Definition:Simultaneous Equations/Solution Set

Consider the system of $m$ simultaneous equations in $n$ variables: :$\mathbb S := \forall i \in \set {1, 2, \ldots, m} : \map {f_i} {x_1, x_2, \ldots x_n} = \beta_i$ Let $\mathbb X$ be the set of ordered $n$-tuples: :$\set {\sequence {x_j}_{j \mathop \in \set {1, 2, \ldots, n} }: \forall i \in \set {1, 2, \ldots, m}: \map {f_i} {\sequence {x_j} } = \beta_i}$ which satisfies each of the equations in $\mathbb S$. Then $\mathbb X$ is called the '''solution set''' of $\mathbb S$.


Definition:Order Completion

Let $\struct {S, \preceq_S}$ be an ordered set. An ordered set $\struct {T, \preceq_T}$ is an '''order completion of $S$''' {{iff}}: :$(1):\quad S \subseteq T$ :$(2):\quad {\preceq_T \restriction_S} = {\preceq_S}$, where $\restriction$ denotes restriction :$(3):\quad \struct {T, \preceq_T}$ is a complete ordered set :$(4):\quad$ For all ordered sets $\struct {T', \preceq_{T'} }$ satisfying $(1), (2)$ and $(3)$, there is a unique order-preserving injection $\phi: T' \to T$


Definition:Complete Lattice/Definition 2

Let $\struct {S, \preceq}$ be an ordered set. Then $\struct {S, \preceq}$ is a '''complete lattice''' {{iff}}: :$\forall S' \subseteq S: \inf S', \sup S' \in S$ That is, {{iff}} all subsets of $S$ have both a supremum and an infimum.


Definition:Left Null Space

Let $R$ be a ring. Let $\mathbf A$ be a matrix in the matrix space $\mathcal M_{m, n}\left({R}\right)$. Let $\mathbf A^\intercal$ be the transpose of $\mathbf A$. The '''left null space''' '''$\mathbf A$''' is defined as the null space of $\mathbf A^\intercal$.


Definition:Extended Real Addition

Let $\overline \R$ denote the extended real numbers. Define '''extended real addition''' or '''addition on $\overline \R$''', denoted $+_{\overline \R}: \overline \R \times \overline \R \to \overline \R$, by: :$\forall x,y \in \R: x +_{\overline \R} y := x +_\R y$ where $+_\R$ denotes real addition :$\forall x \in \R: x +_{\overline \R} \left({+\infty}\right) = \left({+\infty}\right) +_{\overline \R} x := +\infty$ :$\forall x \in \R: x +_{\overline \R} \left({-\infty}\right) = \left({-\infty}\right) +_{\overline \R} x := -\infty$ :$\left({+\infty}\right) +_{\overline \R} \left({+\infty}\right) := +\infty$ :$\left({-\infty}\right) +_{\overline \R} \left({-\infty}\right) := -\infty$ In particular, the expressions: :$\left({+\infty}\right) +_{\overline \R} \left({-\infty}\right)$ :$\left({-\infty}\right) +_{\overline \R} \left({+\infty}\right)$ are considered '''void''' and should be avoided. When no danger of confusion arises, $+_{\overline \R}$ is usually replaced with the more familiar $+$. From the definition of $+_{\overline \R}$ on bona fide real numbers, the name '''extended real addition''' is appropriate: the real addition is indeed extended.


Definition:Extended Real Subtraction

Let $\overline \R$ denote the extended real numbers. Define '''extended real subtraction''' or '''subtraction on $\overline \R$''', denoted $-_{\overline \R}: \overline \R \times \overline \R \to \overline \R$, by: :$\forall x, y \in \R: x -_{\overline \R} y := x -_{\R} y$ where $-_\R$ denotes real subtraction :$\forall x \in \R: x -_{\overline \R} \paren {+\infty} = \paren {-\infty} -_{\overline \R} x := -\infty$ :$\forall x \in \R: x -_{\overline \R} \paren {-\infty} = \paren {+\infty} -_{\overline \R} x := +\infty$ :$\paren {-\infty} -_{\overline \R} \paren {+\infty} := -\infty$ :$\paren {+\infty} -_{\overline \R} \paren {-\infty} := +\infty$ In particular, the expressions: :$\paren {+\infty} -_{\overline \R} \paren {+\infty}$ :$\paren {-\infty} -_{\overline \R} \paren {-\infty}$ are considered '''void''' and should be avoided.


Definition:Extended Real Multiplication

Let $\overline \R$ denote the extended real numbers. Define '''extended real multiplication''' or '''multiplication on $\overline \R$''', denoted $\times_{\overline \R}: \overline \R \times \overline \R \to \overline \R$, by: :$\forall x \in \overline \R: x \times_{\overline \R} 0 = 0 \times_{\overline \R} x = 0$ :$\forall x, y \in \R: x \times_{\overline \R} y := x \times_\R y$ where $\times_\R$ denotes real multiplication :$\forall x \in \R_{>0}: x \times_{\overline \R} \paren {+\infty} = \paren {+\infty} \times_{\overline \R} x := +\infty$ :$\forall x \in \R_{<0}: x \times_{\overline \R} \paren {+\infty} = \paren {+\infty} \times_{\overline \R} x := -\infty$ :$\forall x \in \R_{>0}: x \times_{\overline \R} \paren {-\infty} = \paren {-\infty} \times_{\overline \R} x := -\infty$ :$\forall x \in \R_{<0}: x \times_{\overline \R} \paren {-\infty} = \paren {-\infty} \times_{\overline \R} x := +\infty$ :$\paren {+\infty} \times_{\overline \R} \paren {+\infty} := +\infty$ :$\paren {-\infty} \times_{\overline \R} \paren {-\infty} := +\infty$ :$\paren {+\infty} \times_{\overline \R} \paren {-\infty} := -\infty$ :$\paren {-\infty} \times_{\overline \R} \paren {+\infty} := -\infty$ When no danger of confusion arises, $\times_{\overline \R}$ is usually replaced with the more familiar $\times$, or even suppressed. From the definition of $\times_{\overline \R}$ on bona fide real numbers, the name '''extended real multiplication''' is appropriate: the real multiplication is indeed extended.


Definition:Internal Direct Product/General Definition

Let $\left({S_1, \circ {\restriction_{S_1}}}\right), \ldots, \left({S_n, \circ {\restriction_{S_n}}}\right)$ be closed algebraic substructures of an algebraic structure $\left({S, \circ}\right)$ where $\circ {\restriction_{S_1}}, \ldots, \circ {\restriction_{S_n}}$ are the operations induced by the restrictions of $\circ$ to $S_1, \ldots, S_n$ respectively. The structure $\left({S, \circ}\right)$ is the '''internal direct product of $\left \langle {S_n} \right \rangle$''' if the mapping: :$\displaystyle C: \prod_{k \mathop = 1}^n S_k \to S: C \left({s_1, \ldots, s_n}\right) = \prod_{k \mathop = 1}^n s_k$ is an isomorphism from the cartesian product $\left({S_1, \circ {\restriction_{S_1}}}\right) \times \cdots \times \left({S_n, \circ {\restriction_{S_n}}}\right)$ onto $\left({S, \circ}\right)$. The operation $\circ$ on $S$ is the operation induced on $S$ by $\circ {\restriction_{S_1}}, \circ {\restriction_{S_2}}, \ldots, \circ {\restriction_{S_n}}$.


Definition:Measurable Mapping

Let $\left({X, \Sigma}\right)$ and $\left({X', \Sigma'}\right)$ be measurable spaces. A mapping $f: X \to X'$ is said to be '''$\Sigma \, / \, \Sigma'$-measurable''' iff: :$\forall E' \in \Sigma': f^{-1} \left({E'}\right) \in \Sigma$ That is, iff the preimage of every measurable set under $f$ is again measurable.


Let $\left({S_1, \circ_1}\right)$ and $\left({S_2, \circ_2}\right)$ be algebraic structures. Let $S = S_1 \times S_2$ be the cartesian product of $S_1$ and $S_2$. Then the '''operation induced on $S$ by $\circ_1$ and $\circ_2$''' is the operation $\circ$ defined as: :$\left({s_1, s_2}\right) \circ \left({t_1, t_2}\right) := \left({s_1 \circ_1 t_1, s_2 \circ_2 t_2}\right)$ for all ordered pairs in $S$. === General Definition === {{:Definition:Operation Induced by Direct Product/General Definition}}


Definition:Translation Mapping

=== In an Abelian Group === {{:Definition:Translation Mapping/Abelian Group}} === In a Euclidean Space === {{:Definition:Translation Mapping/Euclidean Space}} === In an Affine Space === {{:Definition:Translation Mapping/Affine Space}}


Definition:Operation Induced by Direct Product/General Definition

Let $\left({S_1, \circ_1}\right), \left({S_2, \circ_2}\right), \ldots, \left({S_n, \circ_n}\right)$ be algebraic structures. Let $\displaystyle \mathcal S_n = \prod_{k \mathop = 1}^n S_k$ be the cartesian product of $S_1, S_2, \ldots, S_n$. Then the '''operation induced on $\mathcal S_n$ by $\circ_1, \ldots, \circ_n$''' is the operation $\circledcirc_n$ defined as: :$\left({s_1, s_2, \ldots, s_n}\right) \circledcirc_n \left({t_1, t_2, \ldots, t_n}\right) := \begin{cases} s_1 \circ_1 t_1 & : n = 1 \\ \left({s_1 \circ_1 t_1, s_2 \circ_2 t_2}\right) & : n = 2 \\ \left({\left({s_1, s_2, \ldots, s_{n-1} }\right) \circledcirc_{n-1} \left({t_1, t_2, \ldots, t_{n-1}}\right), s_n \circ_n t_n}\right) & : n > 2 \end{cases}$ for all ordered $n$-tuples in $\mathcal S_n$. That is: :$\left({s_1, s_2, \ldots, s_n}\right) \circledcirc_n \left({t_1, t_2, \ldots, t_n}\right) := \left({s_1 \circ_1 t_1, s_2 \circ_2 t_2, \ldots, s_n \circ_n t_n}\right)$


Definition:Pre-Image Sigma-Algebra

=== On Domain === {{:Definition:Pre-Image Sigma-Algebra/Domain}} === On Codomain === {{:Definition:Pre-Image Sigma-Algebra/Codomain}}


Definition:Pre-Image Sigma-Algebra/Codomain

Let $X, X'$ be sets, and let $f: X \to X'$ be a mapping. Let $\Sigma$ be a $\sigma$-algebra on $X$. Then the '''pre-image $\sigma$-algebra (of $\Sigma$) on the codomain of $f$''' is defined as: :$\Sigma' := \left\{{E' \subseteq X': f^{-1} \left({E'}\right) \in \Sigma}\right\}$


Definition:Pointwise Operation

Let $S$ be a set. Let $\struct {T, \circ}$ be an algebraic structure. Let $T^S$ be the set of all mappings from $S$ to $T$. Let $f, g \in T^S$, that is, let $f: S \to T$ and $g: S \to T$ be mappings. Then the operation $f \oplus g$ is defined on $T^S$ as follows: :$f \oplus g: S \to T: \forall x \in S: \map {\paren {f \oplus g} } x = \map f x \circ \map g x$ The operation $\oplus$ is called the '''pointwise operation on $T^S$ induced by $\circ$'''. === Induced Structure === {{:Definition:Pointwise Operation/Induced Structure}}


Definition:Sigma-Algebra Generated by Collection of Mappings

Let $\left({X_i, \Sigma_i}\right)$ be measurable spaces, with $i \in I$ for some index set $I$. Let $X$ be a set, and let, for $i \in I$, $f_i: X \to X_i$ be a mapping. Then the '''$\sigma$-algebra generated by $\left({f_i}\right)_{i \in I}$''', $\sigma \left({f_i: i \in I}\right)$, is the smallest $\sigma$-algebra on $X$ such that every $f_i$ is $\sigma \left({f_i: i \in I}\right) \, / \, \Sigma_i$-measurable. That is, $\sigma \left({f_i: i \in I}\right)$ is subject to: :$(1):\quad \forall i \in I: \forall E_i \in \Sigma_i: f_i^{-1} \left({E_i}\right) \in \sigma \left({f_i: i \in I}\right)$ :$(2):\quad \sigma \left({f_i: i \in I}\right) \subseteq \Sigma$ for all $\sigma$-algebras $\Sigma$ on $X$ satisfying $(1)$ In fact, $\sigma \left({f_i: i \in I}\right)$ always exists, and is unique, as proved on Existence and Uniqueness of Sigma-Algebra Generated by Collection of Mappings.


Definition:Gradient Operator/Real Cartesian Space

Let $\R^n$ denote the real Cartesian space of $n$ dimensions. Let $\map f {x_1, x_2, \ldots, x_n}$ denote a real-valued function on $\R^n$. Let $\tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ be the standard ordered basis on $\R^n$. Let $\mathbf u = u_1 \mathbf e_1 + u_2 \mathbf e_2 + \cdots + u_n \mathbf e_n = \displaystyle \sum_{k \mathop = 1}^n u_k \mathbf e_k$ be a vector in $\R^n$. Let the partial derivative of $f$ with respect to $u_k$ exist for all $u_k$. The '''gradient of $f$''' (at $\mathbf u$) is defined as: {{begin-eqn}} {{eqn | l = \grad f       | o = :=       | r = \nabla f }} {{eqn | r = \paren {\sum_{k \mathop = 1}^n \mathbf e_k \dfrac \partial {\partial x_k} } \map f {\mathbf u}       | c = {{Defof|Del Operator}} }} {{eqn | r = \sum_{k \mathop = 1}^n \dfrac {\map {\partial f} {\mathbf u} } {\partial x_k} \mathbf e_k       | c =  }} {{end-eqn}}


Definition:Directional Derivative

Let: :$f: \R^n \to \R, \mathbf x \mapsto \map f {\mathbf x}$  be a real-valued function such that the gradient: :$\nabla \map f {\mathbf x}$ exists. Let: :$\mathbf u$ be a unit vector in $\R^n$. The '''directional derivative of $f$ in the direction of $\mathbf u$''' is defined as: {{begin-eqn}} {{eqn | l = D_{\mathbf u} \map f {\mathbf x}       | r = \nabla \map f {\mathbf x} \cdot \mathbf u }} {{end-eqn}} where $\cdot$ denotes the dot product. {{expand|Geometric Interpretation}}


Definition:Pushforward Measure

Let $\struct {X, \Sigma}$ and $\struct {X', \Sigma'}$ be measurable spaces. Let $\mu$ be a measure on $\struct {X, \Sigma}$. Let $f: X \to X'$ be a $\Sigma \, / \, \Sigma'$-measurable mapping. Then the '''pushforward of $\mu$ under $f$''' is the mapping $f_* \mu: \Sigma' \to \overline \R$ defined by: :$\forall E' \in \Sigma': \map {f_* \mu} {E'} := \map \mu {\map {f^{-1} } {E'} }$ where $\overline \R$ denotes the extended real numbers.


Definition:Orthogonal Group

Let $k$ be a field. The '''($n$th) orthogonal group (on $k$)''', denoted $\map {\mathrm O} {n, k}$, is the following subset of the general linear group $\GL {n, k}$: :$\map {\mathrm O} {n, k} := \set {M \in \GL {n, k}: M^\intercal = M^{-1} }$ where $M^\intercal$ denotes the transpose of $M$.


Definition:Stieltjes Function

Let $f: \R \to \overline \R$ be a real function, where $\overline \R$ denotes the extended real numbers. Then $f$ is said to be a '''Stieltjes function''' {{iff}}: :$(1): \quad f$ is increasing :$(2): \quad f$ is left-continuous.


Definition:Stieltjes Function of Measure on Real Numbers

Let $\mu$ be a measure on $\R$ with the Borel $\sigma$-algebra $\mathcal B \left({\R}\right)$. The '''Stieltjes function of $\mu$''' is the mapping $F_\mu: \R \to \overline \R$ defined by: :$F_\mu \left({x}\right) := \begin{cases} \mu \left({ \left[{0 \,.\,.\, x}\right) \, }\right) & \text{if } x > 0\\ 0 & \text{if } x = 0\\ - \mu \left({ \left[{x \,.\,.\, 0}\right) \, }\right) & \text{if } x < 0 \end{cases}$ where $\overline \R$ denotes the extended real numbers.


Definition:Pre-Measure of Finite Stieltjes Function

Let $\mathcal J_{ho}$ denote the collection of half-open intervals in $\R$. Let $f: \R \to \R$ be a finite Stieltjes function. The '''pre-measure of $f$''' is the mapping $\mu_f: \mathcal J_{ho} \to \overline \R_{\ge 0}$ defined by: :$\mu_f \left({ \left[{a \,.\,.\, b}\right) \, }\right) := \begin{cases} f \left({b}\right) - f\left({a}\right) & \text{if } b \ge a \\ 0 & \text{otherwise} \end{cases}$ where $\overline \R_{\ge 0}$ denotes the set of positive extended real numbers.


Definition:Measure of Finite Stieltjes Function

Let $f: \R \to \R$ be a finite Stieltjes function. Let $\mu_f$ be the pre-measure of $f$. Let $\mu$ be the unique measure extending $\mu_f$ provided on Pre-Measure of Finite Stieltjes Function Extends to Unique Measure. Then $\mu$ is called the '''measure of $f$'''. This definition makes $\mu$ a measure on $\mathcal B \left({\R}\right)$, the Borel $\sigma$-algebra of $\R$.


Definition:Extended Real-Valued Function

Let $S$ be a set, and let $\overline \R$ denote the extended real numbers. A mapping $f: S \to \overline \R$ is said to be an '''extended real-valued function'''.


Definition:Extended Real Sigma-Algebra

Let $\left({\overline{\R}, \tau}\right)$ be the extended real number space. The '''extended real $\sigma$-algebra''' $\overline{\mathcal B}$ is the Borel $\sigma$-algebra $\mathcal B \left({\overline{\R}, \tau}\right)$.


Definition:Space of Measurable Functions

Let $\left({X, \Sigma}\right)$ be a measurable space. Then the '''space of $\Sigma$-measurable, real-valued functions''' $\mathcal M \left({\Sigma}\right)$ is the collection of all $\Sigma$-measurable, real-valued functions: :$\mathcal M \left({\Sigma}\right) := \left\{{f: X \to \R: f \text{ is $\Sigma$-measurable}}\right\}$ Similarly, the '''space of $\Sigma$-measurable, extended real-valued functions''' $\mathcal{M}_{\overline{\R}} \left({\Sigma}\right)$ is the collection of all $\Sigma$-measurable, extended real-valued functions: :$\mathcal{M}_{\overline{\R}} \left({\Sigma}\right) := \left\{{f: X \to \overline{\R}: f \text{ is $\Sigma$-measurable}}\right\}$ === Space of Positive Measurable Functions === The '''space of $\Sigma$-measurable, positive real-valued functions''' $\mathcal{M}^+ \left({\Sigma}\right)$ is the subset of positive $\Sigma$-measurable functions in $\mathcal M \left({\Sigma}\right)$: :$\mathcal{M}^+ \left({\Sigma}\right) := \left\{{f: X \to \R: f \text{ is positive $\Sigma$-measurable}}\right\}$ Analogously, the '''space of $\Sigma$-measurable, positive extended real-valued functions''' $\mathcal{M}_{\overline{\R}}^+ \left({\Sigma}\right)$ is defined as: :$\mathcal{M}_{\overline{\R}}^+ \left({\Sigma}\right) := \left\{{f: X \to \overline{\R}: f \text{ is positive $\Sigma$-measurable}}\right\}$


Definition:Standard Representation of Simple Function

Let $\left({X, \Sigma}\right)$ be a measurable space. Let $f: X \to \R$ be a simple function. A '''standard representation of $f$''' consists of: :a finite sequence $a_1, \ldots, a_n$ of real numbers :a partition $E_0, E_1, \ldots, E_n$ of $\Sigma$-measurable sets subject to: :$f = \displaystyle \sum_{j \mathop = 0}^n a_j \chi_{E_j}$ where $a_0 := 0$, and $\chi$ denotes characteristic function.


Definition:Space of Simple Functions

Let $\left({X, \Sigma}\right)$ be a measurable space. Then the '''space of simple functions on $\left({X, \Sigma}\right)$''', denoted $\mathcal E \left({\Sigma}\right)$, is the collection of all simple functions $f: X \to \R$: :$\mathcal E \left({\Sigma}\right) := \left\{{f: X \to \R: f \text{ is a simple function}}\right\}$ === Space of Positive Simple Functions === The '''space of positive simple functions on $\left({X, \Sigma}\right)$''', denoted $\mathcal{E}^+ \left({\Sigma}\right)$ is the subset of positive simple functions in $\mathcal E \left({\Sigma}\right)$: :$\mathcal{E}^+ \left({\Sigma}\right) := \left\{{f: X \to \R: f \text{ is a positive simple function}}\right\}$


Definition:Positive Part

Let $X$ be a set. Let $f: X \to \overline \R$ be an extended real-valued function. Then the '''positive part of $f$''', $f^+: X \to \overline \R$, is the extended real-valued function defined by: :$\forall x \in X: \map {f^+} x := \max \set {0, \map f x}$ where the maximum is taken with respect to the extended real ordering. That is: :$\forall x \in X: \map {f^+} x := \begin {cases} \map f x & : \map f x \ge 0 \\ 0 & : \map f x < 0 \end {cases}$


Definition:Negative Part

Let $X$ be a set, and let $f: X \to \overline \R$ be an extended real-valued function. Then the '''negative part of $f$''', $f^-: X \to \overline \R$, is the extended real-valued function defined by: :$\forall x \in X: \map {f^-} x := -\min \set {0, \map f x}$ where the minimum is taken with respect to the extended real ordering. That is: :$\forall x \in X: \map {f^-} x := \begin {cases} \map f x & : \map f x \le 0 \\ 0 & : \map f x > 0 \end {cases}$


Definition:Dedekind-Infinite

A set is said to be '''Dedekind-infinite''' {{iff}} it is equivalent to (at least) one of its proper subsets.


Definition:Continuous Real Function/Left-Continuous/Point

Let $x_0 \in A$.  Then $f$ is said to be '''left-continuous at $x_0$''' {{iff}} the limit from the left of $\map f x$ as $x \to x_0$ exists and: :$\displaystyle \lim_{\substack {x \mathop \to x_0^- \\ x_0 \mathop \in A} } \map f x = \map f {x_0}$ where $\displaystyle \lim_{x \mathop \to x_0^-}$ is a limit from the left.


Definition:Continuous Real Function/Right-Continuous/Point

Let $x_0 \in S$.  Then $f$ is said to be '''right-continuous at $x_0$''' {{iff}} the limit from the right of $f \left({x}\right)$ as $x \to x_0$ exists and: :$\displaystyle \lim_{\substack{x \mathop \to x_0^+ \\ x_0 \mathop \in A}} f \left({x}\right) = f \left({x_0}\right)$ where $\displaystyle \lim_{x \mathop \to x_0^+}$ is a limit from the right.


Definition:Pointwise Operation/Real-Valued Functions

Let $\R^S$ be the set of all mappings $f: S \to \R$, where $\R$ is the set of real numbers. Let $\oplus$ be a binary operation on $\R$. Define $\oplus: \R^S \times \R^S \to \R^S$, called '''pointwise $\oplus$''', by: :$\forall f, g \in \R^S: \forall s \in S: \map {\paren {f \oplus g} } s := \map f s \oplus \map g s$ In the above expression, the operator on the {{RHS}} is the given $\oplus$ on the real numbers.


Definition:Pointwise Scalar Multiplication of Mappings/Real-Valued Functions

Let $S$ be a non-empty set, and let $\R^S$ be the set of all mappings $f: S \to \R$. Then '''pointwise ($\R$-)scalar multiplication''' on $\R^S$ is the binary operation $\cdot: \R \times \R^S \to \R^S$ defined by: :$\forall \lambda \in \R: \forall f \in \R^S: \forall s \in S: \map {\paren {\lambda \cdot f} } s := \lambda \cdot \map f s$ where the $\cdot$ on the right is real multiplication.


Definition:Pointwise Addition of Real-Valued Functions

Let $f, g: S \to \R$ be real-valued functions. Then the '''pointwise sum of $f$ and $g$''' is defined as: :$f + g: S \to \R:$ ::$\forall s \in S: \map {\paren {f + g} } s := \map f s + \map g s$ where the $+$ on the {{RHS}} is real-number addition.


Definition:Pointwise Multiplication

The (binary) operation of '''pointwise multiplication''' is defined on $\mathbb F^S$ as: :$\times: \mathbb F^S \times \mathbb F^S \to \mathbb F^S: \forall f, g \in \mathbb F^S:$ ::$\forall s \in S: \map {\paren {f \times g} } s := \map f s \times \map g s$ where the $\times$ on the {{RHS}} is conventional arithmetic multiplication.


Definition:Pointwise Limit

Let $S$ be a set. Let $\sequence {f_n}_{n \mathop \in \N}$, $f_n: S \to \R$ be a sequence of real-valued functions. Suppose that for all $s \in S$, the limit: :$\displaystyle \lim_{n \mathop \to \infty} \map {f_n} s$ exists. {{definition wanted|Incorporate infinite limits}} Then the '''pointwise limit of $\sequence {f_n}_{n \mathop \in \N}$''', denoted $\displaystyle \lim_{n \mathop \to \infty} f_n: S \to \R$, is defined as: :$\forall s \in S: \displaystyle \map {\paren {\lim_{n \mathop \to \infty} f_n} } s := \lim_{n \mathop \to \infty} \map {f_n} s$ '''Pointwise limit''' thence is an instance of a pointwise operation on real-valued functions. {{definition wanted|of course, makes sense in arbitrary metric/topological space. cover this}}


Definition:Absolute Value of Mapping/Real-Valued Function

Let $S$ be a set. Let $f: S \to \R$ be a real-valued function. Then the '''absolute value of $f$''', denoted $\size f: S \to \R$, is defined as: :$\forall s \in S: \map {\size f} s := \size {\map f s}$ where $\size {\map f s}$ denotes the absolute value function on $\R$. '''Absolute value''' thence is an instance of a pointwise operation on real-valued functions.


Definition:Extended Absolute Value

Let $\overline \R$ denote the extended real numbers. Extend the absolute value $\left\vert{\cdot}\right\vert$ on $\R$ to $\overline \R = \R \cup \left\{{+\infty, -\infty}\right\}$ by defining: :$\left\vert{-\infty}\right\vert = \left\vert{+\infty}\right\vert = +\infty$ Thus, the '''extended absolute value''' is a mapping $\left\vert{\cdot}\right\vert: \overline \R \to \overline \R$.


Definition:Absolute Value of Mapping/Extended Real-Valued Function

Let $S$ be a set, and let $f: S \to \overline \R$ be an extended real-valued function. Then the '''absolute value of $f$''', denoted $\size f: S \to \overline \R$, is defined as: :$\forall s \in S: \map {\size f} s := \size {\map f s}$ where $\size {\map f s}$ denotes the extended absolute value function on $\overline \R$. '''Absolute value''' thence is an instance of a pointwise operation on extended real-valued functions. Since extended absolute value coincides on $\R$ with the standard ordering, this definition incorporates the definition for real-valued functions.


Definition:Absolute Value of Mapping

Let $D$ be an ordered integral domain. Let $\size {\, \cdot \,}_D$ denote the absolute value function on $D$. Let $S$ be a set. Let $f: S \to D$ be a mapping. Then the '''absolute value of $f$''', denoted $\size f_D: S \to D$, is defined as: :$\forall s \in S: \map {\size f_D} s := \size {\map f s}_D$ '''Absolute value''' thence is an instance of a pointwise operation on a mapping.


Definition:Pointwise Operation on Extended Real-Valued Functions

Let $S$ be a set, and let $f, g : S \to \overline \R$ be extended real-valued functions. Let $\lambda \in \R$. Then extended real-valued functions can be formed by defining (for all $s \in S$): :$\lambda \cdot f: S \to \R, \map {\paren {\lambda \cdot f} } s := \lambda \cdot \map f s$ :$f + g: S \to \R, \map {\paren {f + g} } s := \map f s + \map g s$ :$f \cdot g: S \to \R, \map {\paren {f \cdot g} } s := \map f s \cdot \map g s$ as is done on Pointwise Scalar Multiplication, Pointwise Addition and Pointwise Multiplication, respectively. More generally, let $\oplus$ be a binary operation on $\overline \R$. Define $f \oplus g: S \to \overline \R$, called '''pointwise $\oplus$''', by: :$\map {\paren {f \oplus g} } s := \map f s \oplus \map g s$ In above expressions, the subscript ${\overline \R}$ of an operator expresses that its operands are extended real numbers. Next, let $\family {f_i}_{i \mathop \in I}, f_i: S \to \overline \R$, be any $I$-indexed collection of extended real-valued functions, where $I$ is some index set. Suppose that $\oplus^I$ is an $I$-ary operation on $\overline \R$. Then define $\oplus^I \family {f_i}_{i \mathop \in I}: S \to \overline \R$, called '''pointwise $\oplus^I$''', by: :$\map {\oplus^I \family {f_i}_{i \mathop \in I} } s := \oplus^I \family {\map {f_i} s}_{i \mathop \in I}$


Definition:Ordered Subsemigroup

Let $\left({S, \circ, \preceq}\right)$ be an ordered structure. Let $T \subseteq S$ be a subset of $S$ such that: :$\left({T, \circ_T, \preceq_T}\right)$ is an ordered semigroup where: :$\circ_T$ is the operation induced on $H$ by $\circ$ :$\preceq_t$ is the restriction of $\preceq$ to $T \times T$. Then $\left({T, \circ_T, \preceq_T}\right)$ is an '''ordered subsemigroup''' of $\left({S, \circ, \preceq}\right)$.


Definition:Ordered Subgroup

An '''ordered subgroup''' $\struct {T, \circ, \preceq}$ of an ordered structure $\struct {S, \circ, \preceq}$ is an ordered group such that the group $\struct {T, \circ}$ is a subgroup of $\struct {S, \circ}$.


Definition:Totally Ordered Structure

When the ordering $\preceq$ is a total ordering, the structure $\left({S, \circ, \preceq}\right)$ is then a '''totally ordered structure'''.


Definition:Integral of Positive Simple Function

Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Let $\mathcal E^+$ denote the space of positive simple functions Let $f: X \to \R, f \in \mathcal E^+$ be a positive simple function. Suppose that $f$ admits the following standard representation: :$\displaystyle f = \sum_{i \mathop = 0}^n a_i \chi_{E_i}$ where $a_0 := 0$, and $\chi$ denotes characteristic function. Then the '''$\mu$-integral of $f$''', denoted $I_\mu \left({f}\right)$, is defined by: :$I_\mu \left({f}\right) := \displaystyle \sum_{i \mathop = 0}^n a_i \mu \left({E_i}\right)$


Definition:Ordered Semigroup Isomorphism

Let $\left({S, \circ, \preceq}\right)$ and $\left({T, *, \preccurlyeq}\right)$ be ordered semigroups. An '''ordered semigroup isomorphism''' from $\left({S, \circ, \preceq}\right)$ to $\left({T, *, \preccurlyeq}\right)$ is a mapping $\phi: S \to T$ that is both: :$(1): \quad$ A semigroup isomorphism from the semigroup $\left({S, \circ}\right)$ to the semigroup $\left({T, *}\right)$ :$(2): \quad$ An order isomorphism from the ordered set $\left({S, \preceq}\right)$ to the ordered set $\left({T, \preccurlyeq}\right)$.


Definition:Isomorphism (Abstract Algebra)/Semigroup Isomorphism

Let $\struct {S, \circ}$ and $\struct {T, *}$ be semigroups. Let $\phi: S \to T$ be a (semigroup) homomorphism. Then $\phi$ is a semigroup isomorphism {{iff}} $\phi$ is a bijection.


Definition:Semigroup Homomorphism

Let $\left({S, \circ}\right)$ and $\left({T, *}\right)$ be semigroups. Let $\phi: S \to T$ be a mapping such that $\circ$ has the morphism property under $\phi$. That is, $\forall a, b \in S$: :$\phi \left({a \circ b}\right) = \phi \left({a}\right) * \phi \left({b}\right)$ Then $\phi: \left({S, \circ}\right) \to \left({T, *}\right)$ is a semigroup homomorphism.


Definition:Semigroup Epimorphism

Let $\left({S, \circ}\right)$ and $\left({T, *}\right)$ be semigroups. Let $\phi: S \to T$ be a (semigroup) homomorphism. Then $\phi$ is a semigroup epimorphism {{iff}} $\phi$ is a surjection.


Definition:Semigroup Monomorphism

Let $\struct {S, \circ}$ and $\struct {T, *}$ be semigroups. Let $\phi: S \to T$ be a (semigroup) homomorphism. Then $\phi$ is a semigroup monomorphism {{iff}} $\phi$ is an injection.


Definition:Integral of Positive Measurable Function

Let $\struct {X, \Sigma, \mu}$ be a measure space. Define the '''$\mu$-integral of positive measurable functions''', denoted $\displaystyle \int \cdot \rd \mu: \MM_{\overline \R}^+ \to \overline \R_{\ge 0}$, as: :$\forall f \in \MM_{\overline \R}^+: \displaystyle \int f \rd \mu := \sup \set {\map {I_\mu} g: g \le f, g \in \EE^+}$ where: :$\MM_{\overline \R}^+$ denotes the space of positive $\Sigma$-measurable functions :$\overline \R_{\ge 0}$ denotes the positive extended real numbers :$\sup$ is a supremum in the extended real ordering :$\map {I_\mu} g$ denotes the $\mu$-integral of the positive simple function $g$ :$g \le f$ denotes pointwise inequality :$\EE^+$ denotes the space of positive simple functions === Integral Sign === {{:Definition:Integral Sign}}


Definition:Semigroup Endomorphism

Let $\left({S, \circ}\right)$ be a semigroups. Let $\phi: S \to S$ be a (semigroup) homomorphism from $S$ to itself. Then $\phi$ is a semigroup endomorphism.


Definition:Semigroup Automorphism

Let $\left({S, \circ}\right)$ be a semigroup. Let $\phi: S \to S$ be a (semigroup) isomorphism from $S$ to itself. Then $\phi$ is a semigroup automorphism.


Definition:Ordered Semigroup Automorphism

Let $\left({S, \circ, \preceq}\right)$ be an ordered semigroup. An '''ordered semigroup automorphism''' from $\left({S, \circ, \preceq}\right)$ to itself is a mapping $\phi: S \to S$ that is both: :$(1): \quad$ A semigroup automorphism, that is, a semigroup isomorphism from the semigroup $\left({S, \circ}\right)$ to itself :$(2): \quad$ An order isomorphism from the ordered set $\left({S, \preceq}\right)$ to itself.


Definition:Totally Ordered Semigroup

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


Definition:Ordered Semigroup Monomorphism

Let $\left({S, \circ, \preceq}\right)$ and $\left({T, *, \preccurlyeq}\right)$ be ordered semigroups. An '''ordered semigroup monomorphism''' from $\left({S, \circ, \preceq}\right)$ to $\left({T, *, \preccurlyeq}\right)$ is a mapping $\phi: S \to T$ that is both: :$(1): \quad$ A (semigroup) monomorphism from the semigroup $\left({S, \circ}\right)$ to the semigroup $\left({T, *}\right)$ :$(2): \quad$ An order embedding from the ordered set $\left({S, \preceq}\right)$ to the ordered set $\left({T, \preccurlyeq}\right)$.


Definition:Zero (Number)/Naturally Ordered Semigroup

Let $\left({S, \circ, \preceq}\right)$ be a naturally ordered semigroup. Then from axiom $(NO 1)$, $\left({S, \circ, \preceq}\right)$ has a smallest element. This smallest element of $\left({S, \circ, \preceq}\right)$ is called '''zero''' and has the symbol $0$. That is: :$\forall n \in S: 0 \preceq n$


Definition:Closed Interval/Integer Interval

The '''integer interval between $m$ and $n$''' is denoted and defined as: : $\closedint m n = \begin{cases} \set {x \in S: m \le x \le n} & : m \le n \\ \O & : n < m \end{cases}$ where $\O$ is the empty set.


Definition:Restriction of Ordering

Let $\left({S, \preceq}\right)$ be an ordered set. Let $T \subseteq S$ be a subset of $S$. Then the '''restriction of $\preceq$ to $T$''', denoted $\preceq \restriction_T$, is defined as: :${\preceq \restriction_T} := {\preceq} \cap \left({T \times T}\right)$ viewing ${\preceq} \subseteq S \times S$ as a relation on $S$. Here, $\times$ denotes Cartesian product. Thence the '''restriction of $\preceq$ to $T$''' is an instance of a restriction of a relation. {{MissingLinks|Undoubtedly there are similar things to this, which need to be put in 'Also see'}}


Definition:Multiplication/Natural Numbers

Let $\N$ be the natural numbers. '''Multiplication''' on $\N$ is the basic operation $\times$ everyone is familiar with. For example: :$3 \times 4 = 12$ :$13 \times 7 = 91$


Definition:Distributive Operation/Distributand

Let $\circ$ be distributive over $*$. Then $*$ is a '''distributand''' of $\circ$.


Definition:Distributive Operation/Distributor

Let $\circ$ be distributive over $*$. Then $\circ$ is a '''distributor''' of $*$.


Definition:Increasing Sequence of Mappings

Let $S$ be a set, and let $\left({T, \preceq}\right)$ be an ordered set. Let $\left({f_n}\right)_{n \mathop \in \N}, f_n: S \to T$ be a sequence of mappings. Then $\left({f_n}\right)_{n \mathop \in \N}$ is said to be an '''increasing sequence (of mappings)''' {{iff}}: :$\forall s \in S: \forall m, n \in \N: m \le n \implies f_m \left({s}\right) \preceq f_n \left({s}\right)$ That is, {{iff}} $m \le n \implies f_m \preceq f_n$, where $\preceq$ denotes pointwise inequality.


Definition:Pointwise Inequality

Let $S$ be a set, and let $\struct {T, \preceq}$ be an ordered set. Let $f, g: S \to T$ be mappings. Then '''$f$ pointwise precedes $g$''', denoted $f \preceq g$, {{iff}}: :$\forall s \in S: \map f s \preceq \map g s$ Thence it can be seen that '''pointwise precedence''' is an instance of an induced relation on mappings. {{wtd|Not entirely happy with the naming}}


Definition:Pointwise Inequality of Real-Valued Functions

Let $S$ be a set. Let $f, g: S \to \R$ be real-valued functions. Then '''pointwise inequality of $f$ and $g$''', denoted $f \le g$, is defined to hold {{iff}}: :$\forall s \in S: f \left({s}\right) \le g \left({s}\right)$ where $\le$ denotes the usual ordering on the real numbers $\R$. Thence '''pointwise inequality''' of real-valued functions is an instance of an induced relation on mappings.


Definition:Pointwise Inequality of Extended Real-Valued Functions

Let $S$ be a set, and let $f,g: S \to \overline{\R}$ be extended real-valued functions. Then '''pointwise inequality of $f$ and $g$''', denoted $f \le g$, is defined to hold iff: :$\forall s \in S: f \left({s}\right) \le g \left({s}\right)$ where $\le$ denotes the usual ordering on the extended real numbers $\overline{\R}$. Thence '''pointwise inequality''' of extended real-valued functions is an instance of an induced relation on mappings.


Definition:Increasing Sequence of Extended Real-Valued Functions

Let $S$ be a set, and let $\left({f_n}\right)_{n \in \N}, f_n: S \to \overline{\R}$ be a sequence of extended real-valued functions. Then $\left({f_n}\right)_{n \in \N}$ is said to be an '''increasing sequence (of extended real-valued functions)''' iff: :$\forall s \in S: \forall m, n \in \N: m \le n \implies f_m \left({s}\right) \le f_n \left({s}\right)$ That is, iff $m \le n \implies f_m \le f_n$, where the second $\le$ denotes pointwise inequality.


Definition:Increasing Sequence of Real-Valued Functions

Let $S$ be a set, and let $\left({f_n}\right)_{n \in \N}, f_n: S \to \R$ be a sequence of real-valued functions. Then $\left({f_n}\right)_{n \in \N}$ is said to be an '''increasing sequence (of real-valued functions)''' {{iff}}: :$\forall s \in S: \forall m, n \in \N: m \le n \implies f_m \left({s}\right) \le f_n \left({s}\right)$ That is, {{iff}} $m \le n \implies f_m \le f_n$, where the second $\le$ denotes pointwise inequality.


Definition:Pointwise Supremum of Extended Real-Valued Functions

Let $S$ be a set. Let $\family {f_i}_{i \mathop \in I}, f_i: S \to \overline \R$ be an $I$-indexed collection of extended real-valued functions. Then the '''pointwise supremum of $\family {f_i}_{i \mathop \in I}$''', denoted $\displaystyle \sup_{i \mathop \in I} f_i: S \to \overline \R$, is defined by: :$\displaystyle \map {\paren {\sup_{i \mathop \in I} f_i} } s := \sup_{i \mathop \in I} \map {f_i} s$ where the latter supremum is taken in the extended real numbers $\overline \R$. By Extended Real Numbers form Complete Poset, this supremum is guaranteed to exist. Thence it can be seen that '''pointwise supremum''' is an instance of a pointwise operation on extended real-valued functions.


Definition:Pointwise Supremum of Real-Valued Functions

Let $S$ be a set. Let $\family {f_i}_{i \mathop \in I}, f_i: S \to \R$ be an $I$-indexed collection of real-valued functions. Then the '''pointwise supremum of $\family {f_i}_{i \mathop \in I}$''', denoted $\displaystyle \sup_{i \mathop \in I} f_i: S \to \overline \R$, is defined by: :$\displaystyle \map {\paren {\sup_{i \mathop \in I} f_i} } s := \sup_{i \mathop \in I} \map {f_i} s$ where the latter supremum is taken in the extended real numbers $\overline \R$. By Extended Real Numbers form Complete Poset, this supremum is guaranteed to exist. Thence it can be seen that '''pointwise supremum''' is an instance of a pointwise operation on real-valued functions. However, mind that this '''pointwise supremum''' need not be a real-valued function.


Definition:Pointwise Supremum

Let $S$ be a set. Let $\struct {T, \preceq}$ be an ordered set. Let $\family {f_i}_{i \mathop \in I}, f_i: S \to T$ be an $I$-indexed collection of mappings. Suppose that for all $s \in S$, it holds that: :$\displaystyle \sup_{i \mathop \in I} \map {f_i} s \in T$ where the supremum is taken in $T$. Then the '''pointwise supremum of $\family {f_i}_{i \mathop \in I}$''', denoted $\displaystyle \sup_{i \mathop \in I} f_i: S \to T$, is defined by: :$\displaystyle \map {\paren {\sup_{i \mathop \in I} f_i} } s := \sup_{i \mathop \in I} \map {f_i} s$ where the latter supremum is again taken in $T$. By assumption, this supremum is guaranteed to exist. Thence it can be seen that '''pointwise supremum''' is an instance of a pointwise operation.


Definition:Ordered Commutative Semigroup

An '''ordered commutative semigroup''' is an ordered semigroup $\left({S, \circ, \preceq}\right)$ such that $\left({S, \circ}\right)$ is a commutative semigroup.


Definition:Clearly

A statement is '''clearly''' true if its proof is apparent. Because of its subjective nature and the high likelihood of mistakes, this website does not endorse using such handwavery in any but the most trivial and boring of cases. Category:Definitions/Language Definitions 1a9f9v01qmxwuwray0m49i2v8ezycv9


Definition:Divisor (Algebra)/Natural Numbers

Let $\N$ be the natural numbers. Let $n \in \N$ and $m \in \N_{>0}$. Then '''$m$ divides $n$''' is defined as: :$m \divides n \iff \exists p \in \N: m \times p = n$


Definition:Ordered Tuple/Defined by Sequence

Let $\sequence {a_k}_{k \mathop \in A}$ be a finite sequence of $n$ terms. Let $\sigma$ be a permutation of $A$. Then the '''ordered $n$-tuple defined by the sequence $\sequence {a_{\map \sigma k} }_{k \mathop \in A}$''' is the ordered $n$-tuple: :$\sequence {a_{\map \sigma {\map \tau j} } }_{1 \mathop \le j \mathop \le n}$ where $\tau$ is the unique isomorphism from the totally ordered set $\closedint 1 n$ onto the totally ordered set $A$.


Definition:Operation/N-Ary Operation

Let $S_1, S_2, \dots, S_n$ be sets. Let $\circ: S_1 \times S_2 \times \ldots \times S_n \to \mathbb U$ be a mapping from the cartesian product $S_1 \times S_2 \times \ldots \times S_n$ to a universal set $\mathbb U$: That is, suppose that: :$\circ: S_1 \times S_2 \times \ldots \times S_n \to \mathbb U: \forall \tuple {s_1, s_2, \ldots, s_n} \in S_1 \times S_2 \times \ldots \times S_n: \map \circ {s_1, s_2, \ldots, s_n} \in \mathbb U$ Then $\circ$ is an '''$n$-ary operation'''.


Definition:Operation/Operation on Set

An '''$n$-ary operation on a set $S$''' is an $n$-ary operation where: :the domain is the cartesian space $S^n$ :the codomain is $S$: :$\odot: S^n \to S: \forall \tuple {s_1, s_2, \ldots, s_n} \in S^n: \map \odot {s_1, s_2, \ldots, s_n} \in S$ That is: :an '''$n$-ary operation on $S$''' needs to be defined for ''all'' tuples in $S^n$ :the image of $\odot$ is itself in $S$.


Definition:Indexing Set/Index

An element of the domain $I$ of $x$ is called an '''index'''.


Definition:Indexing Set/Indexed Set

The image of $x$, that is, $x \sqbrk I$ or $\Img x$, is called an '''indexed set'''. That is, it is the '''set indexed by $I$'''


Definition:Indexing Set/Function

When used in this context, the mapping $x$ is referred to as an '''indexing function for $S$'''.


Definition:Indexing Set/Term

The image of $x$ at an index $i$ is referred to as a '''term''' of the (indexed) family, and is denoted $x_i$.


Definition:Indexing Set/Family of Distinct Elements

Let $x$ be an injection, that is: :$\forall \alpha, \beta \in I: \alpha \ne \beta \implies x_\alpha \ne x_\beta$ Then $\family {x_i} _{i \mathop \in I}$ is called a '''family of distinct elements of $S$'''.


Definition:Kernel (Measure Theory)

Let $\left({X, \Sigma}\right)$ be a measurable space. Let $\overline{\R}_{\ge0}$ be the set of positive extended real numbers. A '''kernel''' is a mapping $N: X \times \Sigma \to \overline{\R}_{\ge0}$ such that: :$(1):\quad \forall x \in X: N_x: \Sigma \to \overline{\R}_{\ge0}, E \mapsto N(x, E)$ is a measure :$(2):\quad \forall E \in \Sigma: N_E: X \to \overline{\R}_{\ge0}, x \mapsto N(x, E)$ is a positive $\Sigma$-measurable function {{expand|I found a more general approach on [http://www.physicsforums.com/showthread.php?t{{=}}352389 this website]}}


Definition:Kernel Transformation of Measure

Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Let $N: X \times \Sigma \to \overline{\R}_{\ge0}$ be a kernel. The '''transformation of $\mu$ by $N$''' is the mapping $\mu N: \Sigma \to \overline{\R}$ defined by: :$\displaystyle \forall E \in \Sigma: \mu N \left({E}\right) := \int N_E \left({x}\right) \, d\mu \left({x}\right)$ where $N_E \left({x}\right) = N \left({x, E}\right)$.


Definition:Kernel Transformation of Positive Measurable Function

Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Let $N: X \times \Sigma \to \overline{\R}_{\ge0}$ be a kernel. Let $f: X \to \overline{\R}$ be a positive measurable function. The '''transformation of $f$ by $N$''' is the mapping $N f: X \to \overline{\R}$ defined by: :$\forall x \in X: N f \left({x}\right) := \displaystyle \int f \, \mathrm dN_x$ where $N_x$ is the measure $E \mapsto N(x, E)$.


Definition:Integrable Function

=== Darboux === {{:Definition:Definite Integral/Darboux}} === Riemann === {{:Definition:Definite Integral/Riemann}} === Unbounded Above Positive Real Function === {{:Definition:Integrable Function/Unbounded Above}} === Unbounded Real Function === {{:Definition:Integrable Function/Unbounded}} === Measure Space === {{:Definition:Integrable Function/Measure Space}} === $p$-Integrable Function === {{:Definition:Integrable Function/p-Integrable}} === Lebesgue === {{:Definition:Integrable Function/Lebesgue}}


Let $\sequence {H_n} = \struct {H_1, \circ {\restriction_{H_1} } }, \ldots, \struct {H_n, \circ {\restriction_{H_n} } }$ be a (finite) sequence of subgroups of a group $\struct {G, \circ}$ where $\circ {\restriction_{H_1} }, \ldots, \circ {\restriction_{H_n} }$ are the restrictions of $\circ$ to $H_1, \ldots, H_n$ respectively. === Definition 1 === {{:Definition:Internal Group Direct Product/General Definition/Definition 1}} === Definition 2 === {{:Definition:Internal Group Direct Product/General Definition/Definition 2}}


Definition:Binomial Coefficient/Real Numbers

Let $r \in \R, k \in \Z$. Then $\dbinom r k$ is defined as: :$\dbinom r k = \begin{cases} \dfrac {r^{\underline k} } {k!} & : k \ge 0 \\ & \\ 0 & : k < 0 \end{cases}$ where $r^{\underline k}$ denotes the falling factorial. That is, when $k \ge 0$: :$\displaystyle \binom r k = \frac {r \paren {r - 1} \cdots \paren {r - k + 1} } {k \paren {k - 1} \cdots 1} = \prod_{j \mathop = 1}^k \frac {r + 1 - j} j$ It can be seen that this agrees with the definition for integers when $r$ is an integer. For most applications the integer form is sufficient.


Definition:Integer/Formal Definition

Let $\struct {\N, +}$ be the commutative semigroup of natural numbers under addition. From Inverse Completion of Natural Numbers, we can create $\struct {\N', +'}$, an inverse completion of $\struct {\N, +}$. From Construction of Inverse Completion, this is done as follows: Let $\boxtimes$ be the cross-relation defined on $\N \times \N$ by: :$\tuple {x_1, y_1} \boxtimes \tuple {x_2, y_2} \iff x_1 + y_2 = x_2 + y_1$ From Cross-Relation is Congruence Relation, $\boxtimes$ is a congruence relation. Let $\struct {\N \times \N, \oplus}$ be the external direct product of $\struct {\N, +}$ with itself, where $\oplus$ is the operation on $\N \times \N$ induced by $+$ on $\N$: :$\tuple {x_1, y_1} \oplus \tuple {x_2, y_2} = \tuple {x_1 + x_2, y_1 + y_2}$ Let the quotient structure defined by $\boxtimes$ be $\struct {\dfrac {\N \times \N} \boxtimes, \oplus_\boxtimes}$ where $\oplus_\boxtimes$ is the operation induced on $\dfrac {\N \times \N} \boxtimes$ by $\oplus$. Let us use $\N'$ to denote the quotient set $\dfrac {\N \times \N} \boxtimes$. Let us use $+'$ to denote the operation $\oplus_\boxtimes$. Thus $\struct {\N', +'}$ is the Inverse Completion of Natural Numbers. As the Inverse Completion is Unique up to isomorphism, it follows that we can ''define'' the structure $\struct {\Z, +}$ which is isomorphic to $\struct {\N', +'}$. An element of $\N'$ is therefore an equivalence class of the congruence relation $\boxtimes$. So an element of $\Z$ is the isomorphic image of an element $\eqclass {\tuple {a, b} } \boxtimes$ of $\dfrac {\N \times \N} \boxtimes$. The set of elements $\Z$ is called '''the integers'''. === Natural Number Difference === {{:Construction of Inverse Completion/Natural Number Difference}}


Definition:Totally Ordered Commutative Semigroup

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


Definition:Multiplication/Integers

The multiplication operation in the domain of integers $\Z$ is written $\times$. Let us define $\eqclass {\tuple {a, b} } \boxtimes$ as in the formal definition of integers. That is, $\eqclass {\tuple {a, b} } \boxtimes$ is an equivalence class of ordered pairs of natural numbers under the congruence relation $\boxtimes$. $\boxtimes$ is the congruence relation defined on $\N \times \N$ by $\tuple {x_1, y_1} \boxtimes \tuple {x_2, y_2} \iff x_1 + y_2 = x_2 + y_1$. In order to streamline the notation, we will use $\eqclass {a, b} {}$ to mean $\eqclass {\tuple {a, b} } \boxtimes$, as suggested. As the set of integers is the Inverse Completion of Natural Numbers, it follows that elements of $\Z$ are the isomorphic images of the elements of equivalence classes of $\N \times \N$ where two tuples are equivalent if the difference between the two elements of each tuple is the same. Thus multiplication can be formally defined on $\Z$ as the operation induced on those equivalence classes as specified in the definition of integers. That is, the integers being defined as all the difference congruence classes, integer multiplication can be defined directly as the operation induced by natural number multiplication on these congruence classes. It follows that: :$\forall a, b, c, d \in \N: \eqclass {a, b} {} \times \eqclass {c, d} {} = \eqclass {a \times c + b \times d, a \times d + b \times c} {}$ or, more compactly, as $\eqclass {a c + b d, a d + b c} {}$. This can also be defined as: :$n \times m = +^n m = \underbrace {m + m + \cdots + m}_{\text{$n$ copies of $m$} }$ and the validity of this is proved in Index Laws for Monoids.


Definition:Ring (Abstract Algebra)/Product

The distributive operation $\circ$ in $\struct {R, *, \circ}$ is known as the '''(ring) product'''.


Definition:Ring (Abstract Algebra)/Ring Less Zero

It is convenient to have a symbol for $R \setminus \set 0$, that is, the set of all elements of the ring without the zero. Thus we usually use: :$R_{\ne 0} = R \setminus \set 0$


Definition:Non-Null Ring

A '''non-null ring''' is a ring with more than one element.


Definition:Integral of Integrable Function

Let $\struct {X, \Sigma, \mu}$ be a measure space. Let $f: X \to \overline \R$, $f \in \map {\LL^1} \mu$ be a $\mu$-integrable function. Then the '''$\mu$-integral of $f$''' is defined by: :$\displaystyle \int f \rd \mu := \int f^+ \rd \mu - \int f^- \rd \mu$ where $f^+$ and $f^-$ are the positive and negative parts of $f$, respectively.


Definition:Space of Integrable Functions

Let $\struct {X, \Sigma, \mu}$ be a measure space. Then the '''space of $\mu$-integrable, real-valued functions''' $\map {\LL^1} \mu$ is the collection of all $\mu$-integrable real-valued functions: :$\map {\LL^1} \mu := \set {f: X \to \R: \text {$f$ is $\mu$-integrable} }$ Similarly, the '''space of $\mu$-integrable, extended real-valued functions''' $\map {\LL^1_{\overline \R} } \mu$ is the collection of all $\mu$-integrable extended real-valued functions: :$\map {\LL^1_{\overline \R} } \mu := \set {f: X \to \overline \R: \text {$f$ is $\mu$-integrable} }$


Definition:Integrable Function/Lebesgue

Let $\lambda^n$ be a Lebesgue measure on $\R^n$ for some $n > 0$. Let $f: \R^n \to \overline \R$ be an extended real-valued function. Then $f$ is said to be '''Lebesgue integrable''' {{iff}} it is $\lambda^n$-integrable. Similarly, for all real numbers $p \ge 1$, $f$ is said to be '''Lebesgue $p$-integrable''' {{iff}} it is $p$-integrable under $\lambda^n$.


Definition:Symmetric Matrix

Let $\mathbf A$ be a square matrix over a set $S$. $\mathbf A$ is '''symmetric''' {{iff}}: :$\mathbf A = \mathbf A^\intercal$ where $\mathbf A^\intercal$ is the transpose of $\mathbf A$.


Definition:Vector Space Axioms

The '''vector space axioms''' consist of the abelian group axioms: {{begin-axiom}} {{axiom | n = \text V 0         | lc= Closure Axiom         | q = \forall \mathbf x, \mathbf y \in G         | m = \mathbf x +_G \mathbf y \in G }} {{axiom | n = \text V 1         | lc= Commutativity Axiom         | q = \forall \mathbf x, \mathbf y \in G         | m = \mathbf x +_G \mathbf y = \mathbf y +_G \mathbf x }} {{axiom | n = \text V 2         | lc= Associativity Axiom         | q = \forall \mathbf x, \mathbf y, \mathbf z \in G         | m = \paren {\mathbf x +_G \mathbf y} +_G \mathbf z = \mathbf x +_G \paren {\mathbf y +_G \mathbf z} }} {{axiom | n = \text V 3         | lc= Identity Axiom         | q = \exists \mathbf 0 \in G: \forall \mathbf x \in G         | m = \mathbf 0 +_G \mathbf x = \mathbf x = \mathbf x +_G \mathbf 0 }} {{axiom | n = \text V 4         | lc= Inverse Axiom         | q = \forall \mathbf x \in G: \exists \paren {-\mathbf x} \in G         | m = \mathbf x +_G \paren {-\mathbf x} = \mathbf 0 }} {{end-axiom}} together with the properties of a unitary module: {{begin-axiom}} {{axiom | n = \text V 5         | lc= Distributivity over Scalar Addition         | q = \forall \lambda, \mu \in K: \forall \mathbf x \in G         | m = \paren {\lambda + \mu} \circ \mathbf x = \lambda \circ \mathbf x +_G \mu \circ \mathbf x }} {{axiom | n = \text V 6         | lc= Distributivity over Vector Addition         | q = \forall \lambda \in K: \forall \mathbf x, \mathbf y \in G         | m = \lambda \circ \paren {\mathbf x +_G \mathbf y} = \lambda \circ \mathbf x +_G \lambda \circ \mathbf y }} {{axiom | n = \text V 7         | lc= Associativity with Scalar Multiplication         | q = \forall \lambda, \mu \in K: \forall \mathbf x \in G         | m = \lambda \circ \paren {\mu \circ \mathbf x} = \paren {\lambda \cdot \mu} \circ \mathbf x }} {{axiom | n = \text V 8         | lc= Identity for Scalar Multiplication         | q = \forall \mathbf x \in G         | m = 1_K \circ \mathbf x = \mathbf x }} {{end-axiom}}


Definition:Pairwise Disjoint/Set of Sets

A set of sets $\Bbb S$ is said to be '''pairwise disjoint''' {{iff}}: :$\forall X, Y \in \Bbb S: X \ne Y \implies X \cap Y = \O$ Here, $\cap$ denotes intersection, and $\O$ denotes the empty set. Hence we can say that the elements of $\Bbb S$ are '''pairwise disjoint'''.


Definition:Pairwise Disjoint/Family

An indexed family of sets $\family {S_i}_{i \mathop \in I}$ is said to be '''pairwise disjoint''' {{iff}}: :$\forall i, j \in I: i \ne j \implies S_i \cap S_j = \O$ Hence the indexed sets $S_i$ themselves, where $i \in I$, are referred to as being '''pairwise disjoint'''.


Definition:Oscillation

Let $X$ be a set. Let $\left({Y, d}\right)$ be a metric space. Let $f: X \to Y$ be a mapping. === Oscillation on a Set === {{:Definition:Oscillation/Oscillation on Set}} === Oscillation at a Point === {{:Definition:Oscillation at Point}}


Definition:Null Ideal

Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$. The null ring $\struct {\set {0_R}, +, \circ}$ is called the '''null ideal''' of $R$.


Definition:Non-Null Ideal

Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$. Let $J$ be an ideal of $R$ such that $J \ne \set {0_R}$. Then $J$ is a '''non-null ideal''' of $R$.


Definition:Only If

To say that: :''a statement $p$ is true '''only if''' $q$ is true'' is to say that: :''$p$ is a sufficient condition for $q$'': Thus: :$p \ \text{only if} \ q \dashv \vdash p \implies q$


Definition:If

To say that: :a statement $p$ is true '''if''' $q$ is true ...is to say that $p$ is a necessary condition for $q$: :$p \ \text{if} \ q \dashv \vdash q \implies p$


Definition:Internal Direct Sum of Rings/Direct Summand

In Conditions for Internal Ring Direct Sum it is proved that for this to be the case, then $S_1, S_2, \ldots, S_n$ must be ideals of $R$. Such ideals are known as '''direct summands''' of $R$.


Definition:Finite Integral Domain

A finite integral domain is an integral domain $\left({D, +, \circ}\right)$ such that $D$ is a finite set. {{SUBPAGENAME}} k6e73sa570prcefg8p2zj9yaswiy4ir


Definition:Standard Ordered Basis

Let $\struct {R, +, \circ}$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $n$ be a positive integer. For each $j \in \closedint 1 n$, let $e_j = \tuple {0_R, 0_R, \ldots, 1_R, \cdots, 0_R}$ be the ordered $n$-tuple of elements of $R$ whose $j$th term is $1_R$ and all of whose other entries is $0_R$. Then the ordered $n$-tuple $\sequence {e_k}_{1 \mathop \le k \mathop \le n} = \tuple {e_1, e_2, \ldots, e_n}$ is called the '''standard ordered basis (of the $R$-module $R^n$)'''. === Vector Space === The concept of a '''standard ordered basis''' is often found in the context of vector spaces. {{:Definition:Standard Ordered Basis/Vector Space}}


Definition:Standard Basis

Let $\struct {R, +, \circ}$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $\sequence {e_k}_{1 \mathop \le k \mathop \le n}$ be the standard ordered basis of the $R$-module $R^n$. The corresponding (unordered) set $\set {e_1, e_2, \ldots, e_n}$ is called the '''standard basis of $R^n$'''. === Vector Space === The concept of a '''standard basis''' is often found in the context of vector spaces. {{:Definition:Standard Basis/Vector Space}}


Definition:Unit Matrix

Let $R$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $\struct {\map {\MM_R} n, +, \times}$ be the ring of order $n$ square matrices over $R$. Then the '''unit matrix (of order $n$)''' of $\struct {\map {\MM_R} n, +, \times}$ is defined as: :$\mathbf I_n := \sqbrk a_n: a_{i j} = \delta_{i j}$ where $\delta_{i j}$ is the Kronecker delta for $\map {\MM_R} n$.


Definition:Positive/Integer

Informally, the '''positive integers''' are the set: :$\Z_{\ge 0} = \set {0, 1, 2, 3, \ldots}$ As the set of integers $\Z$ is the Inverse Completion of Natural Numbers, it follows that elements of $\Z$ are the isomorphic images of the elements of equivalence classes of $\N \times \N$ where two tuples are equivalent if the difference between the two elements of each tuple is the same. Thus '''positive''' can be formally defined on $\Z$ as a relation induced on those equivalence classes as specified in the definition of integers. That is, the integers being defined as all the difference congruence classes, '''positive''' can be defined directly as the relation specified as follows: :The integer $z \in \Z: z = \eqclass {\tuple {a, b} } \boxminus$ is '''positive''' {{iff}} $b \le a$. The set of '''positive integers''' is denoted $\Z_{\ge 0}$. An element of $\Z$ can be specifically indicated as being '''positive''' by prepending a $+$ sign: :$+x := x \in \Z_{\ge 0}$. === Ordering on Integers === {{:Definition:Ordering on Integers}}


Definition:Positive/Number

The concept of positive can be applied to the following sets of numbers: :$(1): \quad$ The integers $\Z$ :$(2): \quad$ The rational numbers $\Q$ :$(3): \quad$ The real numbers $\R$ The Complex Numbers cannot be Ordered Compatibly with Ring Structure, so there is no such concept as a positive complex number. As for the natural numbers, they are all positive by dint of their being the non-negative integers. === Integers === {{:Definition:Positive/Integer}} === Rational Numbers === {{:Definition:Positive/Rational Number}} === Real Numbers === {{:Definition:Positive/Real Number}}


Definition:Ordering on Integers

=== Definition 1 === {{:Definition:Ordering on Integers/Definition 1}} === Definition 2 === {{:Definition:Ordering on Integers/Definition 2}}


Definition:Ordered Integral Domain/Trichotomy Law

The property: :$\forall a \in D: \map P a \lor \map P {-a} \lor a = 0_D$ is known as the '''trichotomy law'''.


Definition:Positive/Rational Number

The '''positive rational numbers''' are the set defined as: :$\Q_{\ge 0} := \set {x \in \Q: x \ge 0}$ That is, all the rational numbers that are greater than or equal to zero.


Definition:Positive/Real Number

The '''positive real numbers''' are the set: :$\R_{\ge 0} = \set {x \in \R: x \ge 0}$ That is, all the real numbers that are greater than or equal to zero.


Definition:Strictly Positive/Number

The concept of strictly positive can be applied to the following sets of numbers: :$(1): \quad$ The integers $\Z$ :$(2): \quad$ The rational numbers $\Q$ :$(3): \quad$ The real numbers $\R$ === Integers === {{:Definition:Strictly Positive/Integer}} === Rational Numbers === {{:Definition:Strictly Positive/Rational Number}} === Real Numbers === {{:Definition:Strictly Positive/Real Number/Definition 1}}


Definition:Strictly Positive/Integer

The '''strictly positive integers''' are the set defined as: :$\Z_{> 0} := \set {x \in \Z: x > 0}$ That is, all the integers that are strictly greater than zero: :$\Z_{> 0} := \set {1, 2, 3, \ldots}$


Definition:Strictly Positive/Rational Number

The '''strictly positive rational numbers''' are the set defined as: :$\Q_{>0} := \set {x \in \Q: x > 0}$ That is, all the rational numbers that are strictly greater than zero.


=== Definition 1 === {{:Definition:Strictly Positive/Real Number/Definition 1}} === Definition 2 === {{:Definition:Strictly Positive/Real Number/Definition 2}}


Definition:Negative/Number

The concept of negative can be applied to the following sets of numbers: :$(1): \quad$ The integers $\Z$ :$(2): \quad$ The rational numbers $\Q$ :$(3): \quad$ The real numbers $\R$ The Complex Numbers cannot be Ordered Compatibly with Ring Structure, so there is no such concept as a negative complex number. === Integers === {{:Definition:Negative/Integer}} === Rational Numbers === {{:Definition:Negative/Rational Number}} === Real Numbers === {{:Definition:Negative/Real Number}} === Complex Numbers === {{:Definition:Negative/Complex Number}}


Definition:Negative/Rational Number

The '''negative rational numbers''' are the set defined as: :$\Q_{\le 0} := \set {x \in \Q: x \le 0}$ That is, all the rational numbers that are less than or equal to zero.


Definition:Negative/Real Number

The '''negative real numbers''' are the set defined as: :$\R_{\le 0} := \set {x \in \R: x \le 0}$ That is, all the real numbers that are less than or equal to zero.


Definition:Negative/Integer

The '''negative integers''' comprise the set: :$\set {0, -1, -2, -3, \ldots}$ As the set of integers is the Inverse Completion of Natural Numbers, it follows that elements of $\Z$ are the isomorphic images of the elements of equivalence classes of $\N \times \N$ where two tuples are equivalent if the difference between the two elements of each tuple is the same. Thus '''negative''' can be formally defined on $\Z$ as a relation induced on those equivalence classes as specified in the definition of integers. That is, the integers being defined as all the difference congruence classes, '''negative''' can be defined directly as the relation specified as follows: The integer $z \in \Z: z = \eqclass {\tuple {a, b} } \boxminus$ is '''negative''' {{iff}} $b > a$. The set of '''negative integers''' is denoted $\Z_{\le 0}$. An element of $\Z$ can be specifically indicated as being '''negative''' by prepending a $-$ sign: :$-x \in \Z_{\le 0} \iff x \in \Z_{\ge 0}$


Definition:Strictly Negative/Number

The concept of strictly negative can be applied to the following sets of numbers: :$(1): \quad$ The integers $\Z$ :$(2): \quad$ The rational numbers $\Q$ :$(3): \quad$ The real numbers $\R$ === Integers === {{:Definition:Strictly Negative/Integer}} === Rational Numbers === {{:Definition:Strictly Negative/Rational Number}} === Real Numbers === {{:Definition:Strictly Negative/Real Number}}


Definition:Strictly Negative/Integer

The '''strictly negative integers''' are the set defined as: {{begin-eqn}} {{eqn | l = \Z_{< 0}       | o = :=       | r = \set {x \in \Z: x < 0} }} {{eqn | r = \set {-1, -2, -3, \ldots} }} {{end-eqn}} That is, all the integers that are strictly less than zero.


Definition:Strictly Negative/Rational Number

The '''strictly negative rational numbers''' are the set defined as: :$\Q_{< 0} := \set {x \in \Q: x < 0}$ That is, all the rational numbers that are strictly less than zero.


Definition:Strictly Negative/Real Number

The '''strictly negative real numbers''' are the set defined as: :$\R_{<0} := \set {x \in \R: x < 0}$ That is, all the real numbers that are strictly less than zero.


Definition:Ordering Induced by Positivity Property

Let $\struct {R, +, \circ, \le}$ be an ordered ring whose zero is $0_R$ and whose unity is $1_R$. Let $P \subseteq R$ such that: :$(1): \quad P + P \subseteq P$ :$(2): \quad P \cap \paren {-P} = \set {0_R}$ :$(3): \quad P \circ P \subseteq P$ Then the ordering $\le$ compatible with the ring structure of $R$ is called the '''ordering induced by (the positivity property) $P$'''.


Definition:Rational Number/Formal Definition

The field $\struct {\Q, +, \times}$ of rational numbers is the field of quotients of the integral domain $\struct {\Z, +, \times}$ of integers. This is shown to exist in Existence of Field of Quotients. In view of Field of Quotients is Unique, we construct the field of quotients of $\Z$, give it a label $\Q$ and call its elements '''rational numbers'''.


Definition:Ordered Ring Isomorphism

Let $\struct {S, +, \circ, \preceq}$ and $\struct {T, \oplus, *, \preccurlyeq}$ be ordered rings. An '''ordered ring isomorphism''' from $\struct {S, +, \circ, \preceq}$ to $\struct {T, \oplus, *, \preccurlyeq}$ is a mapping $\phi: S \to T$ that is both: :$(1): \quad$ An ordered group isomorphism from the ordered group $\struct {S, +, \preceq}$ to the ordered group $\struct {T, \oplus, \preccurlyeq}$ :$(2): \quad$ A semigroup isomorphism from the semigroup $\struct {S, \circ}$ to the semigroup $\struct {T, *}$.


Definition:Ordered Group Isomorphism

Let $\left({S, \circ, \preceq}\right)$ and $\left({T, *, \preccurlyeq}\right)$ be ordered groups. An '''ordered group isomorphism''' from $\left({S, \circ, \preceq}\right)$ to $\left({T, *, \preccurlyeq}\right)$ is a mapping $\phi: S \to T$ that is both: :$(1): \quad$ A group isomorphism from the group $\left({S, \circ}\right)$ to the group $\left({T, *}\right)$ :$(2): \quad$ An order isomorphism from the ordered set $\left({S, \preceq}\right)$ to the ordered set $\left({T, \preccurlyeq}\right)$.


Definition:Ordered Group Monomorphism

Let $\left({S, \circ, \preceq}\right)$ and $\left({T, *, \preccurlyeq}\right)$ be ordered groups. An '''ordered group monomorphism''' from $\left({S, \circ, \preceq}\right)$ to $\left({T, *, \preccurlyeq}\right)$ is a mapping $\phi: S \to T$ that is both: :$(1): \quad$ A group monomorphism from the group $\left({S, \circ}\right)$ to the group $\left({T, *}\right)$ :$(2): \quad$ An order embedding from the ordered set $\left({S, \preceq}\right)$ to the ordered set $\left({T, \preccurlyeq}\right)$.


Definition:Ordered Ring Monomorphism

Let $\left({S, +, \circ, \preceq}\right)$ and $\left({T, \oplus, *, \preccurlyeq}\right)$ be ordered rings. An '''ordered ring monomorphism''' from $\left({S, +, \circ, \preceq}\right)$ to $\left({T, \oplus, *, \preccurlyeq}\right)$ is a mapping $\phi: S \to T$ that is both: :$(1): \quad$ An ordered group monomorphism from the ordered group $\left({S,+, \preceq}\right)$ to the ordered group $\left({T, \oplus, \preccurlyeq}\right)$ :$(2): \quad$ A semigroup monomorphism from the semigroup $\left({S, \circ}\right)$ to the semigroup $\left({T, *}\right)$.


Definition:Ordered Structure Automorphism

Let $\left({S, \circ, \preceq}\right)$ be an ordered structures. Let $\phi: S \to S$ be an ordered structure isomorphism from $S$ to itself. Then $\phi$ is an ordered structure automorphism.


Definition:Ordered Group Automorphism

Let $\left({G, \circ, \preceq}\right)$ be an ordered group. An '''ordered group automorphism''' from $\left({G, \circ, \preceq}\right)$ to itself is a mapping $\phi: G \to G$ that is both: :$(1): \quad$ A group automorphism, that is, a group isomorphism from the group $\left({G, \circ}\right)$ to itself :$(2): \quad$ An order isomorphism from the ordered set $\left({G, \preceq}\right)$ to itself.


Definition:Ordered Ring Automorphism

Let $\struct {R, +, \circ, \preceq}$ be an ordered ring. An '''ordered ring automorphism''' from $\struct {R, +, \circ, \preceq}$ to itself is a mapping $\phi: R \to R$ that is both: :$(1): \quad$ An ordered group automorphism from the ordered group $\struct {R, +, \preceq}$ to itself :$(2): \quad$ A semigroup automorphism from the semigroup $\struct {R, \circ}$ to itself.


Definition:Abelian Group Axioms

{{begin-axiom}} {{axiom | n = \text G 0         | lc= Closure         | q = \forall x, y \in G         | m = x + y \in G }} {{axiom | n = \text G 1         | lc= Associativity         | q = \forall x, y, z \in G         | m = x + \paren {y + z} = \paren {x + y} + z }} {{axiom | n = \text G 2         | lc= Identity         | q = \exists 0 \in G: \forall x \in G         | m = 0 + x = x = x + 0 }} {{axiom | n = \text G 3         | lc= Inverse         | q = \forall x \in G: \exists \paren {-x}\in G         | m = x + \paren {-x} = 0 = \paren {-x} + x }} {{axiom | n = \text C         | lc= Commutativity         | q = \forall x, y \in G         | m = x + y = y + x }} {{end-axiom}}


Definition:Scalar/R-Algebraic Structure

The elements of the scalar ring $\struct {R, +_R, \times_R}$ are called '''scalars'''.


Definition:Scalar Multiplication/R-Algebraic Structure

Let $\struct {S, *_1, *_2, \ldots, *_n, \circ}_R$ be an $R$-algebraic structure with $n$ operations, where: :$\struct {R, +_R, \times_R}$ is a ring :$\struct {S, *_1, *_2, \ldots, *_n}$ is an algebraic structure with $n$ operations The operation $\circ: R \times S \to S$ is called '''scalar multiplication'''.


Definition:Scalar Ring/Zero Scalar

The zero of the scalar ring is called the '''zero scalar''' and usually denoted $0$, or, if it is necessary to distinguish it from the identity of $\struct {G, +_G}$, by $0_R$.


Definition:Vector Subspace/Proper Subspace

If $T$ is a proper subset of $S$, then $\struct {T, +_T, \circ_T}_K$ is a '''proper (vector) subspace''' of $\struct {S, +, \circ}_K$.


Definition:Vector Subspace/Hilbert Spaces

When considering Hilbert spaces, one wants to deal with projections onto subspaces. These projections however require the linear subspace to be closed in topological sense in order to be well-defined. Therefore, in treatises of Hilbert spaces, one encounters the terminology '''linear manifold''' for the concept of '''vector subspace''' defined above. The adapted definition of '''linear subspace''' is then that it is a topologically closed '''linear manifold'''.


Definition:Polynomial over Ring as Function on Free Monoid on Set

Let $M$ be the free commutative monoid on the indexed set $\left\{{X_j: j \in J}\right\}$. A '''polynomial''' in $\left\{{X_j: j \in J}\right\}$ can be defined as a mapping $f: M \to R$ of finite support. That is, it is an element of the ring of polynomial forms.


Definition:Polynomial Equation

A '''polynomial equation''' is an equation in the form: :$\map f x = 0$ where $f$ is a polynomial function.


Definition:Strongly Additive Function

Let $\mathcal S$ be an algebra of sets. Let $f: \mathcal S \to \overline{\R}$ be a function, where $\overline{\R}$ denotes the extended set of real numbers. Then $f$ is defined to be '''strongly additive''' {{iff}}: :$\forall S, T \in \mathcal S: f \left({S \cup T}\right) + f \left({S \cap T}\right) = f \left({S}\right) + f \left({T}\right)$


Definition:Polynomial Function/General Definition

Let $f = a_1 \mathbf X^{k_1} + \cdots + a_r \mathbf X^{k_r}$ be a polynomial form over $R$ in the indeterminates $\left\{{X_j: j \in J}\right\}$. For each $x = \left({x_j}\right)_{j \in J} \in R^J$, let $\phi_x: R \left[{\left\{{X_j: j \in J}\right\}}\right] \to R$ be the evaluation homomorphism from the ring of polynomial forms at $x$. Then the set: :$\left\{{\left({x, \phi_x \left({f}\right)}\right): x \in R^J}\right\} \subseteq R^J \times R$ defines a '''polynomial function''' $R^J \to R$.


Definition:Linear Combination/Sequence

Let $G$ be an $R$-module. Let $\left \langle {a_n} \right \rangle := \left \langle {a_j} \right \rangle_{1 \mathop \le j \mathop \le n}$ be a sequence of elements of $G$ of length $n$. An element $b \in G$ is a '''linear combination''' of $\left \langle {a_n} \right \rangle$ {{iff}}: : $\displaystyle \exists \left \langle {\lambda_n} \right \rangle \subseteq R: b = \sum_{k \mathop = 1}^n \lambda_k a_k$


Definition:Linear Combination/Subset

Let $G$ be an $R$-module. Let $\O \subset S \subseteq G$. Let $b \in G$ be a linear combination of some sequence $\sequence {a_n}$ of elements of $S$. Then $b$ is a '''linear combination of $S$'''.


Definition:Linear Combination/Empty Set

Let $G$ be an $R$-module. $b$ is a linear combination of $\varnothing$ {{iff}}: : $b = e_G$


Let $\left \langle {a_n} \right \rangle$ be a sequence of elements of $G$ such that: : $\displaystyle \forall \left \langle {\lambda_n} \right \rangle \subseteq R: \sum_{k \mathop = 1}^n \lambda_k \circ a_k = e \implies \lambda_1 = \lambda_2 = \cdots = \lambda_n = 0_R$ That is, the only way to make $e$ with a linear combination of $\left \langle {a_n} \right \rangle$ is by making all the elements of $\left \langle {\lambda_n} \right \rangle$ equal to $0_R$. Such a sequence is '''linearly independent'''. === Linearly Independent Sequence on a Real Vector Space === {{:Definition:Linearly Independent/Sequence/Real Vector Space}}


Definition:Linearly Independent/Set

Let $S \subseteq G$. Then $S$ is a '''linearly independent set (over $R$)''' {{iff}} every finite sequence of distinct terms in $S$ is a linearly independent sequence. That is, such that: : $\displaystyle \forall \sequence {\lambda_n} \subseteq R: \sum_{k \mathop = 1}^n \lambda_k \circ a_k = e \implies \lambda_1 = \lambda_2 = \cdots = \lambda_n = 0_R$ where $a_1, a_2, \ldots, a_k$ are distinct elements of $S$. === Linearly Independent Set on a Real Vector Space === {{:Definition:Linearly Independent/Set/Real Vector Space}} === Linearly Independent Set on a Complex Vector Space === {{:Definition:Linearly Independent/Set/Complex Vector Space}}


Definition:Linearly Independent/Sequence/Real Vector Space

Let $\struct {\R^n, +, \cdot}_\R$ be a real vector space. Let $\sequence {\mathbf v_n}$ be a sequence of vectors in $\R^n$. Then $\sequence {\mathbf v_n}$ is '''linearly independent''' {{iff}}: :$\ds \forall \sequence {\lambda_n} \subseteq \R: \sum_{k \mathop = 1}^n \lambda_k \mathbf v_k = \mathbf 0 \implies \lambda_1 = \lambda_2 = \cdots = \lambda_n = 0$ where $\mathbf 0 \in \R^n$ is the zero vector and $0 \in \R$ is the zero scalar.


Definition:Linearly Independent/Set/Real Vector Space

Let $\struct {\R^n, +, \cdot}_\R$ be a real vector space. Let $S \subseteq \R^n$. Then $S$ is a '''linearly independent set of real vectors''' if every finite sequence of distinct terms in $S$ is a linearly independent sequence. That is, such that: :$\displaystyle \forall \set {\lambda_k: 1 \le k \le n} \subseteq \R: \sum_{k \mathop = 1}^n \lambda_k \mathbf v_k = \mathbf 0 \implies \lambda_1 = \lambda_2 = \cdots = \lambda_n = 0$ where $\mathbf v_1, \mathbf v_2, \ldots, \mathbf v_n$ are distinct elements of $S$.


Definition:Linearly Dependent/Sequence

Let $\left \langle {a_k} \right \rangle_{1 \mathop \le k \mathop \le n}$ be a sequence of elements of $G$ such that: : $\displaystyle \exists \left \langle {\lambda_k} \right \rangle_{1 \mathop \le k \mathop \le n} \subseteq R: \sum_{k \mathop = 1}^n \lambda_k \circ a_k = e$ where not all of $\lambda_k$ are equal to $0_R$. That is, it is possible to find a linear combination of $\left \langle {a_k} \right \rangle_{1 \mathop \le k \mathop \le n}$ which equals $e$. Such a sequence is '''linearly dependent'''. === Linearly Dependent Sequence on a Real Vector Space === {{:Definition:Linearly Dependent/Sequence/Real Vector Space}}


Definition:Linearly Dependent/Sequence/Real Vector Space

Let $\left({\R^n,+,\cdot}\right)_{\R}$ be a real vector space. Let $\mathbf 0 \in \R^n$ be the zero vector. Let $\left \langle {\mathbf v_k} \right \rangle_{1 \mathop \le k \mathop \le n}$ be a sequence of vectors in $\R^n$. Then $\left \langle {\mathbf v_k} \right \rangle_{1 \mathop \le k \mathop \le n}$ is '''linearly dependent''' iff: : $\displaystyle \exists \left \langle {\lambda_k} \right \rangle_{1 \mathop \le k \mathop \le n} \subseteq \R: \sum_{k \mathop = 1}^n \lambda_k \mathbf v_k = \mathbf 0$ where not all $\lambda_k$ are equal to $0$. That is, it is possible to find a linear combination of $\left \langle {\mathbf v_k} \right \rangle_{1 \mathop \le k \mathop \le n}$ which equals $\mathbf 0$.


Definition:Linearly Dependent/Set

Let $S \subseteq G$. Then $S$ is a '''linearly dependent set''' if there exists a sequence of distinct terms in $S$ which is a linearly dependent sequence. That is, such that: :$\displaystyle \exists \set {\lambda_k: 1 \le k \le n} \subseteq R: \sum_{k \mathop = 1}^n \lambda_k \circ a_k = e$ where $a_1, a_2, \ldots, a_n$ are distinct elements of $S$, and where at least one of $\lambda_k$ is not equal to $0_R$.


Definition:Linearly Dependent/Set/Real Vector Space

Let $\struct {\R^n, +, \cdot}_\R$ be a real vector space. Let $S \subseteq \R^n$. Then $S$ is a '''linearly dependent set''' if there exists a sequence of distinct terms in $S$ which is a linearly dependent sequence. That is, such that: :$\displaystyle \exists \set {\lambda_k: 1 \le k \le n} \subseteq \R: \sum_{k \mathop = 1}^n \lambda_k \mathbf v_k = \mathbf 0$ where $\set {\mathbf v_1, \mathbf v_2, \ldots, \mathbf v_n} \subseteq S$, and such that at least one of $\lambda_k$ is not equal to $0$.


Definition:Atom of Measure

Let $\struct {X, \Sigma, \mu}$ be a measure space. An element $x \in X$ is said to be an '''atom (of $\mu$)''' {{iff}}: :$(1): \quad \set x \in \Sigma$ :$(2): \quad \map \mu {\set x} > 0$


Definition:Standard Ordered Basis/Vector Space

Let $\struct {\mathbf V, +, \circ}_{\mathbb F}$ be a vector space over a field $\mathbb F$, as defined by the vector space axioms. Let the unity of $\mathbb F$ be denoted $1_{\mathbb F}$, and its zero $0_{\mathbb F}$. Let $\mathbf e_i$ be a vector whose $i$th term is $1_{\mathbb F}$ and with entries $0_{\mathbb F}$ elsewhere. Then the ordered $n$-tuple $\tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ is the '''standard ordered basis of $\mathbf V$'''.


Definition:Standard Basis/Vector Space

Let $\left({\mathbf V, +, \circ}\right)_{\mathbb F}$ be a vector space over $\mathbb F$. Let $\left({\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}\right)$ be the standard ordered basis on $\mathbf V$. The corresponding (unordered) set $\left\{{\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}\right\}$ is called the '''standard basis of $\mathbf V$'''


Definition:Coordinate System

Let $R$ be a ring with unity. Let $\sequence {a_k}_{1 \mathop \le k \mathop \le n}$ be an ordered basis of a free $R$-module $G$. Then $\sequence {a_k}_{1 \mathop \le k \mathop \le n}$ can be referred to as a '''coordinate system'''.


Definition:Dimension of Module

Let $R$ be a ring with unity. Let $G$ be a free $R$-module which has a basis of $n$ elements. Then $G$ is said to have a '''dimension of $n$''' or to be '''$n$-dimensional'''. The dimension of a free $R$-module $G$ is denoted $\map {\dim_R} G$, or just $\map \dim G$.


Definition:Dimension of Vector Space

Let $K$ be a division ring. Let $V$ be a vector space over $K$. === Definition 1 === {{:Definition:Dimension of Vector Space/Definition 1}} === Definition 2 === {{:Definition:Dimension of Vector Space/Definition 2}}


Definition:Dimension of Module/Finite

Let $G$ be a (unitary) module which is $n$-dimensional for some $n \in \N_{>0}$. Then $G$ is '''finite dimensional'''.


Definition:Dimension of Vector Space/Finite

Let $V$ be a vector space which is $n$-dimensional for some $n \in \N_{>0}$. Then $V$ is '''finite dimensional'''. The '''dimension''' of a '''finite-dimensional $K$-vector space''' $V$ is denoted $\map {\dim_K} V$, or just $\map \dim V$.


Definition:Homomorphism (Abstract Algebra)/Image

As a homomorphism is a mapping, the '''homomorphic image''' of $\phi$ is defined in the same way as the image of a mapping: :$\Img \phi = \set {t \in T: \exists s \in S: t = \map \phi s}$


Definition:Linear Operator/Vector Space

A '''linear operator''' on a vector space is a linear transformation from a vector space into itself.


Definition:Linear Transformation/Vector Space

Let $V, W$ be vector spaces over a field (or, more generally, division ring) $K$. A mapping $A: V \to W$ is a '''linear transformation''' {{iff}}: :$\forall v_1, v_2 \in V, \lambda \in K: \map A {\lambda v_1 + v_2} = \lambda \map A {v_1} + \map A {v_2}$ That is, a homomorphism from one vector space to another.


Definition:Kernel of Linear Transformation/Vector Space

Let $\struct {\mathbf V, +, \times}$ be a vector space. Let $\struct {\mathbf V', +, \times}$ be a vector space whose zero vector is $\mathbf 0'$. Let $T: \mathbf V \to \mathbf V'$ be a linear transformation. Then the '''kernel''' of $T$ is defined as: :$\map \ker T := T^{-1} \sqbrk {\set {\mathbf 0'} } = \set {\mathbf x \in \mathbf V: \map T {\mathbf x} = \mathbf 0'}$


Definition:Rank/Linear Transformation

Let $\phi$ be a linear transformation from one vector space to another. Let the image of $\phi$ be finite-dimensional. Then its dimension is called the '''rank of $\phi$''' and is denoted $\map \rho \phi$.


Definition:Rank/Matrix

=== Definition 1 === {{:Definition:Rank/Matrix/Definition 1}} === Definition 2 === {{:Definition:Rank/Matrix/Definition 2}} === Definition 3 === {{:Definition:Rank/Matrix/Definition 3}}


Definition:Nullity/Linear Transformation

Let $K$ be a division ring. Let $V$ and $W$ be $K$-vector spaces. Let $\phi: V \to W$ be a linear transformation. Let the kernel $\ker \phi$ be finite dimensional. Then the '''nullity of $\phi$''' is the dimension of $\ker \phi$ and is denoted $\map \nu \phi$.


Definition:Nullity/Matrix

Let $\mathbf A$ be a matrix. Then the '''nullity of $\mathbf A$''' is defined to be the dimension of the null space of $\mathbf A$.


Definition:Vector Notation

Several conventions are found in the literature for annotating a general vector quantity in a style that distinguishes it from a scalar quantity, as follows. Let $\set {x_1, x_2, \ldots, x_n}$ be a collection of scalars which form the components of an $n$-dimensional vector. The vector $\tuple {x_1, x_2, \ldots, x_n}$ can be annotated as: {{begin-eqn}} {{eqn | l = \bsx       | r = \tuple {x_1, x_2, \ldots, x_n} }} {{eqn | l = \vec x       | r = \tuple {x_1, x_2, \ldots, x_n} }} {{eqn | l = \hat x       | r = \tuple {x_1, x_2, \ldots, x_n} }} {{eqn | l = \underline x       | r = \tuple {x_1, x_2, \ldots, x_n} }} {{eqn | l = \tilde x       | r = \tuple {x_1, x_2, \ldots, x_n} }} {{end-eqn}} To emphasize the arrow interpretation of a vector, we can write: :$\bsv = \sqbrk {x_1, x_2, \ldots, x_n}$ or: :$\bsv = \sequence {x_1, x_2, \ldots, x_n}$ In printed material the '''boldface''' $\bsx$ or $\mathbf x$ is common. This is the style encouraged and endorsed by {{ProofWiki}}. However, for handwritten material (where boldface is difficult to render) it is usual to use the '''underline''' version $\underline x$.  Also found in handwritten work are the '''tilde''' version $\tilde x$ and '''arrow''' version $\vec x$, but as these are more intricate than the simple underline (and therefore more time-consuming and tedious to write), they will only usually be found in fair copy. It is also noted that the '''tilde''' over $\tilde x$ does not render well in {{MathJax}} under all browsers, and differs little visually from an '''overline''': $\overline x$. The '''hat''' version $\hat x$ usually has a more specialized meaning, namely to symbolize a unit vector. In computer-rendered materials, the '''arrow''' version $\vec x$ is popular, as it is descriptive and relatively unambiguous, and in $\LaTeX$ it is straightforward. However, it does not render well in all browsers, and is therefore (reluctantly) not recommended for use on this website.


Definition:Annihilator on Algebraic Dual

Let $R$ be a commutative ring. Let $G$ be a module over $R$. Let $G^*$ be the algebraic dual of $G$. Let $M$ be a submodule of $G$. The '''annihilator of $M$''', denoted $M^\circ$, is defined as: :$\ M^\circ := \set {t' \in G^*: \forall x \in M: \map {t'} x = 0}$


Definition:Matrix/Order

Let $\sqbrk a_{m n}$ be an $m \times n$ matrix. Then the parameters $m$ and $n$ are known as the '''order''' of the matrix.


Definition:Matrix/Row

Let $\mathbf A$ be an $m \times n$ matrix. For each $i \in \closedint 1 m$, the '''rows''' of $\mathbf A$ are the ordered $n$-tuples: :$r_i = \tuple {a_{i 1}, a_{i 2}, \ldots, a_{i n} }$ where $r_i$ is called the '''$i$th row of $\mathbf A$'''. A '''row''' of an $m \times n$ matrix can also be treated as a $1 \times n$ row matrix in its own right: :$r_i = \begin {bmatrix} a_{i 1} & a_{i 2} & \cdots & a_{i n} \end {bmatrix}$ for $i = 1, 2, \ldots, m$.


Definition:Matrix/Column

Let $\mathbf A$ be an $m \times n$ matrix. For each $j \in \closedint 1 n$, the '''columns''' of $\mathbf A$ are the ordered $m$-tuples: : $c_j = \tuple {a_{1 j}, a_{2 j}, \ldots, a_{m j} }$ where $c_j$ is called the '''$j$th column of $\mathbf A$'''. A '''column''' of an $m \times n$ matrix can also be treated as a $m \times 1$ column matrix in its own right: :$c_j = \begin {bmatrix} a_{1 j} \\ a_{2 j} \\ \vdots \\ a_{m j} \end {bmatrix}$ for $j = 1, 2, \ldots, n$.


Definition:Matrix/Element

Let $\mathbf A$ be an $m \times n$ matrix over a set $S$. The individual $m \times n$ elements of $S$ that go to form $\mathbf A = \sqbrk a_{m n}$ are known as the '''elements of the matrix'''. The '''element''' at row $i$ and column $j$ is called '''element $\tuple {i, j}$ of $\mathbf A$''', and can be written $a_{i j}$, or $a_{i, j}$ if $i$ and $j$ are of more than one character. If the indices are still more complicated coefficients and further clarity is required, then the form $a \tuple {i, j}$ can be used. Note that the first subscript determines the row, and the second the column, of the matrix where the '''element''' is positioned.


Definition:Matrix/Square Matrix

An $n \times n$ matrix is called a '''square matrix'''. That is, a '''square matrix''' is a matrix which has the same number of rows as it has columns. A '''square matrix''' is usually denoted $\sqbrk a_n$ in preference to $\sqbrk a_{n n}$. In contrast, a non-'''square matrix''' can be referred to as a '''rectangular matrix'''.


Definition:Matrix/Diagonal Elements

Let $\mathbf A = \sqbrk a_{m n}$ be a matrix. The elements $a_{j j}: j \in \closedint 1 {\min \set {m, n} }$ constitute the '''main diagonal''' of the matrix. The elements themselves are called the '''diagonal elements'''.


Definition:Matrix/Underlying Structure

Let $\mathbf A$ be a matrix over a set $S$. The set $S$ can be referred to as the '''underlying set of $\mathbf A$'''. In the context of matrices, however, it is usual for $S$ itself to be the underlying set of an algebraic structure in its own right. If this is the case, then the structure $\struct {S, \circ_1, \circ_2, \ldots, \circ_n}$ (which may also be an ordered structure) can be referred to as the '''underlying structure of $\mathbf A$'''. When the '''underlying structure''' is not specified, it is taken for granted that it is one of the standard number systems, usually the real numbers $\R$.


Definition:Matrix/Zero Row or Column

Let $\mathbf A = \sqbrk a_{m n}$ be an $m \times n$ matrix whose underlying structure is a ring or field (usually numbers). If a row or column of $\mathbf A$ contains only zeroes, then it is a '''zero row''' or a '''zero column'''.


Definition:Matrix/Indices

Let $\mathbf A$ be an $m \times n$ matrix. Let $a_{i j}$ be the element in row $i$ and column $j$ of $\mathbf A$. Then the subscripts $i$ and $j$ are referred to as the '''indices''' (singular: '''index''') of $a_{i j}$.


Definition:Module Automorphism

Let $\left({G, +_G, \circ}\right)_R$ be an $R$-module. Let $\phi: G \to G$ be a module isomorphism to itself. Then $\phi$ is a '''module automorphism'''.


Definition:Vector Space Automorphism

Let $V$ be a $K$-vector space. Let $\phi: V \to V$ be a vector space isomorphism to itself. Then $\phi$ is a '''vector space automorphism'''.


Definition:Pointwise Maximum of Mappings/Extended Real-Valued Functions

Let $X$ be a set, and let $f, g: X \to \overline{\R}$ be extended real-valued functions. Let $\max$ be the max operation on $\overline{\R}$ (Ordering on Extended Real Numbers is Total Ordering ensures it is in fact defined). Then the '''pointwise maximum of $f$ and $g$''', denoted $\max \left({f, g}\right)$, is defined by: :$\max \left({f, g}\right): X \to \overline{\R}: \max \left({f, g}\right) \, \left({x}\right) := \max \left({f \left({x}\right), g \left({x}\right)}\right)$ '''Pointwise maximum''' thence is an instance of a pointwise operation on extended real-valued functions. Since the ordering on $\overline{\R}$ coincides on $\R$ with the standard ordering, this definition incorporates the definition for real-valued functions.


Definition:Pointwise Maximum of Mappings/Real-Valued Functions

Let $S$ be a set. Let $f, g: S \to \R$ be real-valued functions. Let $\max$ be the max operation on $\R$ (Ordering on Real Numbers is Total Ordering ensures it is in fact defined). Then the '''pointwise maximum of $f$ and $g$''', denoted $\map \max {f, g}$, is defined by: :$\map \max {f, g}: S \to \R: \map {\map \max {f, g} } x := \map \max {\map f x, \map g x}$ '''Pointwise maximum''' thence is an instance of a pointwise operation on real-valued functions.


Definition:Non-Invertible Matrix

{{:Definition:Non-Invertible Matrix/Definition 1}}


Definition:Pointwise Maximum of Mappings

Let $X$ be a set. Let $\struct {S, \preceq}$ be a toset. Let $f, g: X \to S$ be mappings. Let $\max$ be the max operation on $\struct {S, \preceq}$. Then the '''pointwise maximum of $f$ and $g$''', denoted $\map \max {f, g}$, is defined by: :$\map \max {f, g}: X \to S: \map {\map \max {f, g} } x := \map \max {\map f x, \map g x}$ Hence '''pointwise maximum''' is an instance of a pointwise operation on mappings.


Definition:Pointwise Minimum of Mappings

Let $X$ be a set. Let $\struct {S, \preceq}$ be a toset. Let $f, g: X \to S$ be mappings. Let $\min$ be the min operation on $\struct {S, \preceq}$. Then the '''pointwise minimum of $f$ and $g$''', denoted $\map \min {f, g}$, is defined by: :$\map \min {f, g}: X \to S: \map {\map \min {f, g} } x := \map \min {\map f x, \map g x}$ Hence '''pointwise minimum''' is an instance of a pointwise operation on mappings.


Definition:Pointwise Minimum of Mappings/Real-Valued Functions

Let $X$ be a set, and let $f, g: X \to \R$ be real-valued functions. Let $\min$ be the min operation on $\R$ (Ordering on Real Numbers is Total Ordering ensures it is in fact defined). Then the '''pointwise minimum of $f$ and $g$''', denoted $\min \left({f, g}\right)$, is defined by: :$\min \left({f, g}\right): X \to \R: \min \left({f, g}\right) \, \left({x}\right) := \min \left({f \left({x}\right), g \left({x}\right)}\right)$ '''Pointwise minimum''' thence is an instance of a pointwise operation on real-valued functions.


Definition:Pointwise Minimum of Mappings/Extended Real-Valued Functions

Let $X$ be a set, and let $f, g: X \to \overline{\R}$ be extended real-valued functions. Let $\min$ be the min operation on $\overline{\R}$ (Ordering on Extended Real Numbers is Total Ordering ensures it is in fact defined). Then the '''pointwise minimum of $f$ and $g$''', denoted $\min \left({f, g}\right)$, is defined by: :$\min \left({f, g}\right): X \to \overline{\R}: \min \left({f, g}\right) \, \left({x}\right) := \min \left({f \left({x}\right), g \left({x}\right)}\right)$ '''Pointwise minimum''' thence is an instance of a pointwise operation on extended real-valued functions. Since the ordering on $\overline{\R}$ coincides on $\R$ with the standard ordering, this definition incorporates the definition for real-valued functions.


Definition:Simultaneous Equations/Linear Equations

A '''system of simultaneous linear equations''' is a set of equations: :$\displaystyle \forall i \in \set {1, 2, \ldots, m} : \sum_{j \mathop = 1}^n \alpha_{i j} x_j = \beta_i$ That is: {{begin-eqn}} {{eqn | l = \beta_1       | r = \alpha_{1 1} x_1 + \alpha_{1 2} x_2 + \cdots + \alpha_{1 n} x_n }} {{eqn | l = \beta_2       | r = \alpha_{2 1} x_1 + \alpha_{2 2} x_2 + \cdots + \alpha_{2 n} x_n }} {{eqn | o = \cdots}} {{eqn | l = \beta_m       | r = \alpha_{m 1} x_1 + \alpha_{m 2} x_2 + \cdots + \alpha_{m n} x_n }} {{end-eqn}}


Definition:Set of Residue Classes

The quotient set of congruence modulo $m$ denoted $\Z_m$ is: :$\Z_m = \dfrac \Z {\RR_m}$


Definition:Pointwise Scalar Multiplication of Extended Real-Valued Functions

Let $S$ be a non-empty set, and let ${\overline \R}^S$ be the set of all mappings $f: S \to \overline \R$. Here $\overline \R$ denotes the extended set of real numbers. Then '''pointwise ($\overline \R$-)scalar multiplication''' on ${\overline \R}^S$ is the binary operation $\cdot: \overline \R \times {\overline \R}^S \to {\overline \R}^S$ defined by: :$\forall \lambda \in \overline \R: \forall f \in {\overline \R}^S: \forall s \in S: \map {\paren {\lambda \cdot f} s := \lambda \cdot \map f s$ where the $\cdot$ on the right is extended real multiplication. '''Pointwise scalar multiplication''' thence is an instance of a pointwise operation on extended real-valued functions.


Definition:Addition/Real Numbers

The addition operation in the domain of real numbers $\R$ is written $+$. From the definition, the real numbers are the set of all equivalence classes $\eqclass {\sequence {x_n} } {}$ of Cauchy sequences of rational numbers. Let $x = \eqclass {\sequence {x_n} } {}, y = \eqclass {\sequence {y_n} } {}$, where $\eqclass {\sequence {x_n} } {}$ and $\eqclass {\sequence {y_n} } {}$ are such equivalence classes. Then $x + y$ is defined as: :$\eqclass {\sequence {x_n} } {} + \eqclass {\sequence {y_n} } {} = \eqclass {\sequence {x_n + y_n} } {}$


Definition:Multiplication/Real Numbers

The multiplication operation in the domain of real numbers $\R$ is written $\times$. From the definition, the real numbers are the set of all equivalence classes $\eqclass {\sequence {x_n} } {}$ of Cauchy sequences of rational numbers. Let $x = \eqclass {\sequence {x_n} } {}, y = \eqclass {\sequence {y_n} } {}$, where $\eqclass {\sequence {x_n} } {}$ and $\eqclass {\sequence {y_n} } {}$ are such equivalence classes. Then $x \times y$ is defined as: : $\eqclass {\sequence {x_n} } {} \times \eqclass {\sequence {y_n} } {} = \eqclass {\sequence {x_n \times y_n} } {}$


Definition:Ring (Abstract Algebra)/Addition

The distributand $*$ of a ring $\struct {R, *, \circ}$ is referred to as '''ring addition''', or just '''addition'''. The conventional symbol for this operation is $+$, and thus a general ring is usually denoted $\struct {R, +, \circ}$.


Definition:Rig

A '''rig''' is an additive semiring $\struct {S, *, \circ}$ in which $\struct {S, *}$ is a monoid. Alternatively, this is a semiring in which $\struct {S, *}$ is a commutative monoid. That is, $\struct {S, *, \circ}$ has the following properties: {{begin-axiom}} {{axiom | n = \text A 0         | q = \forall a, b \in S         | m = a * b \in S         | lc= Closure under $*$ }} {{axiom | n = \text A 1         | q = \forall a, b, c \in S         | m = \paren {a * b} * c = a * \paren {b * c}         | lc= Associativity of $*$ }} {{axiom | n = \text A 2         | q = \forall a, b \in S         | m = a * b = b * a         | lc= Commutativity of $*$ }} {{axiom | n = \text A 3         | q = \exists 0_S \in S: \forall a \in F         | m = a * 0_S = a = 0_S * a         | lc= Identity element for $*$: the zero }} {{axiom | n = \text M 0         | q = \forall a, b \in S         | m = a \circ b \in S         | lc= Closure under $\circ$ }} {{axiom | n = \text M 1         | q = \forall a, b, c \in S         | m = \paren {a \circ b} \circ c = a \circ \paren {b \circ c}         | lc= Associativity of $\circ$ }} {{axiom | n = \text M 2         | q = \forall a \in S         | m = a \circ 0_S = 0_S = 0_S \circ a         | lc= The zero is a zero element for $\circ$ }} {{axiom | n = \text D         | q = \forall a, b, c \in S         | m = a \circ \paren {b * c} = \paren {a \circ b} * \paren {a \circ c}, \paren {a * b} \circ c = \paren {a \circ c} * \paren {a \circ c}         | lc= $\circ$ is distributive over $*$ }} {{end-axiom}} Note that the zero element needs to be specified here as an axiom: $\text M 2$. By Ring Product with Zero, in a ring, the property $\text M 2$ of the zero element follows as a consequence of the ring axioms.


Definition:Congruence (Number Theory)/Modulo Operation

Let $\bmod$ be defined as the modulo operation: :$x \bmod y := \begin{cases} x - y \left \lfloor {\dfrac x y}\right \rfloor & : y \ne 0 \\ x & : y = 0 \end{cases}$ Then '''congruence modulo $z$''' is the relation on $\R$ defined as: :$\forall x, y \in \R: x \equiv y \pmod z \iff x \bmod z = y \bmod z$


Definition:Congruence (Number Theory)/Integer Multiple

Let $x, y \in \R$. Then '''$x$ is congruent to $y$ modulo $z$''' {{iff}} their difference is an integer multiple of $z$: :$x \equiv y \pmod z \iff \exists k \in \Z: x - y = k z$


Let $m \in \Z_{> 0}$. === Definition by Remainder after Division === {{:Definition:Congruence (Number Theory)/Integers/Remainder after Division}} === Definition by Modulo Operation === {{:Definition:Congruence (Number Theory)/Integers/Modulo Operation}} === Definition by Integer Multiple === We also see that $a$ is congruent to $b$ modulo $m$ if their difference is a multiple of $m$: {{:Definition:Congruence (Number Theory)/Integers/Integer Multiple}}


Definition:Congruence (Number Theory)/Residue

Let $a, b \in \Z$. Let $a \equiv b \pmod m$. Then $b$ is a '''residue of $a$ modulo $m$'''. '''Residue''' is another word meaning remainder, and is ''any'' integer congruent to $a$ modulo $m$.


Definition:Congruence (Number Theory)/Integers/Modulo Operation

Let $\bmod$ be defined as the modulo operation: :$x \bmod m := \begin{cases} x - m \left \lfloor {\dfrac x m}\right \rfloor & : m \ne 0 \\ x & : m = 0 \end{cases}$ Then '''congruence modulo $m$''' is the relation on $\Z$ defined as: :$\forall x, y \in \Z: x \equiv y \pmod m \iff x \bmod m = y \bmod m$


Definition:Congruence (Number Theory)/Integers/Integer Multiple

Let $x, y \in \Z$. '''$x$ is congruent to $y$ modulo $m$''' {{iff}} their difference is an integer multiple of $m$: :$x \equiv y \pmod m \iff \exists k \in \Z: x - y = k m$


Definition:Integral Multiple/Rings and Fields

Let $\struct {F, +, \times}$ be a ring or a field. Let $a \in F$. Let $n \in \Z$ be an integer. Then $n \cdot a$ is an integral multiple of $a$ where $n \cdot a$ is defined as in Powers of Ring Elements: :$n \cdot a := \begin {cases} 0_F & : n = 0 \\ \paren {\paren {n - 1} \cdot a} + a & : n > 1 \\ \size n \cdot \paren {-a} & : n < 0 \\ \end {cases}$ where $\size n$ is the absolute value of $n$.


Definition:Integral Multiple/Real Numbers

Let $x, y \in \R$ be real numbers. Then $x$ is an '''integral multiple''' of $y$ {{iff}} $x$ is congruent to $0$ modulo $y$: :$x \equiv 0 \pmod y$ That is: :$\exists k \in \Z: x = 0 + k y$


Definition:Trivial Annihilator

From Annihilator of Ring Always Contains Zero, we have that $0 \in \map {\mathrm {Ann} } R$ whatever the ring $R$ is. $R$ is said to have a '''trivial annihilator''' {{iff}} its annihilator $\map {\mathrm {Ann} } R$ consists '''only''' of the integer $0$.


Definition:Pendant Vertex

Let $G$ be a graph. A vertex $v$ of $G$ is said to be a '''pendant vertex''' {{iff}} it has degree $1$.


Definition:Pendant Edge

Let $G$ be a graph. An edge $e$ of $G$ is said to be a '''pendant edge''' {{iff}} it is incident on a pendant vertex.


Definition:Parallel Edges

Let $G$ be a graph or a digraph. A pair of edges $e$ and $e'$ of $G$ are said to be a '''parallel''' {{iff}} they are incident on precisely the same vertices.


Definition:Even Vertex (Graph Theory)

If the degree of $v$ is even, then $v$ is called an '''even vertex'''.


Definition:Odd Vertex (Graph Theory)

If the degree of $v$ is odd, then $v$ is an '''odd vertex'''.


Definition:Out-Degree

The '''out-degree of $v$ in $G$''' is the number of arcs which are incident from $v$. It is denoted $\map {\operatorname {outdeg}_G} v$, or just $\map {\operatorname {outdeg} } v$ if it is clear from the context which digraph is being referred to.


Definition:In-Degree

The '''in-degree of $v$ in $G$''' is the number of arcs which are incident to $v$.  It is denoted $\map {\operatorname {indeg}_G} v$, or just $\map {\operatorname {indeg} } v$ if it is clear from the context which digraph is being referred to.


Definition:Non-Empty Set

Let $S$ be a set. Then $S$ is said to be '''non-empty''' {{iff}} $S$ has at least one element. By the Axiom of Extension, this may also be phrased as: :$S \ne \O$ where $\O$ denotes the empty set. Many mathematical theorems and definitions require sets to be '''non-empty''' in order to avoid erratic results and inconsistencies.


Definition:Left Cancellable Operation

The operation $\circ$ in $\struct {S, \circ}$ is '''left cancellable''' {{iff}}: :$\forall a, b, c \in S: a \circ b = a \circ c \implies b = c$ That is, {{iff}} all elements of $\struct {S, \circ}$ are left cancellable.


Definition:Right Cancellable Operation

The operation $\circ$ in $\struct {S, \circ}$ is '''right cancellable''' {{iff}}: :$\forall a, b, c \in S: a \circ c = b \circ c \implies a = b$ That is, {{iff}} all elements of $\struct {S, \circ}$ are right cancellable.