: using Local Bases === {{:Definition:Locally Path-Connected Space/Definition 1}} === Definition 2: using Neighborhood Bases === {{:Definition:Locally Path-Connected Space/Definition 2}} === Definition 3: using (Global) Basis === {{:Definition:Locally Path-Connected Space/Definition 3}} === Definition 4: using Open Path Components === {{:Definition:Locally Path-Connected Space/Definition 4}}


Definition:Locally Arc-Connected Space

A topological space $T = \struct {S, \tau}$ is '''locally arc-connected''' {{iff}} it has a basis consisting entirely of arc-connected sets.


Let $X$ be a topological space. Let $\AA = \set {\phi_\alpha : \alpha \in A}$ be a collection of continuous mappings $X \to \R$ such that: :$(1): \quad$ The set $\set {\map {\operatorname {supp} } {\phi_\alpha}^\circ: \alpha \in A}$ of interiors of the supports is a locally finite cover of $X$ :$(2): \quad \forall x \in X: \forall \alpha \in A: \map {\phi_\alpha} x \ge 0$ :$(3): \quad \displaystyle \forall x \in X: \sum_{\alpha \mathop \in A} \map {\phi_\alpha} x = 1$ Then $\set {\phi_\alpha: \alpha \in A}$ is a '''partition of unity''' on $X$. === Subordinate to Cover === {{:Definition:Partition of Unity (Topology)/Subordinate}} Category:Definitions/Topology j29zrl0omgdcxalwqgdxp3rcvl5o32m


Definition:Vector Space on Field Extension

Let $\struct {K, +, \times}$ be a field. Let $L / K$ be a field extension over $K$. Then $\struct {L, +, \times}_K$ is the '''vector space on $L$ over $K$'''.


=== Definition 1 === {{:Definition:Totally Pathwise Disconnected Space/Definition 1}} === Definition 2 === {{:Definition:Totally Pathwise Disconnected Space/Definition 2}}


Definition:Characteristic Polynomial

Let $K$ be a field. Let $L / K$ be a finite field extension of $K$. Then by Vector Space on Field Extension is Vector Space, $L$ is naturally a vector space over $K$. Let $\alpha \in L$, and $\theta_\alpha$ be the linear operator: :$\theta_\alpha: L \to L : \beta \mapsto \alpha \beta$ The '''characteristic polynomial''' of $\alpha$ with respect to the extension $L / K$ is: :$\det \sqbrk {X I_L - \theta_\alpha}$ where: :$\det$ denotes the determinant of a linear operator :$X$ is an indeterminate  :$I_L$ is the identity mapping on $L$.


Definition:Determinant/Linear Operator

Let $V$ be a finite-dimensional vector space over a field $K$. Let $A: V \to V$ be a linear operator of $V$. The '''determinant''' $\det \left({A}\right)$ of $A$ is defined to be the determinant of any matrix of $A$ relative to some basis.


Definition:Trace (Linear Algebra)

=== Matrix === {{:Definition:Trace (Linear Algebra)/Matrix}} === Linear Operator === {{:Definition:Trace (Linear Algebra)/Linear Operator}} Category:Definitions/Linear Algebra Category:Definitions/Matrix Algebra iwd100re67e8wjvyj3pv3ss0y9xwzxv


Definition:Totally Separated Space

=== Definition 1 === {{:Definition:Totally Separated Space/Definition 1}} === Definition 2 === {{:Definition:Totally Separated Space/Definition 2}}


Definition:Grothendieck Universe

A '''Grothendieck universe''' is a set (not a class) which has the properties expected of the universe $\mathbb U$ of sets in the sense of the Zermelo-Fraenkel axioms with the following properties: :$(1): \quad \mathbb U$ is a transitive set: if $u \in \mathbb U$ and $x \in u$ then $x \in \mathbb U$ :$(2): \quad$ If $ u, v \in \mathbb U$ then $\set {u, v} \in \mathbb U$ :$(3): \quad$ If $u \in \mathbb U$ then the power set $\powerset u \in \mathbb U$ :$(4): \quad$ If $A \in \mathbb U$, and $\set {u_\alpha: \alpha \in A}$ is a collection of elements of $\mathbb U$, then $\displaystyle \bigcup_{\alpha \mathop \in A} u_\alpha \in \mathbb U$ {{explain|Tidy up language of that last axiom -- when "collection" is used, does this mean set? If so, say so -- if not, then link to exactly what concept is meant. This area of mathematics is such that differences between sets and e.g. classes is important to distinguish, and undefined terms such as "collection" are not to be tolerated.}}


Definition:Morphism

Let $\mathbf C$ be a metacategory. A '''morphism''' of $\mathbf C$ is an object $f$, together with: * A domain $\operatorname{dom} f$, which is an object of $\mathbf C$ * A codomain $\operatorname{cod} f$, also an object of $\mathbf C$


Definition:Hom Class

Let $\mathbf C$ be a metacategory. Let $C$ and $D$ be objects of $\mathbf C$. The collection of morphisms $f: C \to D$ is called a '''hom class''' and is denoted $\operatorname{Hom}_{\mathbf C} \left({C, D}\right)$.


Definition:Small Category

Let $\mathbf C$ be a metacategory. Then $\mathbf C$ is said to be '''small''' {{iff}} both of the following hold: :The collection of objects $\mathbf C_0$ is a set; :The collection of morphisms $\mathbf C_1$ is a set.


Definition:Functor

Informally, a '''functor''' is a morphism of categories. It may be described as what one must define in order to define a '''natural transformation'''. This is formalized by defining the category of categories. === Covariant Functor === {{:Definition:Functor/Covariant}} === Contravariant Functor === {{:Definition:Functor/Contravariant}}


Definition:Composition of Morphisms

Let $\mathbf C$ be a metacategory. Let $\left({g, f}\right)$ be a pair of composable morphisms. Then the '''composition of $f$ and $g$''' is a morphism $g \circ f$ of $\mathbf C$ subject to: :$\operatorname{dom} \left({g \circ f}\right) = \operatorname{dom} f$ :$\operatorname{cod} \left({g \circ f}\right) = \operatorname{cod} g$ This '''composition of morphisms''' can be thought of as an abstraction of both composition of mappings and transitive relations.


Definition:Identity Morphism

Let $\mathbf C$ be a metacategory. Let $X$ be an object of $\mathbf C$. The '''identity morphism''' of $X$, denoted $\operatorname{id}_X$, is a morphism of $\mathbf C$ subject to: :$\operatorname{dom} \operatorname{id}_X = \operatorname{cod} \operatorname{id}_X = X$ :$f \circ \operatorname{id}_X = f$ :$\operatorname{id}_X \circ g = g$ whenever $X$ is the domain of $f$ or the codomain of $g$, respectively. In most metacategories, the '''identity morphisms''' can be viewed as a representation of "doing nothing", in a sense suitable to the metacategory under consideration.


Definition:Product Category

Let $\mathbf C$ and $\mathbf D$ be metacategories. The '''product category''' $\mathbf C \times \mathbf D$ is the category with: {{DefineCategory | ob = $\tuple {X, Y}$, for all $X \in \operatorname {ob} \mathbf C$, $Y \in \operatorname {ob} \mathbf D$ | mor = $\tuple {f, g}: \tuple {X, Y} \to \tuple {X', Y'}$ for all $f: X \to X'$ in $\mathbf C_1$ and $g: Y \to Y'$ in  $\mathbf D_1$ | comp = $\tuple {f, g} \circ \tuple {h, k} := \tuple {f \circ h, g \circ k}$, whenever this is defined | id = $\operatorname {id}_{\tuple {X, Y} } := \tuple {\operatorname {id}_X, \operatorname {id}_Y}$ }}


=== Definition 1 === {{:Definition:Extremally Disconnected Space/Definition 1}} === Definition 2 === {{:Definition:Extremally Disconnected Space/Definition 2}} === Definition 3 === {{:Definition:Extremally Disconnected Space/Definition 3}}


Definition:Zero Dimensional Space

Let $T = \struct {S, \tau}$ be a topological space. Then $T$ is zero dimensional {{iff}} it has a basis whose sets are all both closed and open.


Definition:Scattered Space

Let $T = \struct {S, \tau}$ be a topological space. === Definition 1 === {{:Definition:Scattered Space/Definition 1}} === Definition 2 === {{:Definition:Scattered Space/Definition 2}}


Definition:Operator Domain

An operator domain is a notational tool for indexing a collection of operations on a set. Formally, an '''operator domain''' is a set $\Omega$  together with a mapping $a : \Omega \to \N_0$. We also use the notation, for $n \in \N_0$, $\Omega(n) = \{\omega \in \Omega : a(\omega) = n \}$. The elements of $\Omega$ are '''operators'''. If $\omega \in \Omega$, then $a(\omega)$ is the '''arity''' of the operator $\omega$. The elements of $\Omega(n)$ are the '''$n$-ary''' operators of $\Omega$. Note that an operator domain is an abstract labelling set, so the definitions here are purely formal; we have not identified any concrete operations. The latter is done using an Omega algebra. {{SUBPAGENAME}} 16yueicrvs4xcp7n0b9p6dmg2elbm7e


Definition:Biconnected Set

Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$ be a connected set in $T$. Then $H$ is '''biconnected''' {{iff}} it is not the union of two disjoint non-degenerate connected sets.


Definition:Cut Point

Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$ be a connected set in $T$ and let $p \in H$. Let $p \in H$ such that $H \setminus \set p$ is disconnected, where $\setminus$ denotes set difference. Then $p$ is a cut point of $H$.


Definition:Dispersion Point

Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$ be a connected set in $T$ and let $p \in H$. Let $p \in H$ such that $H \setminus \set p$ is totally disconnected, where $\setminus$ denotes set difference. Then $p$ is a dispersion point of $H$.


Definition:Continuum (Topology)

Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$ be both compact and connected in $T$. Then $H$ is a '''continuum'''.


Definition:Degenerate Continuum

Let $T = \left({S, \tau}\right)$ be a topological space. A '''degenerate continuum''' of $T$ is a continuum in $T$ containing exactly one element. === Non-Degenerate Continuum === {{:Definition:Degenerate Continuum/Non-Degenerate}} Category:Definitions/Continua nt93zsnk4vlipk91bmmucao1s968w4j


Definition:Subcontinuum

Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$ be a continuum in $T$. Let $K \subseteq H$ also be a continuum in $T$. Then $K$ is a '''subcontinuum''' of $H$. === Proper Subcontinuum === {{:Definition:Subcontinuum/Proper Subcontinuum}}


Definition:Indecomposable Continuum

Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$ be a continuum in $T$. Then $H$ is indecomposable {{iff}} it is not the union of two different non-degenerate proper subcontinua.


Definition:Composant

Let $T = \left({S, \tau}\right)$ be a topological space. Let $H \subseteq S$ be a continuum in $T$. Let $C \subseteq H$ be a subset of $H$. === Composant of a Continuum === {{:Definition:Composant/Continuum}} === Composant of a Point === {{:Definition:Composant/Point}} Category:Definitions/Continua aho2kjjba1m5v0etttg9qp3d19e5alq


Definition:Punctiform Space

Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$ be a subset of $T$. Then $H$ is punctiform if it contains no non-degenerate continua.


Definition:Metric Space Axioms

{{begin-axiom}} {{axiom | n = \text M 1         | q = \forall x \in A         | m = \map d {x, x} = 0 }} {{axiom | n = \text M 2         | q = \forall x, y, z \in A         | m = \map d {x, y} + \map d {y, z} \ge \map d {x, z} }} {{axiom | n = \text M 3         | q = \forall x, y \in A         | m = \map d {x, y} = \map d {y, x} }} {{axiom | n = \text M 4         | q = \forall x, y \in A         | m = x \ne y \implies \map d {x, y} > 0 }} {{end-axiom}}


Definition:Quasimetric

A '''quasimetric''' on a set $X$ is a real-valued function $d: A \times A \to \R$ which satisfies the following conditions: {{begin-axiom}} {{axiom | n = \text M 1         | q = \forall x \in A         | m = \map d {x, x} = 0 }} {{axiom | n = \text M 2         | q = \forall x, y, z \in A         | m = \map d {x, y} + \map d {y, z} \ge \map d {x, z} }} {{axiom | n = \text M 4         | q = \forall x, y \in A         | m = x \ne y \implies \map d {x, y} > 0 }} {{end-axiom}} Note the numbering system of these conditions. They are numbered this way so as to retain consistency with the metric space axioms, of which these are a subset. The difference between a '''quasimetric''' and a metric is that a '''quasimetric''' does not insist that the distance function between distinct elements is commutative, that is, that $\map d {x, y} = \map d {y, x}$. === Quasimetric Space === {{:Definition:Quasimetric/Quasimetric Space}}


Definition:Trivial Character

Let $G$ be a finite abelian group. The character $\chi_0: G \to \C_{\ne 0}$ defined as: :$\forall g \in G: \chi_0 \left({g}\right) = 1$ is the '''trivial character on $G$'''.


Definition:Bounded Metric Space

Let $M = \struct {A, d}$ be a metric space. Let $M' = \struct {B, d_B}$ be a subspace of $M$. === Definition 1 === {{:Definition:Bounded Metric Space/Definition 1}} === Definition 2 === {{:Definition:Bounded Metric Space/Definition 2}}


Definition:Indicator of Group Element

Let $G$ be a finite group. Let $a \in G$. Let $H$ be a subgroup of $G$. The '''indicator of $a$ in $H$''' is the least strictly positive integer $n$ such that $a^n \in H$.


Definition:Nested Sequence

Let $S$ be a set. Let $\SS = \powerset S$ be the power set of $S$. Let $\family {S_k}_{k \mathop \in \N}$ indexed family of subsets of $S$ such that either: :$\forall k \in \N: S_k \subseteq S_{k + 1}$ or: :$\forall k \in \N: S_k \supseteq S_{k + 1}$ Then $\family {S_k}_{k \mathop \in \N}$ is a '''nested sequence''' of sets.


Definition:Topologically Complete Space

Let $T = \struct {S, \tau}$ be a topological space. Let $M = \struct {S, d}$ be a complete metric space such that $\struct {S, \tau}$ is the topological space induced by $d$. If there exists such a complete metric space, then $T$ is described as '''topologically complete'''.


Definition:Sigma-Locally Finite Basis

Let $T = \struct {S, \tau}$ be a topological space. A '''$\sigma$-locally finite basis''' for $T$ is a basis which is the countable union of locally finite covers. {{explain|Is this the same thing as "countably locally finite"?}}


Definition:Quasiuniformity

Let $S$ be a set. A '''quasiuniformity''' on $S$ is a set of subsets $\UU$ of the cartesian product $S \times S$ satisfying the following '''quasiuniformity axioms''': {{:Definition:Quasiuniformity Axioms}} That is, a '''quasiuniformity''' on $S$ is a filter on the cartesian product $S \times S$ (from $(\text U 1)$ to $(\text U 3)$) which also fulfils the condition: :$\forall u \in \UU: \exists v \in \UU$ such that whenever $\tuple {x, y} \in v$ and $\tuple {y, z} \in v$, then $\tuple {x, z} \in u$ which can be seen to be an equivalent statement to $(\text U 4)$. $u \circ v$ in this context can be seen to be equivalent to composition of relations. Thus a '''quasiuniformity''' on $S$ is a filter on $S \times S$ which also fulfils the condition that every element is the composition of another element with itself.


Definition:Uniformity

Let $S$ be a set. A '''uniformity''' on $S$ is a set of subsets $\UU$ of the cartesian product $S \times S$ satisfying the '''quasiuniformity axioms''': {{:Definition:Quasiuniformity Axioms}} and also: :$(\text U 5): \forall u \in \UU: u^{-1} \in \UU$ where $u^{-1}$ is defined as: ::$u^{-1} := \set {\tuple {y, x}: \tuple {x, y} \in u}$ :That is, all elements of $\UU$ are symmetric.


Definition:Entourage

Let $S$ be a set. Let $\UU \subseteq \powerset {S \times S}$ be a quasiuniformity on $S$. An element $u \in \UU$ of such a quasiuniformity is known as an '''entourage''' of $\UU$'''. Note that an entourage of $\UU$ actually ''is'' a relation on $S$.


Definition:Separated Quasiuniformity

Let $\UU$ be a quasiuniformity on a set $S$. Then $\UU$ is '''separated''' {{iff}}: :$\forall u, v \in \UU: u \cap v = \Delta_S$ That is, {{iff}} the intersection of all its entourages is the diagonal relation.


Definition:Symmetric Filter Basis

Let $S$ be a set. Let $\UU$ be a quasiuniformity on $S$. From the definition, a quasiuniformity on $S$ is also a filter on the cartesian product $S \times S$. Let $\BB \subset \powerset {S \times S}$ be a filter basis of $\UU$. Then $\BB$ is a '''symmetric filter basis''' of $\UU$ {{iff}} every element of $\BB$ is symmetric.


Definition:Inverse Entourage

Let $S$ be a set. Let $\UU$ be a quasiuniformity on $S$. Let $u \in \UU$ be an entourage of $\UU$. Then the '''inverse entourage''' $u^{-1}$ is defined as: :$u^{-1} := \set {\tuple {y, x}: \tuple {x, y} \in u}$


Definition:Symmetric Entourage

Let $S$ be a set. Let $\UU$ be a quasiuniformity on $S$. Let $u \in \UU$ be an entourage of $\UU$ such that: :$u = u^{-1}$ where $u^{-1}$ is the inverse of $u$. Then $u$ is '''symmetric'''.


Definition:Quasiuniform Space

Let $\UU$ be a quasiuniformity on a set $S$. Then a topology $\tau$ can be created from $\UU$ by: :$\tau := \set {\map u x: u \in \UU, x \in S}$ where: :$\forall x \in S: \map u x := \set {y: \tuple {x, y} \in u}$ The resulting topological space $T = \struct {S, \tau}$ is called a quasiuniform space. It can be denoted $\struct {\struct {S, \UU}, \tau}$, or just $\struct {S, \UU}$ if it is understood that $\tau$ is the topology created from $\UU$.


Definition:Compatible Quasiuniformities

Let $\UU_1$ and $\UU_2$ be quasiuniformities on a set $S$. Let $\struct {\struct {S, \UU_1}, \tau_1}$ and $\struct {\struct {S, \UU_2}, \tau_2}$ be the quasiuniform spaces generated by $\UU_1$ and $\UU_2$. Then $\UU_1$ and $\UU_2$ are '''compatible (with each other)''' {{iff}} their topologies are equal. That is, {{iff}} $\tau_1 = \tau_2$.


Definition:Quasiuniformizable Space

Let $T = \struct {S, \tau}$ be a topological space. Then $T$ is '''quasiuniformizable''' if there exists a quasiuniformity $\UU$ on $S$ such that $\struct {\struct {S, \UU}, \tau}$ is a quasiuniform space.


Definition:Uniform Space

Let $\UU$ be a uniformity on a set $S$. The quasiuniform space $\struct {\struct {S, \UU}, \tau}$ generated from $\UU$ is a '''uniform space'''.


Definition:Uniformizable Space

Let $T = \struct {S, \tau}$ be a topological space. Then $T$ is '''uniformizable''' if there exists a uniformity $\UU$ on $S$ such that $\struct {\struct {S, \UU}, \tau}$ is a uniform space.


Definition:Metrizable Uniformity

Let $M = \struct {A, d}$ be a metric space. Let $\UU$ be the uniformity on $X$ defined as: :$\UU := \set {u_\epsilon: \epsilon \in \R_{>0} }$ where: :$\R_{>0}$ is the set of strictly positive real numbers :$u_\epsilon$ is defined as: ::$u_\epsilon := \set {\paren {x, y}: \map d {x, y} < \epsilon}$ Then $\UU$ is defined as '''metrizable'''.


Definition:Superfunction

Let $C, D \subseteq \C$ with $z \in C \implies z + 1 \in C$. Let $F: C \to D$ and $H: D \to D$ be holomorphic functions. Let $\map H {\map F z} = \map F {z + 1}$ for all $z \in C$. Then $F$ is said to be a '''superfunction''' of $H$, and $H$ is called a '''transfer function''' of $F$.