hcju/proofwikips huggingface dataset:
Definition:Regular Expression
A '''regular expression''' is an algebraic structure on an ''alphabet'' $\Sigma$ defined as follows: * The empty-set regular expression, $\varnothing$, is a regular expression. * The empty-word regular expression, $\epsilon$, is a regular expression. * Every $\sigma \in \Sigma$ is a regular expression. (These are called ''literals''.) * If $R_1$ and $R_2$ are regular expressions, $R_1 R_2$ is a regular expression (''concatenation''). * If $R_1$ and $R_2$ are regular expressions, $R_1 \mid R_2$ is a regular expression (''alternation''). * If $R$ is a regular expression, $R^*$ is a regular expression (''Kleene star''). Every regular expression $R$ on an alphabet $\Sigma$ defines a language $L \left({R}\right) \subseteq \Sigma^*$, where $\Sigma^*$ is the set of all (finite-length) ''words'' (sequences) of symbols in $\Sigma$: * $ L \left({\varnothing}\right) = \varnothing $ (the empty set). * $ L \left({\epsilon}\right) = \left\{ { \left[{}\right] }\right\} $ (the set containing only the empty word). * If $R$ is a ''literal'' $\sigma$, $ L \left({R}\right) = \left\{ { \left[{ \sigma }\right] }\right\} $ (i.e., the set containing only the single-symbol word “$\sigma$”). * If $R$ is a ''concatenation'' $R_1 R_2$, $ L \left({R}\right) = \left\{ { w_1 w_2 : w_1 \in L \left({R_1}\right), w_2 \in L \left({R_2}\right) }\right\} $, where $w_1 w_2$ is the concatenation of the words $w_1$ and $w_2$. * If $R$ is an ''alternation'' $R_1 \mid R_2$, $ L \left({R}\right) = L \left({R_1}\right) \cup L \left({R_2}\right)$. * If $R$ is a ''Kleene star'' $R_1^*$, $ L\left({R}\right) $ is the smallest set satisfying the following: ** $\left[{}\right] \in L \left({R}\right) $ (the empty word is in the set); ** if $w_1 \in L \left({R}\right)$ and $w_2 \in L \left({R_1}\right)$, then $w_1 w_2 \in L \left({R}\right)$.
Definition:Directed Smooth Curve
Let $\R^n$ be a real cartesian space of $n$ dimensions. Let $\rho: \left[{a \,.\,.\, b}\right] \to \R^n$ be a smooth path in $\R^n$. The '''directed smooth curve''' with '''parameterization''' $\rho$ is defined as an equivalence class of smooth paths as follows: A smooth path $\sigma: \left[{a \,.\,.\, b}\right] \to \R^n$ belongs to the equivalence class of $\rho$ : : there exists a bijective differentiable strictly increasing real function: :: $\phi: \left[{c \,.\,.\, d}\right] \to \left[{a \,.\,.\, b}\right]$ : such that $\sigma = \rho \circ \phi$.  It follows from Directed Smooth Curve Relation is Equivalence and Fundamental Theorem on Equivalence Relations that this does in fact define an equivalence class. If a '''directed smooth curve''' is only defined by a smooth path $\rho$, then it is often denoted with the same symbol $\rho$. === Parameterization === {{:Definition:Directed Smooth Curve/Parameterization}} === Endpoints === {{:Definition:Directed Smooth Curve/Endpoints}}
Definition:Smooth Path/Simple/Complex Plane
Let $\gamma : \left[{a \,.\,.\, b}\right] \to \C$ be a '''smooth path in $\C$'''. $\gamma$ is a '''simple smooth path''' : :$(1): \quad \gamma$ is injective on the half-open interval $\left[{a \,.\,.\, b}\right)$ :$(2): \quad \forall t \in \left({a \,.\,.\, b}\right): \gamma \left({t}\right) \ne \gamma \left({b}\right)$ That is, if $t_1, t_2 \in \left({a \,.\,.\, b}\right)$ with $t_1 \ne t_2$, then $\gamma \left({a}\right) \ne \gamma \left({t_1}\right) \ne \gamma \left({t_2}\right) \ne \gamma \left({b}\right)$.
Definition:Arc-Connected/Points
Let $T = \struct {S, \tau}$ be a topological space. Let $a, b \in S$ be such that there exists an arc from $a$ to $b$. That is, there exists a continuous injection $f: \closedint 0 1 \to S$ such that $\map f 0 = a$ and $\map f 1 = b$. Then $a$ and $b$ are '''arc-connected'''. It is also declared that any point $a$ is '''arc-connected''' to itself.
Definition:Arc-Connected/Topological Space
Let $T = \struct {S, \tau}$ be a topological space. Then $T$ is '''arc-connected'''  every two points in $T$ are arc-connected in $T$. That is, $T$ is '''arc-connected''' : :for every $x, y \in X, \exists$ a continuous injection $f: \closedint 0 1 \to X$ such that $\map f 0 = x$ and $\map f 1 = y$.
Definition:Arc-Connected/Subset
Let $T = \left({S, \tau}\right)$ be a topological space. Let $U \subseteq S$ be a subset of $T$. Let $T\,' = \left({U, \tau_U}\right)$ be the subspace of $T$ induced by $U$. Then $U$ is '''arc-connected in $T$'''  every two points in $U$ are arc-connected in $T\,'$. That is, $U$ is '''arc-connected''' : :for every $x, y \in U$, there exists a continuous injection $f: \left[{0 \,.\,.\, 1}\right] \to S$ such that $f \left({0}\right) = x$ and $f \left({1}\right) = y$.
Definition:Mistake/Spello
A '''spello''' (abbreviation for '''spelling mistake''') is a mistake caused by inadequate spelling skills of the writer. These can sometimes be corrected by the writer having the self-doubt necessary to occasion the act of referring to a reliable manual, for example, a dictionary. A competent proof-reader (that is, one whose spelling skills are more comprehensive than the writer) is usually an adequate resource for reducing spelloes.
Definition:Mistake/Typo
A '''typo''' (abbreviation for '''typographical error''') is a mistake arising from pressing the wrong button on the machine via which you are transcribing your thoughts onto a storage medium. These can be corrected by assiduous checking by the writer.
Definition:Mistake/Homophone Horror
A '''homophone horror''' is a mistake caused by the use of a word which sounds the same as, but is spelled differently to, the word intended. A proficient proof-reader is needed to reduce instances of this sort of mistake.
Definition:Directed Smooth Curve/Parameterization/Complex Plane
Let $C$ be a directed smooth curve in the complex plane $\C$. Let $\gamma: \left[{a \,.\,.\, b}\right] \to \C$ be a smooth path in $\C$. Then $\gamma$ is a '''parameterization''' of $C$  $\gamma$ is an element of the equivalence class that constitutes $C$.
Definition:Directed Smooth Curve/Endpoints/Complex Plane
Let $C$ be a directed smooth curve in the complex plane $\C$. Let $C$ be parameterized by a smooth path $\gamma: \left[{a \,.\,.\, b}\right] \to \C$. Then: : $\gamma \left({a}\right)$ is the '''start point''' of $C$ : $\gamma \left({b}\right)$ is the '''end point''' of $C$. Collectively, $\gamma \left({a}\right)$ and $\gamma \left({b}\right)$ are known as the '''endpoints''' of $\rho$.
Definition:Integral Domain/Definition 1
An '''integral domain''' $\struct {D, +, \circ}$ is: :a commutative ring which is non-null :with a unity :in which there are no (proper) zero divisors, that is: ::: $\forall x, y \in D: x \circ y = 0_D \implies x = 0_D \text{ or } y = 0_D$ that is (from the Cancellation Law of Ring Product of Integral Domain) in which all non-zero elements are cancellable.
Definition:Integral Domain/Definition 2
An '''integral domain''' $\left({D, +, \circ}\right)$ is a commutative ring such that $\left({D^*, \circ}\right)$ is a monoid, all of whose elements are cancellable. In this context, $D^*$ denotes the ring $D$ without zero: $D \setminus \left\{{0_D}\right\}$.
Definition:Integral Domain Axioms
An integral domain is an algebraic structure $\struct {D, *, \circ}$, on which are defined two binary operations $\circ$ and $*$, which satisfy the following conditions: {{begin-axiom}} {{axiom | n = A0         | q = \forall a, b \in D         | m = a * b \in D         | lc= Closure under addition }} {{axiom | n = A1         | q = \forall a, b, c \in D         | m = \paren {a * b} * c = a * \paren {b * c}         | lc= Associativity of addition }} {{axiom | n = A2         | q = \forall a, b \in D         | m = a * b = b * a         | lc= Commutativity of addition }} {{axiom | n = A3         | q = \exists 0_D \in D: \forall a \in D         | m = a * 0_D = a = 0_D * a         | lc= Identity element for addition: the zero }} {{axiom | n = A4         | q = \forall a \in D: \exists a' \in D         | m = a * a' = 0_D = a' * a         | lc= Inverse elements for addition: negative elements }} {{axiom | n = M0         | q = \forall a, b \in D         | m = a \circ b \in D         | lc= Closure under product }} {{axiom | n = M1         | q = \forall a, b, c \in D         | m = \paren {a \circ b} \circ c = a \circ \paren {b \circ c}         | lc= Associativity of product }} {{axiom | n = D         | q = \forall a, b, c \in D         | m = a \circ \paren {b * c} = \paren {a \circ b} * \paren {a \circ c}         | lc= Product is distributive over addition }} {{axiom | m = \paren {a * b} \circ c = \paren {a \circ c} * \paren {b \circ c} }} {{axiom | n = C         | q = \forall a, b \in D         | m = a \circ b = b \circ a         | lc= Product is commutative }} {{axiom | n = U         | q = \exists 1_D \in D: \forall a \in D         | m = a \circ 1_D = a = 1_D \circ a         | lc= Identity element for product: the unity }} {{axiom | n = ZD         | q = \forall a, b \in D         | m = a \circ b = 0_D \iff a = 0 \lor b = 0         | lc= No proper zero divisors }} {{end-axiom}} These criteria are called the '''integral domain axioms'''.
Definition:Triangular Matrix/Upper Triangular Matrix
An '''upper triangular matrix''' is a matrix in which all the lower triangular elements are zero. That is, all the non-zero elements are on the main diagonal or in the upper triangle. That is, $\mathbf U$ is '''upper triangular''' : :$\forall a_{ij} \in \mathbf U: i > j \implies a_{ij} = 0$
Definition:Triangular Matrix/Lower Triangular Matrix
A '''lower triangular matrix''' is a matrix in which all the upper triangular elements are zero. That is, all the non-zero elements are on the main diagonal or in the lower triangle. That is, $\mathbf L$ is '''lower triangular''' : :$\forall a_{i j} \in \mathbf U: i < j \implies a_{i j} = 0$
Definition:Contour/Image/Complex Plane
Let $C$ be a '''contour''' in $\C$ defined by the (finite) sequence $\left\langle{C_1, \ldots, C_n}\right\rangle$ of directed smooth curves in $\C$. Let $C_i$ be parameterized by the smooth path $\gamma_i: \left[{a_i\,.\,.\,b_i}\right] \to \C$ for all $i \in \left\{ {1, \ldots, n}\right\}$. The '''image of $C$''' is defined as: :$\displaystyle \operatorname{Im} \left({C}\right) := \bigcup_{i \mathop = 1}^n \operatorname{Im} \left({\gamma_i}\right)$ where $\operatorname{Im} \left({\gamma_i}\right)$ denotes the image of $\gamma_i$. If $\operatorname{Im} \left({C}\right) \subseteq D$, where $D$ is a subset of $\C$, we say that $C$ is a '''contour in $D$'''.
Definition:Contour/Parameterization/Complex Plane
Let $C_1, \ldots, C_n$ be directed smooth curves in the complex plane $\C$. Let $C_i$ be parameterized by the smooth path $\gamma_i: \left[{a_i \,.\,.\, b_i}\right] \to \C$ for all $i \in \left\{ {1, \ldots, n}\right\}$. Let $C$ be the contour defined by the finite sequence $C_1, \ldots, C_n$. The '''parameterization of $C$''' is defined as the function $\gamma: \left[{a_1 \,.\,.\, c_n}\right] \to \C$ with: :$\gamma \restriction_{\left[{c_i \,.\,.\, c_{i + 1} }\right] } \left({t}\right) = \gamma_i \left({t}\right)$ where $\displaystyle c_i = a_1 + \sum_{j \mathop = 1}^i b_j - \sum_{j \mathop = 1}^i a_j$ for $i \in \left\{ {0, \ldots, n}\right\}$. Here, $\gamma \restriction_{\left[{c_i \,.\,.\, c_{i + 1} }\right] }$ denotes the restriction of $\gamma$ to $\left[{c_i \,.\,.\, c_{i + 1} }\right]$.
Definition:Contour/Endpoints/Complex Plane
Let $C_1, \ldots, C_n$ be directed smooth curves in $\C$. Let $C_i$ be parameterized by the smooth path $\gamma_i: \left[{a_i \,.\,.\, b_i}\right] \to \C$ for all $i \in \left\{ {1, \ldots, n}\right\}$. Let $C$ be the contour defined by the finite sequence $C_1, \ldots, C_n$. The '''start point''' of $C$ is $\gamma_1 \left({a_1}\right)$. The '''end point''' of $C$ is $\gamma_n \left({b_n}\right)$. Collectively, $\gamma_1 \left({a_1}\right)$ and $\gamma_n \left({b_n}\right)$ are referred to as the '''endpoints''' of $C$.
Definition:Contour/Simple/Complex Plane
Let $C_1, \ldots, C_n$ be directed smooth curves in the complex plane $\C$. Let $C_i$ be parameterized by the smooth path $\gamma_i: \left[{a_i \,.\,.\, b_i}\right] \to \C$ for all $i \in \left\{ {1, \ldots, n}\right\}$. Let $C$ be the contour defined by the finite sequence $C_1, \ldots, C_n$. $C$ is a '''simple contour''' : :$(1): \quad$ For all $i,j \in \left\{ {1, \ldots, n}\right\}, t_1 \in \left[{a_i \,.\,.\, b_i}\right), t_2 \in \left[{a_j \,.\,.\, b_j}\right)$ with $t_1 \ne t_2$, we have $\gamma_i \left({t_1}\right) \ne \gamma_j \left({t_2}\right)$. :$(2): \quad$ For all $k \in \left\{ {1, \ldots, n}\right\}, t \in \left[{a_k \,.\,.\, b_k}\right)$ where either $k \ne 1$ or $t \ne a_1$, we have $\gamma_k \left({t}\right) \ne \gamma_n \left({b_n}\right)$.
Definition:Quotient Epimorphism/Group
Let $G$ be a group. Let $N$ be a normal subgroup of $G$. Let $G / N$ be the quotient group of $G$ by $N$. The mapping $q_N: G \to G / N$ defined as: :$\forall x \in G: \map {q_N} x = x N$ is known as the '''quotient (group) epimorphism''' from $G$ to $G / N$.
Definition:Quotient Epimorphism/Ring
Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$ and whose unity is $1_R$. Let $J$ be an ideal of $R$. Let $\struct {R / J, +, \circ}$ be the quotient ring defined by $J$. The mapping $\phi: R \to R / J$ given by: : $\forall x \in R: \map \phi x = x + J$ is known as the '''quotient (ring) epimorphism from $\struct {R, +, \circ}$ (on)to $\struct {R / J, +, \circ}$'''.
Definition:Integrable Function/p-Integrable
Let $\struct {X, \Sigma, \mu}$ be a measure space. Let $f \in \MM_{\overline \R}, f: X \to \overline \R$ be a measurable function. Let $p \ge 1$ be a real number. Then $f$ is said to be '''$p$-integrable in respect to $\mu$''' : :$\displaystyle \int \size f^p \rd \mu < +\infty$ is $\mu$-integrable.
Definition:Derivative of Smooth Path/Complex Plane
Let $\gamma: \left[{a \,.\,.\, b}\right] \to \C$ be a '''smooth path in $\C$'''. Define the real function $x : \left[{a \,.\,.\, b}\right] \to \R$ by: :$\forall t \in \left[{a \,.\,.\, b}\right]: x \left({t}\right) = \operatorname{Re} \left({\gamma \left({t}\right)}\right)$ Define the real function $y: \left[{a \,.\,.\, b}\right] \to \R$ by: :$\forall t \in \left[{a \,.\,.\, b}\right]: y \left({t}\right) = \operatorname{Im} \left({\gamma \left({t}\right)}\right)$ where: : $\operatorname{Re} \left({\gamma \left({t}\right)}\right)$ denotes the real part of the complex number $\gamma \left({t}\right)$ : $\operatorname{Im} \left({\gamma \left({t}\right)}\right)$ denotes the imaginary part of $\gamma \left({t}\right)$. It follows from the definition of a  smooth path that both $x$ and $y$ are continuously differentiable. Let $x' \left({t}\right)$ and $y' \left({t}\right)$ denote the derivative of $x$ and $y$ {{WRT|Differentiation}} $t$. The '''derivative of $\gamma$''' is the continuous complex function $\gamma': \left[{a \,.\,.\, b}\right] \to \C$ defined by: :$\forall t \in \left[{a \,.\,.\, b}\right]: \gamma' \left({t}\right) = x' \left({t}\right) + i y' \left({t}\right)$
Definition:Ring Direct Product
{{:Definition:Ring Direct Product/Finite Case}}
Definition:Reversed Contour
Let $\R^n$ be a real cartesian space of $n$ dimensions. Let $C$ be a contour in $\R^n$. Then $C$ is defined as a concatenation of a finite sequence $C_1, \ldots, C_n$ of directed smooth curves in $\R^n$. The '''reversed contour of $C$''' is denoted $-C$ and is defined as the concatenation of the finite sequence: :$-C_n, -C_{n - 1}, \ldots, -C_1$ where $-C_i$ is the reversed directed smooth curve of $C_i$ for all $i \in \left\{ {1, \ldots, n}\right\}$.
Definition:Reversed Directed Smooth Curve
Let $\R^n$ be a real cartesian space of $n$ dimensions. Let $C$ be a directed smooth curves in $\R^n$. Let $C$ be parameterized by the smooth path $\gamma: \left[{a \,.\,.\, b}\right] \to \R^n$. The '''reversed directed smooth curve of $C$''' is denoted $-C$ and is defined as the directed smooth curve that is parameterized by: :$\rho: \left[{a \,.\,.\, b}\right] \to \R^n: \rho \left({t}\right) = \gamma \left({a + b - t}\right)$
Definition:Universal Net
Let $S$ be a non-empty set. Let $H$ be a set directed by $\preceq$. Let $f: H \to S$ be a net in $S$. Then $f$ is a '''universal net''' iff for each subset $A \subseteq S$, $f$ is eventually in $A$ or $f$ is eventually in $S \setminus A$. {{explain|"eventually"}}
Definition:Subnet
Let $\left({E, \le}\right)$ and $\left({D,\preceq}\right)$ be directed sets. Let $A$ be a non-empty set. Let $T: E \to A$ and $S: D \to A$ be nets in $A$. Then $T$ is a '''subnet''' of $S$ iff there exists a cofinal mapping $N: E \to D$ such that $T = S \circ N$. That is, for each $m$ in $D$, there is an $n$ in $E$ such that for each $p \in E$, $p \ge n \implies N \left({p}\right) \succeq m$.
Definition:Ordered Tuple as Ordered Set/Ordered Triple
The '''ordered triple''' $\tuple {a, b, c}$ of elements $a$, $b$ and $c$ can be defined either as the ordered pair: :$\tuple {a, \tuple {b, c} }$ or as the ordered pair: :$\tuple {\tuple {a, b}, c}$ where $\tuple {a, b}$ and $\tuple {b, c}$ are themselves ordered pairs.
Definition:Ordered Tuple as Ordered Set/Ordered Quadruple
The '''ordered quadruple''' $\tuple {a, b, c, d}$ of elements $a$, $b$, $c$ and $d$ is defined either as the ordered pair: :$\tuple {a, \tuple {b, c, d} }$ or: :$\tuple {\tuple {a, b, c}, d}$ where $\tuple {a, b, c}$ and $\tuple {b, c, d}$ are themselves ordered triples.
Definition:Ordered Tuple as Ordered Set/Ordered Tuple
The '''ordered tuple''' $\tuple {a_1, a_2, \ldots, a_n}$ of elements $a_1, a_2, \ldots, a_n$ is defined as either the ordered pair: :$\tuple {a_1, \tuple {a_2, a_3, \ldots, a_n} }$ or as the ordered pair: :$\tuple {\tuple {a_1, a_2, \ldots, a_{n - 1} }, a_n}$ where $\tuple {a_2, a_3, \ldots, a_n}$ and $\tuple {a_1, a_2, \ldots, a_{n - 1} }$ are themselves ordered tuples.
Definition:Relation/Relation as Ordered Pair
Some sources define a relation between $S$ and $T$ as an ordered pair: :$\struct {S \times T, \map P {s, t} }$ where: :$S \times T$ is the Cartesian product of $S$ and $T$ :$\map P {s, t}$ is a propositional function on ordered pairs $\tuple {s, t}$ of $S \times T$. Note that this approach leaves the domain and codomain inadequately defined. This situation arises in the case that $S$ or $T$ are empty, whence it follows that $S \times T$ is empty, but $T$ or $S$ are not themselves uniquely determined.
Definition:Relation/Relation as Mapping
It is possible to define a relation as a mapping from the cartesian product $S \times T$ to the set of truth values $\set {\text {true}, \text {false} }$: :$\RR: S \times T \to \set {\text {true}, \text {false} }: \forall \tuple {s, t} \in S \times T: \map \RR {s, t} = \begin{cases} \text {true} & : \tuple {s, t} \in \RR \\ \text {false} & : \tuple {s, t} \notin \RR \end{cases}$ This is called the characteristic function of the relation. However, care needs to be taken that a mapping then cannot be defined as a special relation, as this would be circular.
Definition:Meet (Order Theory)
Let $\struct {S, \preceq}$ be an ordered set. Let $a, b \in S$, and suppose that their infimum $\inf \set {a, b}$ exists in $S$. Then $a \wedge b$, the '''meet of $a$ and $b$''', is defined as: :$a \wedge b = \inf \set {a, b}$ Expanding the definition of infimum, one sees that $c = a \wedge b$ : :$c \preceq a$ and $c \preceq b$ and $\forall s \in S: s \preceq a \land s \preceq b \implies s \preceq c$
Definition:Join (Order Theory)
Let $\left({S, \preceq}\right)$ be an ordered set. Let $a, b \in S$. Let their supremum $\sup \left\{{a, b}\right\}$ exist in $S$. Then the '''join of $a$ and $b$''' is defined as: :$a \vee b = \sup \left\{{a, b}\right\}$ Expanding the definition of supremum, one sees that $c = a \vee b$ : :$a \preceq c$ and $b \preceq c$ and $\forall s \in S: a \preceq s \land b \preceq s \implies c \preceq s$
Definition:Join Semilattice
{{:Definition:Join Semilattice/Definition 1}}
Definition:Meet Semilattice
Let $\struct {S, \preceq}$ be an ordered set. Suppose that for all $a, b \in S$: :$a \wedge b \in S$, where $a \wedge b$ is the meet of $a$ and $b$. Then the ordered structure $\struct {S, \wedge, \preceq}$ is called a '''meet semilattice'''.
Definition:Semilattice
Let $\struct {S, \circ}$ be a semigroup. Then $\struct {S, \circ}$ is called a '''semilattice'''  $\circ$ is a commutative and idempotent operation.
Definition:Lattice
=== Definition 1 === {{:Definition:Lattice/Definition 1}} === Definition 2 === {{:Definition:Lattice/Definition 2}} === Definition 3 === {{:Definition:Lattice/Definition 3}}
Definition:Lattice/Definition 1
Let $\struct {S, \preceq}$ be an ordered set. Suppose that $S$ admits all finite non-empty suprema and finite non-empty infima. Denote with $\vee$ and $\wedge$ the join and meet operations on $S$, respectively. Then the ordered structure $\struct {S, \vee, \wedge, \preceq}$ is called a '''lattice'''.
Definition:Lattice/Definition 2
Let $\struct {S, \vee, \wedge, \preceq}$ be an ordered structure. Then $\struct {S, \vee, \wedge, \preceq}$ is called a '''lattice''' : :$\struct {S, \vee, \preceq}$ is a join semilattice :$\struct {S, \wedge, \preceq}$ is a meet semilattice
Definition:Lattice/Definition 3
Let $\struct {S, \vee}$ and $\struct {S, \wedge}$ be semilattices on a set $S$. Suppose that $\vee$ and $\wedge$ satisfy the absorption laws, that is, for all $a, b \in S$: :$a \vee \paren {a \wedge b} = a$ :$a \wedge \paren {a \vee b} = a$ Let $\preceq$ be the ordering on $S$ defined by: :$\forall a, b \in S: a \preceq b$  $a \vee b = b$ as on Semilattice Induces Ordering. Then the ordered structure $\struct {S, \vee, \wedge, \preceq}$ is called a '''lattice'''.
Definition:Bounded Lattice
=== Definition 1 === {{:Definition:Bounded Lattice/Definition 1}} === Definition 2 === {{:Definition:Bounded Lattice/Definition 2}} === Definition 3 === {{:Definition:Bounded Lattice/Definition 3}}
Definition:Bounded Lattice/Definition 1
Let $\left({S, \preceq}\right)$ be an ordered set. Let $S$ admit all finite suprema and finite infima. Let $\vee$ and $\wedge$ be the join and meet operations on $S$, respectively. Then the ordered structure $\left({S, \vee, \wedge, \preceq}\right)$ is a '''bounded lattice'''.
Definition:Bounded Lattice/Definition 2
Let $\left({S, \vee, \wedge, \preceq}\right)$ be a lattice. Let $\vee$ and $\wedge$ have identity elements $\bot$ and $\top$ respectively. Then $\left({S, \vee, \wedge, \preceq}\right)$ is a '''bounded lattice'''.
Definition:Lowest Common Multiple/Integers
For all $a, b \in \Z: a b \ne 0$, there exists a smallest $m \in \Z: m > 0$ such that $a \divides m$ and $b \divides m$. This $m$ is called the '''lowest common multiple of $a$ and $b$''', and denoted $\lcm \set {a, b}$.
Definition:Lowest Common Multiple/Integral Domain
Let $D$ be an integral domain and let $a, b \in A$ be nonzero. $l$ is the '''lowest common multiple''' of $a$ and $b$ : :$(1): \quad$ both $a$ and $b$ divide $l$ :$(2): \quad$ if $m$ is another element such that $a$ and $b$ divide $m$, then $l$ divides $m$.
Definition:Operation/Binary Operation/Product
Let $z = x \circ y$. Then $z$ is called the '''product''' of $x$ and $y$. This is an extension of the normal definition of product that is encountered in conventional arithmetic.
Definition:Diophantine Equation/Linear Diophantine Equation
A '''linear Diophantine equation''' is a Diophantine equation in which all the arguments appear to no higher than the first degree. For example: :$ax + by + c = 0$ :$a_1 x_1 + a_2 x_2 + \cdots + a_n x_n = b$
Definition:Absorption Law
Let $\struct {S, \circ, *}$ be an algebraic structure. Let both $\circ$ and $*$ be commutative. Then '''$\circ$ absorbs $*$''' : :$\forall a, b \in S: a \circ \paren {a * b} = a$ This equality is called the '''absorption law of $\circ$ for $*$'''.
Definition:Supremum of Set/Finite Supremum
If $T$ is finite, $\sup T$ is called a '''finite supremum'''.
Definition:Infimum of Set/Finite Infimum
If $T$ is finite, $\inf T$ is called a '''finite infimum'''.
Definition:Empty Supremum
Let $\struct {S, \preceq}$ be an ordered set. Then the '''empty supremum''' is the supremum $\sup \O$.
Definition:Empty Infimum
Let $\struct {S, \preceq}$ be an ordered set. Then the '''empty infimum''' is the infimum $\inf \O$. By Infimum of Empty Set is Greatest Element, it exists  $\struct {S, \preceq}$ has a greatest element.