hcju/proofwikips huggingface dataset:
Definition:Indexing Set/Notation
The family of elements $x$ of $S$ indexed by $I$ is often seen with one of the following notations: :$\family {x_i}_{i \mathop \in I}$ :$\paren {x_i}_{i \mathop \in I}$ :$\set {x_i}_{i \mathop \in I}$ There is little consistency in the literature, but $\paren {x_i}_{i \mathop \in I}$ is perhaps most common. The preferred notation on  is $\family {x_i}_{i \mathop \in I}$. The subscripted $i \in I$ is often left out, if it is obvious in the particular context. Note the use of $x_i$ to denote the image of the index $i$ under the indexing function $x$. As $x$ is actually a mapping, one would expect the conventional notation $\map x i$. However, this is generally not used, and $x_i$ is used instead.
Definition:Indexing Set/Family of Subsets
Let $S$ be a set. Let $I$ be an indexing set. For each $i \in I$, let $S_i$ be a corresponding subset of $S$. Let $\family {S_i}_{i \mathop \in I}$ be a family of subsets of $S$ indexed by $I$. Then $\family {S_i}_{i \mathop \in I}$ is referred to as an '''indexed family of subsets (of $S$ by $I$)'''.
Definition:Cartesian Product/Cartesian Space/Real Cartesian Space/Countable
The countable cartesian product defined as: :$\displaystyle \R^\omega := \R \times \R \times \cdots = \prod_\N \R$ is called the '''countable-dimensional real cartesian space'''. Thus, $\R^\omega$ can be defined as the set of all real sequences: :$\R^\omega = \left\{{\left({x_1, x_2, \ldots}\right): x_1, x_2, \ldots \in \R}\right\}$
Definition:Composite Defined by Permutation
Let $\oplus$ be an $n$-ary operation on a set $S$. Let $\left\langle{a_k}\right\rangle_{k \mathop \in A}$ be a sequence of $n$ terms of $S$. Let $\sigma: A \to A$ be a permutation of $A$. Then the '''composite of the ordered $n$-tuple defined by the sequence $\left\langle{a_{\sigma \left({k}\right)}}\right\rangle_{k \mathop \in A}$''' is defined as: {{begin-eqn}} {{eqn | l = \bigoplus_{k \mathop \in A} a_{\sigma \left({k}\right)}       | r = \begin{cases} a_{\sigma \left({1}\right)} & : n = 1 \\ \oplus \left({a_{\sigma \left({1}\right)}, a_{\sigma \left({2}\right)}, \ldots, a_{\sigma \left({m}\right)} }\right) \oplus a_{\sigma \left({n + 1}\right)} & : n = m + 1 \end{cases}       | c =  }} {{eqn | r = a_{\sigma \left({1}\right)} \oplus a_{\sigma \left({2}\right)} \oplus \cdots  \oplus a_{\sigma \left({n}\right)}       | c =  }} {{end-eqn}}
Definition:Power of Element/Semigroup
Let $\struct {S, \circ}$ be a semigroup which has no identity element. Let $a \in S$. For $n \in \N_{>0}$, the $n$th power of $a$ (under $\circ$) is defined as: :$\circ^n a = \begin{cases} a & : n = 1 \\ \paren {\circ^m a} \circ a & : n = m + 1 \end{cases}$ That is: :$a^n = \underbrace {a \circ a \circ \cdots \circ a}_{n \text{ copies of } a}$ which from the General Associativity Theorem is unambiguous.
Definition:Incircle/Incenter
The center of an incircle of a polygon is called an '''incenter of the polygon'''.
Definition:Incircle/Inradius
A radius of an incircle of a polygon is called an '''inradius of the polygon'''. In the above diagram, $OG$ is an inradius of the polygon $ABCDEF$.
Definition:Independent Subgroups/Definition 2
The subgroups $H_1, H_2, \ldots, H_n$ are '''independent''' : :$\displaystyle \forall k \in \set {2, 3, \ldots, n}: \paren {\prod_{j \mathop = 1}^{k - 1} H_j} \cap H_k = \set e$
Definition:Independent Subgroups/Definition 1
The subgroups $H_1, H_2, \ldots, H_n$ are '''independent''' : :$\displaystyle \prod_{k \mathop = 1}^n h_k = e \iff \forall k \in \set {1, 2, \ldots, n}: h_k = e$ where $h_k \in H_k$ for all $k \in \set {1, 2, \ldots, n}$.
Definition:Factorial/Historical Note
The symbol $!$ used on  for the factorial, which is now universal, was introduced by {{AuthorRef|Christian Kramp}} in his $1808$ work {{BookLink|Élémens d'arithmétique universelle|C. Kramp}}. Before that, various symbols were used whose existence is now of less importance. Notations for $n!$ in history include the following: :$\sqbrk n$ as used by {{AuthorRef|Leonhard Paul Euler|Euler}} :$\mathop{\Pi} n$ as used by {{AuthorRef|Carl Friedrich Gauss|Gauss}} :$\left\lvert {\kern-1pt \underline n} \right.$ and $\left. {\underline n \kern-1pt} \right\rvert$, once popular in England and Italy. {{AuthorRef|Augustus De Morgan}} declared his reservations about {{AuthorRef|Christian Kramp|Kramp}}'s notation thus: :''Amongst the worst barbarisms is that of introducing symbols which are quite new in mathematical, but perfectly understood in common, language. Writers have borrowed from the Germans the abbreviation $n!$ ... which gives their pages the appearance of expressing admiration  that $2$, $3$, $4$, etc., should be found in mathematical results.'' ::::-- {{BookReference|A History of Mathematical Notations|1929|Florian Cajori|volume = 2}} The use of $n!$ for non-integer $n$ is uncommon, as the Gamma function tends to be used instead.
Definition:Factorial/Multiindices
Let $\alpha$ be a multiindex, indexed by a set $J$ such that for each $j \in J$, $\alpha_j \geq 0$. Then we define: :$\displaystyle\alpha! = \prod_{j \mathop \in J} \alpha_j!$ where the factorial on the right is a factorial of natural numbers. Note that by definition, all by finitely many of the $\alpha_j$ are zero, so the product over $J$ is convergent.
Definition:Empty Topological Space
Let $\varnothing$ denote the empty set. The topological space $\left({\varnothing, \left\{ {\varnothing}\right\} }\right)$ is called the '''empty topological space'''.
Definition:Topology/Definition 1
Let $S$ be a set. A '''topology on $S$''' is a subset $\tau \subseteq \powerset S$ of the power set of $S$ that satisfies the open set axioms: {{:Definition:Open Set Axioms}} If $\tau$ is a '''topology''' on $S$, then $\struct {S, \tau}$ is called a topological space. The elements of $\tau$ are called the open sets of $\struct {S, \tau}$.
Definition:Topology/Definition 2
Let $S$ be a set. A '''topology on $S$''' is a subset $\tau \subseteq \powerset S$ of the power set of $S$ that satisfies the following axioms: {{begin-axiom}} {{axiom | n = \text O 1'         | t = The union of an arbitrary subset of $\tau$ is an element of $\tau$. }} {{axiom | n = \text O 2'         | t = The intersection of any finite subset of $\tau$ is an element of $\tau$. }} {{end-axiom}}
Definition:Neighborhood (Topology)/Neighborhood defined as Open
Some authorities define a neighborhood of a set $A$ as what  defines as an open neighborhood: :$N_A$ is a '''neighborhood of $A$'''  $N_A$ is an open set of $T$ which itself contains $A$. That is, in order to be a neighborhood of $A$ in $T$, $N_A$ must not only be a subset of $T$, but also be an open set of $T$. However, this treatment is less common, and considered by many to be old-fashioned. When the term '''neighborhood''' is used on this site, it is assumed to be not necessarily open unless so specified.
Definition:Basis (Topology)/Analytic Basis/Definition 1
Let $\struct {S, \tau}$ be a topological space. An '''analytic basis for $\tau$''' is a subset $\BB \subseteq \tau$ such that: :$\displaystyle \forall U \in \tau: \exists \AA \subseteq \BB: U = \bigcup \AA$ That is, such that for all $U \in \tau$, $U$ is a union of sets from $\BB$.
Definition:Basis (Topology)/Analytic Basis/Definition 2
Let $\left({S, \tau}\right)$ be a topological space. Let $\mathcal B \subseteq \tau$. Then $\mathcal B$ is an analytic basis for $\tau$ : :$\forall U \in \tau: \forall x \in U: \exists V \in \mathcal B: x \in V \subseteq U$
Definition:Isolated Point (Topology)/Subset/Definition 1
$x \in H$ is an '''isolated point of $H$''' : :$\exists U \in \tau: U \cap H  = \set x$ That is,  there exists an open set of $T$ containing no points of $H$ other than $x$.
Definition:Isolated Point (Topology)/Subset/Definition 2
$x \in H$ is an '''isolated point of $H$'''  $x$ is not a limit point of $H$. That is,  $x$ is not in the derived set of $H$.
Definition:Addition in Minimal Infinite Successor Set
Let $\omega$ be the minimal infinite successor set. The binary operation $+$ is defined on $\omega$ as follows: :$\forall m,n \in \omega: \begin{cases} m + 0 &= m \\ m + n^+ &= \left({m + n}\right)^+\end{cases}$ where $m^+$ is the successor set of $m$. This operation is called '''addition'''.
Definition:Addition for Natural Numbers in Real Numbers
Let $\struct {\R, +, \times, \le}$ be the field of real numbers. Let $\N$ be the natural numbers in $\R$. Then the restriction of $+$ to $\N$ is called '''addition'''.
Definition:Closed Set/Topology/Definition 1
'''$H$ is closed (in $T$)'''  its complement $S \setminus H$ is open in $T$.  That is, $H$ is '''closed'''  $\paren {S \setminus H} \in \tau$. That is,  $S \setminus H$ is an element of the topology of $T$.
Definition:Closed Set/Topology/Definition 2
'''$H$ is closed (in $T$)'''  every limit point of $H$ is also a point of $H$. That is, by the definition of the derived set: :'''$H$ is closed (in $T$)'''  $H' \subseteq H$ where $H'$ denotes the derived set of $H$.
Definition:Scaled Euclidean Metric
Let $\R_{>0}$ be the set of strictly positive real numbers. Let $\delta: \R_{>0} \times \R_{>0} \to \R$ be the metric on $\R_{>0}$ defined as: :$\forall x, y \in \R_{>0}: \map \delta {x, y} = \dfrac {\size {x - y} } {x y}$ Then $\delta$ is the '''scaled Euclidean metric''' on $\R_{>0}$.
Definition:Alexandroff Extension
Let $T = \struct {S, \tau}$ be a non-empty topological space. Let $p$ be a new element not in $S$. Let $S^* := S \cup \set p$. Let $\tau^*$ be the topology on $S^*$ defined such that $U \subseteq S^*$ is open : :$U$ is an open set of $T$ or :$U$ is the complement in $T^*$ of a closed and compact subset of $T$. This topology is called the '''Alexandroff extension''' on $S$.
Definition:Centroid of Triangle
Let $\triangle ABC$ be a triangle. The '''centroid''' of $\triangle ABC$ is where its three medians meet. :500px
Definition:One-Sided Limit of Real Function
A '''one-sided limit of real function''' is a right-hand limit of real function or a left-hand limit of real function: === Limit from Right === {{:Definition:Limit of Real Function/Right}} === Limit from Left === {{:Definition:Limit of Real Function/Left}}
Definition:Functor/Contravariant/Definition 1
Let $\mathbf C$ and $\mathbf D$ be metacategories. A '''contravariant functor''' $F : \mathbf C \to \mathbf D$ consists of: * An '''object functor''' $F_0$ that assigns to each object $X$ of $\mathbf C$ an object $FX$ of $\mathbf D$. * An '''arrow functor''' $F_1$ that assigns to each arrow $f : X \to Y$ of $\mathbf C$ an arrow $Ff : FY \to FX$ of $\mathbf D$. These '''functors''' must satisfy, for any morphisms $X \stackrel f \longrightarrow Y \stackrel g \longrightarrow Z$ in $\mathbf C$: :$\map F {g \circ f} = F f \circ F g$ and: :$\map F {\operatorname {id}_X} = \operatorname {id}_{F X}$ where: :$\operatorname {id}_W$ denotes the identity arrow on an object $W$ and: :$\circ$ is the composition of morphisms.
Definition:Functor/Contravariant/Definition 2
Let $\mathbf C$ and $\mathbf D$ be metacategories. A '''contravariant functor''' $F : \mathbf C \to \mathbf D$ is a covariant functor: :$F: \mathbf C^{\text{op}} \to \mathbf D$ where $\mathbf C^{\text{op}}$ is the dual category of $\mathbf C$.
Definition:Normal Extension/Definition 1
Let $L / K$ be a field extension. Then $L / K$ is a '''normal extension''' : : for every irreducible polynomial $f \in K \left[{x}\right]$ with at least one root in $L$, $f$ splits completely in $L$.
Definition:Normal Extension/Definition 2
Let $L / K$ be a field extension. Let $\overline K$ be the algebraic closure of $K$. Let $\operatorname{Gal} \left({L / K}\right)$ denote the set of embeddings of $L$ in $\overline K$ which fix $K$ pointwise. Then $L/K$ is a '''normal extension''' : : $\sigma \left({L}\right) = L$ for each $\sigma \in \operatorname{Gal} \left({L / K}\right)$.
Definition:Decomposable Element
Let $\left({D, +, \circ}\right)$ be an integral domain. Let $x$ be a non-zero non-unit element of $D$. Let there exist a complete factorization of $x$ in $D$. Then $x$ is '''decomposable'''.
Definition:Zariski Topology/Affine Space
Let $k$ be a field. Let $\mathbb A^n \left({k}\right) = k^n$ denote the standard affine space of dimension $n$ over $k$. The '''Zariski topology''' on $\mathbb A^n \left({k}\right)$ is the topology on the direct product $k^n$ whose closed sets are the affine algebraic sets in $\mathbb A^n \left({k}\right)$.
Definition:Zariski Topology/Spectrum of Ring
Let $A$ be a commutative ring with unity. Let $\operatorname{Spec} \left({A}\right)$ be the prime spectrum of $A$. The '''Zariski topology''' on $\operatorname{Spec} A$ is the topology with closed sets the vanishing sets $V \left({S}\right)$ for $S \subseteq A$.
Definition:Logarithm/Base
Let $\log_a$ denote the logarithm function on whatever domain: $\R$ or $\C$. The constant $a$ is known as the '''base''' of the logarithm.
Definition:Tangent Vector/Definition 1
A '''tangent vector $X_m$ on $M$ at $m$''' is a linear transformation: :$X_m: C^\infty \left({V, \R}\right) \to \R$ which satisfies the Leibniz law: :$\displaystyle X_m \left({f g}\right) = X_m \left({f}\right) \, g \left({m}\right) + f \left({m}\right) \, X_m \left({g}\right)$
Definition:Tangent Vector/Definition 2
Let $I$ be an open real interval with $0 \in I$.  Let $\gamma: I \to M$ be a smooth curve with $\gamma \left({0}\right) = m$.  Then a '''tangent vector''' $X_m$ at a point $m \in M$ is a mapping  :$X_m: C^\infty \left({V, \R}\right) \to \R$ defined by: :$X_m \left({f} \right) := \dfrac {\mathrm d} {\mathrm d \tau} {\restriction_0} \, f \circ \gamma \left({\tau}\right)$ for all $f \in C^\infty \left({V, \R}\right)$.
Definition:Inverse Laplace Transform
=== Definition 1 === {{:Definition:Inverse Laplace Transform/Definition 1}} === Definition 2 === {{:Definition:Inverse Laplace Transform/Definition 2}}
Definition:Inverse Laplace Transform/Definition 2
Let $\map f s: S \to \R$ be a complex function, where $S \subset \R$. {{mistake|How can it be a complex function when both its domain and codomain are wholly real?}} The '''inverse Laplace transform''' of $f$, denoted $\map F t: \R \to S$, is defined as: :$\map F t = \dfrac 1 {2 \pi i} \PV_{c \mathop - i \, \infty}^{c \mathop + i \, \infty} e^{s t} \map f s \rd s = \frac 1 {2 \pi i} \lim_{T \mathop \to \infty} \int_{c \mathop - i \, T}^{c \mathop + i \, T} e^{s t} \map f s \rd s$ where: :$\PV$ is the Cauchy principal value of the integral :$c$ is any real constant such that all the singular points of $\map f s$ lie to the left of the line $\map \Re s = c$ in the complex $s$ plane.
Definition:Inverse Laplace Transform/Definition 1
Let $f \left({s}\right) : S \to \R$ be a function, where $S \subset \R$. Let $F$ be another function such that $F$ is the Laplace transform of $f$. Then, $f$ is the '''inverse Laplace transform''' of $F$.
Definition:Topological Surface
A '''surface''', in the context of topology, is a smooth topological manifold whose dimension is $2$. More briefly, a '''surface''' is a smooth $2$-manifold.
Definition:Taylor Polynomial
Let $f$ be a real function which is continuous on the closed interval $\closedint a b$ and $n + 1$ times differentiable on the open interval $\openint a b$. Let $\xi \in \openint a b$. The polynomial $T_n$ defined as:  :$\displaystyle \map {T_n} x = \sum_{i \mathop = 0}^i \frac {\paren {x - \xi}^i} {i!} \map {f^{\paren i} } \xi$ is known as the '''Taylor polynomial of degree $n$ for $f$ about $\xi$'''. That is, a '''Taylor polynomial''' is a Taylor series taken for $n$ initial terms.
Definition:Bessel Function
The '''Bessel functions''' are solutions to Bessel's equation: :$x^2 \dfrac {\d^2 y} {\d x^2} + x \dfrac {\d y} {\d x} + \paren {x^2 - n^2} y = 0$ These solutions have two main classes: :the Bessel functions of the first kind $J_n$ and: :the Bessel functions of the second kind $Y_n$.
Definition:Monster Group
A group $G$ is a '''Monster group''' and the largest sporadic simple group  it has the order:   :$808017424794512875886459904961710757005754368000000000 = 2^{46}.3^{20}.5^9.7^6.11^2.13.17.19.23.29.31.41.47.59.71$
Definition:Dirac Delta Function
Let $\epsilon \in \R_{>0}$ be a (strictly) positive real number. Consider the real function $F_\epsilon: \R \to \R$ defined as: :$\map {F_\epsilon} x := \begin{cases} 0 & : x < 0 \\ \dfrac 1 \epsilon & : 0 \le x \le \epsilon \\ 0 & : x > \epsilon \end{cases}$ The '''Dirac delta function''' is defined as: :$\map \delta x := \displaystyle \lim_{\epsilon \mathop \to 0} \map {F_\epsilon} x$
Definition:Symmetric Set/Real Numbers
Let $\R$ be the set of real numbers. Let $S \subseteq \R$ such that: :$\forall x \in S: -x \in S$ That is, for every element in $S$, its negative is also in $S$. Then $S$ is a '''symmetric subset of $\R$''', or (if $\R$ is implicit) $S$ is a '''symmetric set'''.
Definition:Additive Inverse/Ring
Let $\struct {R, +, \circ}$ be a ring whose ring addition operation is $+$. Let $a \in R$ be any arbitrary element of $R$. The '''additive inverse''' of $a$ is its inverse under ring addition, denoted $-a$: :$a + \paren {-a} = 0_R$ where $0_R$ is the zero of $R$.
Definition:Additive Inverse/Number
Let $\Bbb F$ be one of the standard number systems: $\N$, $\Z$, $\Q$, $\R$, $\C$. Let $a \in \Bbb F$ be any arbitrary number. The '''additive inverse''' of $a$ is its inverse under addition, denoted $-a$: :$a + \paren {-a} = 0$
Definition:Mellin Transform
Let $\map f t: \R_{\ge 0} \to \C$ be a function of a real variable $t$. The '''Mellin Transform''' of $f$, denoted $\map \MM f$ or $\phi$, is defined as: :$\displaystyle \map {\MM \set {\map f t} } s = \map \phi s = \int_0^{\to +\infty} t^{s - 1} \map f t \rd t$ wherever this improper integral exists. Here $\map \MM f$ is a complex function of the variable $s$.
Definition:Quaternion Modulus
Let $\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ be a quaternion, where $a, b, c, d \in \R$. Then the '''(quaternion) modulus of''' $\mathbf x$ is written as $\vert \mathbf x \vert$ and is defined as: :$\vert \mathbf x \vert := \sqrt{a^2 + b^2 + c^2 + d^2}$ The '''quaternion modulus''' is a real-valued function, and as when appropriate is referred to as the '''quaternion modulus function'''.
Definition:Quaternion Conjugate/Quaternion Conjugation
The operation of '''quaternion conjugation''' is the mapping: : $\overline \cdot: \mathbb H \to \mathbb H: \mathbf x \mapsto \overline{\mathbf x}$. where $\overline{\mathbf x}$ is the quaternion conjugate of $x$. That is, it maps a quaternion to its quaternion conjugate.
Definition:Natural Numbers/Axiomatization
=== Peano's Axioms === {{:Axiom:Peano's Axioms}} === Naturally Ordered Semigroup === {{:Definition:Naturally Ordered Semigroup}} === 1-Based Natural Numbers === The following axioms are intended to capture the behaviour of $\N_{>0}$, the element $1 \in \N_{>0}$, and the operations $+$ and $\times$ as they pertain to $\N_{>0}$: {{:Axiom:Axiomatization of 1-Based Natural Numbers}}
Definition:Natural Numbers/Construction
=== Von Neumann Construction of Natural Numbers === {{:Definition:Natural Numbers/Von Neumann Construction}} === Inductive Set Definition for Natural Numbers === {{:Definition:Natural Numbers/Inductive Set Definition}} === Inductive Set Definition for Natural Numbers in Real Numbers === {{:Definition:Natural Numbers/Inductive Sets in Real Numbers}}
Definition:Ordering on Natural Numbers
=== Ordering on Peano Structure === {{:Definition:Ordering on Natural Numbers/Peano Structure}} === Ordering on Naturally Ordered Semigroup === {{:Definition:Ordering on Natural Numbers/Naturally Ordered Semigroup}} === Ordering on $1$-Based Natural Numbers === {{:Definition:Ordering on Natural Numbers/1-Based}} === Ordering on Minimal Infinite Successor Set === {{:Definition:Ordering on Natural Numbers/Minimal Infinite Successor Set}} === Ordering on Von Neumann Construction of Natural Numbers === {{:Definition:Ordering on Natural Numbers/Von Neumann Construction}} === Ordering on Natural Numbers in Real Numbers === {{:Definition:Ordering on Natural Numbers/Restriction of Real Numbers}}
Definition:Ordering on Natural Numbers/1-Based
Let $\N_{>0}$ be the axiomatised $1$-based natural numbers. The '''strict ordering of $\N_{>0}$''', denoted $<$, is defined as follows: :$\forall a, b \in \N_{>0}: a < b \iff \exists c \in \N_{>0}: a + c = b$ The '''(weak) ordering of $\N_{>0}$''', denoted $\le$, is defined as: :$\forall a, b \in \N_{>0}: a \le b \iff a = b \lor a < b$
Definition:Ordering on Natural Numbers/Naturally Ordered Semigroup
Let $\struct {S, \circ, \preceq}$ be a naturally ordered semigroup. The relation $\preceq$ in $\struct {S, \circ, \preceq}$ is called the '''ordering'''.
Definition:Ordering on Natural Numbers/Peano Structure
Let $\struct {P, 0, s}$ be a Peano structure. The '''ordering''' of $P$ is the relation $\le$ defined by: :$\forall m, n \in P: m \le n \iff \exists p \in P: m + p = n$ where $+$ denotes addition in $\struct {P, 0, s}$.
Definition:Ordering on Natural Numbers/Minimal Infinite Successor Set
Let $\omega$ be the minimal infinite successor set. The '''strict ordering''' of $\omega$ is the relation $<$ defined by: :$\forall m, n \in \omega: m < n \iff m \in n$ The '''(weak) ordering''' of $\omega$ is the relation $\le$ defined by: :$\forall m, n \in \omega: m \le n \iff m < n \lor m = n$
Definition:Ordering on Natural Numbers/Restriction of Real Numbers
Let $\struct {\R, +, \times, \le}$ be the field of real numbers. Let $\N$ be the natural numbers in $\R$. Then the '''ordering''' of $\N$ is the restriction of $\le$ to $\N$.
Definition:Addition on 1-Based Natural Numbers
Let $\N_{>0}$ be the $1$-based natural numbers, axiomatized by: {{:Axiom:Axiomatization of 1-Based Natural Numbers}} The operation $+$ in this axiomatizaton is called '''addition'''.
Definition:Octonion/Addition
The '''sum''' of two octonions $x=\left({a, b}\right)$ and $y=\left({c, d}\right)$ where $a, b, c, d \in \mathbb H$ are quaternions is defined as: :$x + y = \left({a, b}\right) + \left({c, d}\right) = \left({a + c, b + d}\right)$
Definition:Figurate Number
A '''figurate number''' is loosely defined as a (natural) number corresponding to a set of objects which can be arranged in a geometric pattern (in an arbitrary number of dimensions, but usually $2$ or $3$).
Definition:Periodic Continued Fraction/Cycle
The repeating block in a periodic (or purely periodic) continued fraction $F$ is called the '''cycle''' of $F$.
Definition:Theory (Logic)/Complete
$T$ is '''complete''' : : for every $\mathcal L$-sentence $\phi$, either $T \models \phi$ or $T \models \neg \phi$ where $T \models \phi$ denotes semantic entailment.
Definition:Theory (Logic)/Maximal
$T$ is '''maximal''' : :for every $\mathcal L$-sentence $\phi$, either $\phi \in T$ or $\neg \phi \in T$.
Definition:Theory (Logic)/Structure
Let $\mathcal M$ be an $\mathcal L$-structure. The '''$\mathcal L$-theory of $\mathcal M$''' is the $\mathcal L$-theory consisting of those $\mathcal L$-sentences $\phi$ such that: :$\mathcal M \models \phi$ where $\models$ denotes that $\mathcal M$ is a model for $\phi$. This theory can be denoted $\operatorname{Th} \left({\mathcal M}\right)$ when the language $\mathcal L$ is understood.
Definition:Stability (Model Theory)/Kappa-Stable Theory
$T$ is $\kappa$-'''stable''' : : for all models $\mathcal M$ of $T$ : for all subsets $A \subseteq \mathcal M$ of cardinality $\kappa$ and: : for all $n \in \N$ the cardinality $\left\vert{ {S_n}^{\mathcal M} \left({A}\right) }\right\vert$ of the set ${S_n}^{\mathcal M} \left({A}\right)$ of complete $n$-types over $A$ is $\kappa$.
Definition:Stability (Model Theory)/Stable Theory
$T$ is '''stable'''  it is $\kappa$-stable for some $\kappa \ge \aleph_0$.
Definition:Stability (Model Theory)/Kappa-Stable Structure
Let $\mathcal M$ be an $\mathcal L$-structure. Let $\operatorname {Th} \left({\mathcal M}\right)$ be the $\mathcal L$-theory of $\mathcal M$. $\mathcal M$ is '''$\kappa$-stable''' if $\operatorname{Th} \left({\mathcal M}\right)$ is $\kappa$-stable.
Definition:Affine Monoid
An '''affine monoid''' is a monoid that is: : finitely generated and: : isomorphic to a submonoid of a free abelian group $\Z^d$, for some $d \in \Z_{\ge 0}$.