hcju/proofwikips huggingface dataset:
Definition:Polish Notation/Reverse Polish Notation
For stack-based programming languages, '''reverse Polish language''' is a useful variant of Polish notation, because it naturally coincides with how the input is to be structured for the language. As the name suggests, a string $\mathsf P$ is in '''reverse Polish language'''  reversing it gives a string $\tilde {\mathsf P}$ in Polish notation. Thus the '''reverse Polish language''' equivalents of these examples of Polish notation: :$\Box p q \ldots$ :$\Box {\diamond} p {\diamond} q \ldots$ are: :$\ldots q p \Box$ :$\ldots q {\diamond} p {\diamond} \Box$
Definition:Top (Logic)
'''Top''' is a constant of propositional logic interpreted to mean the canonical, undoubted tautology whose truth nobody could possibly ever question. The symbol used is $\top$.
Definition:Bottom (Logic)
'''Bottom''' is a constant of propositional logic interpreted to mean the canonical, undoubted contradiction whose falsehood nobody could possibly ever question. The symbol used is $\bot$.
Definition:Top (Logic)/Boolean Interpretation
There is only one boolean interpretation for $\top$: :$\map v \top = \T$ where $\T$ symbolises true.
Definition:Bottom (Logic)/Boolean Interpretation
There is only one boolean interpretation for $\bot$: :$\map v \bot = \F$ where $\F$ symbolises false.
Definition:Order Topology/Definition 2
Let $\struct {S, \preceq}$ be a totally ordered set. Define: :$\map {\Uparrow} S = \set {s^\succ: s \in S}$ :$\map {\Downarrow} S = \set {s^\prec: s \in S}$ where $s^\succ$ and $s^\prec$ denote the strict upper closure and strict lower closure of $s$, respectively. The '''order topology''' $\tau$ on $S$ is the topology on $S$ generated by $\map {\Uparrow} S \cup \map {\Downarrow} S$.
Definition:Interval/Ordered Set/Open
The '''open interval between $a$ and $b$''' is the set: :$\openint a b := a^\succ \cap b^\prec = \set {s \in S: \paren {a \prec s} \land \paren {s \prec b} }$ where: :$a^\succ$ denotes the strict upper closure of $a$ :$b^\prec$ denotes the strict lower closure of $b$.
Definition:Interval/Ordered Set/Left Half-Open
The '''left half-open interval between $a$ and $b$''' is the set: :$\hointl a b := a^\succ \cap b^\preccurlyeq = \set {s \in S: \paren {a \prec s} \land \paren {s \preccurlyeq b} }$ where: : $a^\succ$ denotes the strict upper closure of $a$ : $b^\preccurlyeq$ denotes the lower closure of $b$.
Definition:Interval/Ordered Set/Right Half-Open
The '''right half-open interval between $a$ and $b$''' is the set: :$\hointr a b := a^\succcurlyeq \cap b^\prec = \set {s \in S: \paren {a \preccurlyeq s} \land \paren {s \prec b} }$ where: :$a^\succcurlyeq$ denotes the upper closure of $a$ :$b^\prec$ denotes the strict lower closure of $b$.
Definition:Interval/Ordered Set/Closed
The '''closed interval between $a$ and $b$''' is the set: :$\closedint a b := a^\succcurlyeq \cap b^\preccurlyeq = \set {s \in S: \paren {a \preccurlyeq s} \land \paren {s \preccurlyeq b} }$ where: : $a^\succcurlyeq$ denotes the upper closure of $a$ : $b^\preccurlyeq$ denotes the lower closure of $b$.
Definition:Interval/Ordered Set
Let $\struct {S, \preccurlyeq}$ be an ordered set. Let $a, b \in S$. The '''intervals between $a$ and $b$''' are defined as follows: === Open Interval === {{:Definition:Interval/Ordered Set/Open}} === Left Half-Open Interval === {{:Definition:Interval/Ordered Set/Left Half-Open}} === Right Half-Open Interval === {{:Definition:Interval/Ordered Set/Right Half-Open}} === Closed Interval === {{:Definition:Interval/Ordered Set/Closed}}
Definition:Upper Closure/Set
Let $\struct {S, \preceq}$ be an ordered set or preordered set. Let $T \subseteq S$. The '''upper closure of $T$ (in $S$)''' is defined as: :$T^\succeq := \bigcup \set {t^\succeq: t \in T}$ where $t^\succeq$ denotes the upper closure of $t$ in $S$. That is: :$T^\succeq := \set {u \in S: \exists t \in T: t \preceq u}$
Definition:Order Topology/Definition 1
Let $\struct {S, \preceq}$ be a totally ordered set. Let $\XX$ be the set of open rays in $S$. Let $\tau$ be the topology on $S$ generated by $\XX$. Then $\tau$ is called the '''order topology''' on $S$.
Definition:Lower Closure/Element
Let $\struct {S, \preccurlyeq}$ be an ordered set. Let $a \in S$. The '''lower closure of $a$ (in $S$)''' is defined as: :$a^\preccurlyeq := \set {b \in S: b \preccurlyeq a}$ That is, $a^\preccurlyeq$ is the set of all elements of $S$ that precede $a$.
Definition:Lower Closure/Set
Let $\struct {S, \preccurlyeq}$ be an ordered set or preordered set. Let $T \subseteq S$. The '''lower closure of $T$ (in $S$)''' is defined as: :$T^\preccurlyeq := \bigcup \set {t^\preccurlyeq: t \in T}$ where $t^\preccurlyeq$ is the lower closure of $t$. That is: :$T^\preccurlyeq := \set {l \in S: \exists t \in T: l \preccurlyeq t}$
Definition:Upper Set/Definition 2
$U$ is an '''upper set''' in $S$ : :$U^\succeq \subseteq U$ where $U^\succeq$ is the upper closure of $U$.
Definition:Upper Set/Definition 3
$U$ is an '''upper set''' in $S$ : :$U^\succeq = U$ where $U^\succeq$ is the upper closure of $U$.
Definition:Closed Set/Closure Operator
Let $S$ be a set. Let $\cl: \powerset S \to \powerset S$ be a closure operator. Let $T \subseteq S$ be a subset. === Definition 1 === The subset $T$ is '''closed''' (with respect to $\cl$) : :$\map \cl T = T$ === Definition 2 === The subset $T$ is '''closed''' (with respect to $\cl$)  $T$ is in the image of $\cl$: :$T \in \Img \cl$
Definition:Kuratowski Closure Operator
=== Definition 1 === {{:Definition:Kuratowski Closure Operator/Definition 1}} === Definition 2 === {{:Definition:Kuratowski Closure Operator/Definition 2}}
Definition:Kuratowski Closure Operator/Definition 1
Let $S$ be a set. Let $\cl: \powerset S \to \powerset S$ be a mapping from the power set of $S$ to itself. Then $\cl$ is a '''Kuratowski closure operator'''  it satisfies the following '''Kuratowski closure axioms''' for all $A, B \subseteq S$: {{begin-axiom}} {{axiom | n =1         | m = A \subseteq \map \cl A         | rc= $\cl$ is inflationary }} {{axiom | n = 2         | m = \map \cl {\map \cl A} = \map \cl A         | rc= $\cl$ is idempotent }} {{axiom | n = 3         | m = \map \cl {A \cup B} = \map \cl A \cup \map \cl B         | rc= $\cl$ preserves binary unions }} {{axiom | n = 4         | m = \map \cl \O = \O }} {{end-axiom}}
Definition:Kuratowski Closure Operator/Definition 2
Let $S$ be a set. Let $\cl: \powerset S \to \powerset S$ be a mapping from the power set of $S$ to itself. Then $\cl$ is a '''Kuratowski closure operator'''  it satisfies the following axioms for all $A, B \subseteq X$: {{begin-axiom}} {{axiom | n = 1         | t = $\cl$ is a closure operator }} {{axiom | n = 2         | t = $\map \cl {A \cup B} = \map \cl A \cup \map \cl B$         | rc= $\cl$ preserves binary unions }} {{axiom | n = 3         | t = $\map \cl \O = \O$ }} {{end-axiom}}
Definition:Supremum of Mapping
Let $S$ be a set. Let $\struct {T, \preceq}$ be an ordered set. Let $f: S \to T$ be a mapping from $S$ to $T$. Let $f \sqbrk S$, the image of $f$, admit a supremum. Then the '''supremum''' of $f$ (on $S$) is defined by: :$\displaystyle \sup_{x \mathop \in S} \map f x = \sup f \sqbrk S$ === Real-Valued Function === {{:Definition:Supremum of Mapping/Real-Valued Function}}
Definition:Projective Space/Over a Field
Let $V$ be a vector space over a field $K$ of dimension $n + 1 \ge 1$.   Let $\sim$ be the equivalence relation defined on the set $V \setminus \left\{{0}\right\}$ by: :$x, y \in V \setminus \left\{{0}\right\}: x \sim y \iff \exists \lambda \in K: x = \lambda y$ The '''projective space''' associated to $V$ of dimension $n$ over $K$ is the quotient set $\left({V \setminus \left\{{0}\right\}}\right) / \sim$ and is denoted $\mathbb P \left({ V }\right)$. If $V = K^{n+1}$ for $n \ge 0$ a natural number, projective space is sometimes denoted $\mathbb P\left( K^{n+1} \right) = \mathbb P^n\left( K \right)$. This is because while $K^{n+1}$ is an $\left(n+1\right)$-dimensional vector space, the projective space $\mathbb P\left( K^{n+1} \right)$ has dimension $n$. The notation $\mathbb P\left( K^{n+1} \right) = K\mathbb P^n$ is also in use.
Definition:Universal Bounds
Let $\left({S, \preceq}\right)$ be an ordered set. Let $S$ have a smallest element, $\bot$. Let $S$ have a greatest element, $\top$. Then $\bot$ and $\top$ are the '''universal bounds''' of $S$.
Definition:Reflexive Closure/Union with Diagonal
Let $\mathcal R$ be a relation on a set $S$. The '''reflexive closure''' of $\mathcal R$ is denoted $\mathcal R^=$, and is defined as: :$\mathcal R^= := \mathcal R \cup \set {\tuple {x, x}: x \in S}$ That is: :$\mathcal R^= := \mathcal R \cup \Delta_S$ where $\Delta_S$ is the diagonal relation on $S$.
Definition:Reflexive Closure/Intersection of Reflexive Supersets
Let $\mathcal R$ be a relation on a set $S$. Let $\mathcal Q$ be the set of all reflexive relations on $S$ that contain $\mathcal R$. The '''reflexive closure''' of $\mathcal R$ is denoted $\mathcal R^=$, and is defined as: :$\mathcal R^= := \bigcap \mathcal Q$ That is: :$\mathcal R^=$ is the intersection of all reflexive relations on $S$ containing $\mathcal R$.
Definition:Reflexive Closure/Smallest Reflexive Superset
Let $\RR$ be a relation on a set $S$. The '''reflexive closure''' of $\RR$ is defined as the smallest reflexive relation on $S$ that contains $\RR$ as a subset. The '''reflexive closure''' of $\RR$ is denoted $\RR^=$.
Definition:Transitive Closure (Relation Theory)/Smallest Transitive Superset
Let $\RR$ be a relation on a set $S$. The '''transitive closure of $\RR$''' is defined as the smallest transitive relation on $S$ which contains $\RR$ as a subset.
Definition:Transitive Closure (Relation Theory)/Intersection of Transitive Supersets
Let $\mathcal R$ be a relation on a set $S$. The '''transitive closure''' of $\mathcal R$ is defined as the intersection of all transitive relations on $S$ which contain $\mathcal R$.
Definition:Transitive Closure (Relation Theory)/Finite Chain
Let $\mathcal R$ be a relation on a set or class $S$. The '''transitive closure''' of $\mathcal R$ is the relation $\mathcal R^+$ defined as follows: For $x, y \in S$, $x \mathrel {\mathcal R^+} y$  for some $n \in \N_{>0}$ there exist $s_0, s_1, \dots, s_n \in S$ such that $s_0 = x$, $s_n = y$, and: {{begin-eqn}} {{eqn | l = s_0       | o = \mathcal R       | r = s_1 }} {{eqn | l = s_1       | o = \mathcal R       | r = s_2 }} {{eqn | o = \vdots }} {{eqn | l = s_{n - 1}       | o = \mathcal R       | r = s_n }} {{end-eqn}}
Definition:Transitive Closure (Relation Theory)/Union of Compositions
Let $\mathcal R$ be a relation on a set $S$. Let: :$\mathcal R^n := \begin{cases} \mathcal R & : n = 0 \\ \mathcal R^{n-1} \circ \mathcal R & : n > 0 \end{cases}$ where $\circ$ denotes composition of relations. Finally, let: :$\displaystyle \mathcal R^+ = \bigcup_{i \mathop \in \N} \mathcal R^i$ Then $\mathcal R^+$ is called the '''transitive closure''' of $\mathcal R$.
Definition:Affine Frame
Let $\mathcal E$ be an affine space with difference space $V$. === Definition 1 === {{:Definition:Affine Frame/Definition 1}} === Definition 2 === {{:Definition:Affine Frame/Definition 2}}
Definition:Coordinate System/Coordinates on Affine Space
Let $\mathcal E$ be an affine space of dimension $n$ over a field $k$. Let $\mathcal R = \left({p_0, e_1, \ldots, e_n}\right)$ be an affine frame in $\mathcal E$. Let $p \in \mathcal E$ be a point. Since Affine Coordinates are Well-Defined, there exists a unique ordered tuple $\left({\lambda_1, \ldots, \lambda_n}\right) \in k^n$ such that: :$\displaystyle p = p_0 + \sum_{i \mathop = 1}^n \lambda_i e_i$ The numbers $\lambda_1, \ldots, \lambda_n$ are the '''coordinates''' of $p$ in the frame $\mathcal R$.
Definition:Barycenter
Let $\mathcal E$ be an affine space over a field $k$. Let $p_1,\ldots,p_n \in \mathcal E$ be points. Let $\lambda_1,\ldots,\lambda_n \in k$ such that $\displaystyle \sum_{i \mathop = 1}^n \lambda_i = 1$. The '''barycenter''' of $p_1,\ldots,p_n$ with '''weights''' $\lambda_1,\ldots,\lambda_n$ is the unique point $q$ of $\mathcal E$ such that for every point $r \in \mathcal E$ :$\displaystyle q = r + \sum_{i \mathop = 1}^n\lambda_i \vec{r p_i}$
Definition:Negatively Transitive Relation
Let $\RR$ be a relation on a set $S$. Then $\RR$ is '''negatively transitive''' : :$\forall x, y, z \in S: \neg \paren {x \mathrel \RR y} \land \neg \paren {y \mathrel \RR z} \implies \neg \paren {x \mathrel \RR z}$ By De Morgan's Laws, this can be given the alternative form: :$\forall x, y, z \in S: \paren {x \mathrel \RR z} \implies \paren {x \mathrel \RR y} \lor \paren {y \mathrel \RR z}$
Definition:Reflexive Transitive Closure/Transitive Closure of Reflexive Closure
The '''reflexive transitive closure''' of $\mathcal R$ is denoted $\mathcal R^*$, and is defined as the transitive closure of the reflexive closure of $\mathcal R$: :$\mathcal R^* = \left({\mathcal R^=}\right)^+$
Definition:Reflexive Transitive Closure/Reflexive Closure of Transitive Closure
The '''reflexive transitive closure''' of $\mathcal R$ is denoted $\mathcal R^*$, and is defined as the reflexive closure of the transitive closure of $\mathcal R$: :$\mathcal R^* = \left({\mathcal R^+}\right)^=$
Definition:Reflexive Transitive Closure/Smallest Reflexive Transitive Superset
The '''reflexive transitive closure''' of $\mathcal R$ is denoted $\mathcal R^*$, and is defined as the smallest reflexive and transitive relation on $S$ which contains $\mathcal R$.
Definition:Closure Operator/Ordering
{{:Definition:Closure Operator/Ordering/Definition 1}}
Definition:Closed Element
Let $\left({S, \preceq}\right)$ be an ordered set. Let $\operatorname{cl}$ be a closure operator on $S$. Let $x \in S$. === Definition 1 === {{:Definition:Closed Element/Definition 1}} === Definition 2 === {{:Definition:Closed Element/Definition 2}}
Definition:Affine Subspace
Let $\mathcal E$ be an affine space with tangent space $E$. Let $\mathcal F \subseteq \mathcal E$ be a subset of $\mathcal E$. Then $\mathcal F$ is an '''affine subspace''' of $\mathcal E$  there exists a point $p \in \mathcal E$ such that: :$F_p := \set {q - p: q \in \mathcal F}$ is a vector subspace of the vector space $E$.
Definition:Product of Affine Spaces
Let $\mathcal E, \mathcal F$ be affine spaces with difference spaces $E, F$ respectively. Let $\mathcal G = \mathcal E \times \mathcal F$ be the cartesian product of the sets $\mathcal E,\mathcal F$. Let $G = E \times F$ be the direct product of the vector spaces $E, F$. Define sum and difference operations $+ : \mathcal G \times G \to \mathcal G$ and $- : \mathcal G \times \mathcal G \to G$ by, for all $p, p' \in \mathcal E$ and $q, q' \in \mathcal F$: :$\left({p, q}\right) + \left({p', q'}\right) := \left({p + p', q + q'}\right)$ :$\left({p, q}\right) - \left({p', q'}\right) := \left({p - p', q - q'}\right)$ Then the set $\mathcal G$ together with the vector space $G$ and the operations $+, -$ is called the '''product''' of the affine spaces $\mathcal E$ and $\mathcal F$.
Definition:Standard Affine Structure on Vector Space
Let $E$ be a vector space. Let $\mathcal E$ be the underlying set of $E$. Let $+$ denote the addition operation $E \times E \to E$, viewed as a mapping $\mathcal E \times E \to \mathcal E$. Let $-$ denote the subtraction operation $E \times E \to E$, viewed as a mapping $\mathcal E \times \mathcal E \to E$. Then the set $\mathcal E$, together with the vector space $E$ and the operations $+,-$, is called the '''standard affine structure''' on the vector space $E$.
Definition:Vectorialization of Affine Space
Let $\mathcal E$ be an affine space over a field $k$ with difference space $E$. Let $\mathcal R = \left({p_0, e_1, \ldots, e_n}\right)$ be an affine frame in $\mathcal E$. Define a mapping $\Theta_{\mathcal R} : k^n \to \mathcal E$ by: :$\displaystyle \Theta_{\mathcal R} \left({\lambda_1, \ldots, \lambda_n}\right) = p_0 + \sum_{i \mathop = 1}^n \lambda_i e_i$. By Affine Coordinates are Well-Defined, $\Theta_{\mathcal R}$ is a bijection. For any $\mu \in k$, $p, q \in \mathcal E$ let: :$\mu \cdot p = \Theta_{\mathcal R}\left({ \mu \cdot \Theta_{\mathcal R}^{-1} \left({p}\right) }\right)$ and: :$p + q = \Theta_{\mathcal R} \left({ \Theta_{\mathcal R}^{-1} \left({p}\right) + \Theta_{\mathcal R}^{-1} \left({q}\right) }\right)$ We call the set $\mathcal E$, together with the operations $\cdot,+$ the '''vectorialization of $\mathcal E$ with origin $p_0$'''.
Definition:Strict Ordering/Asymmetric and Transitive
Let $\RR$ be a relation on a set $S$. Then $\RR$ is a '''strict ordering (on $S$)'''  the following two conditions hold: {{begin-axiom}} {{axiom | n = 1         | lc= Asymmetry         | q = \forall a, b \in S         | ml= a \mathrel \RR b         | mo= \implies         | mr= \neg \paren {b \mathrel \RR a} }} {{axiom | n = 2         | lc= Transitivity         | q = \forall a, b, c \in S         | ml= \paren {a \mathrel \RR b} \land \paren {b \mathrel \RR c}         | mo= \implies         | mr= a \mathrel \RR c }} {{end-axiom}}
Definition:Strict Ordering/Antireflexive and Transitive
Let $\mathcal R$ be a relation on a set $S$. Then $\mathcal R$ is a '''strict ordering (on $S$)'''  the following two conditions hold: {{begin-axiom}} {{axiom | n = 1         | lc= Antireflexivity         | q = \forall a \in S         | m = \neg \paren {a \mathrel {\mathcal R} a} }} {{axiom | n = 2         | lc= Transitivity         | q = \forall a, b, c \in S         | m = \paren {a \mathrel {\mathcal R} b} \land \paren {b \mathrel {\mathcal R} c} \implies a \mathrel {\mathcal R} c }} {{end-axiom}}
Definition:Closure Operator
=== Ordering === {{:Definition:Closure Operator/Ordering}} === Power Set === When the ordering in question is the subset relation on a power set, the definition can be expressed as follows: {{:Definition:Closure Operator/Power Set}}
Definition:Subband
Let $\struct {S, \circ}$ be an algebraic structure. Let $T \subseteq S$ such that $\struct {T, \circ {\restriction_T} }$ is a band, where $\circ {\restriction_T}$ denotes the restriction of $\circ$ to $T$. Then $\struct {T, \circ {\restriction_T} }$ is a '''subband''' of $S$.
Definition:Closure Operator/Ordering/Definition 1
Let $\struct {S, \preceq}$ be an ordered set. A '''closure operator''' on $S$ is a mapping: :$\cl: S \to S$ which satisfies the following conditions for all elements $x, y \in S$: {{begin-axiom}} {{axiom | lc= $\cl$ is inflationary         | ml= x         | mo= \preceq          | mr= \map \cl x }} {{axiom | lc= $\cl$ is increasing         | ml= x \preceq y         | mo= \implies         | mr= \map \cl x \preceq \map \cl y }} {{axiom | lc= $\cl$ is idempotent         | ml= \map \cl {\map \cl x}          | mo= =         | mr= \map \cl x }} {{end-axiom}}
Definition:Closure Operator/Ordering/Definition 2
Let $\struct {S, \preceq}$ be an ordered set. A '''closure operator''' on $S$ is a mapping: :$\cl: S \to S$ which satisfies the following condition for all elements $x, y \in S$: :$x \preceq \map \cl y \iff \map \cl x \preceq \map \cl y$
Definition:Closed Element/Definition 1
The element $x$ is a '''closed element of $S$ (with respect to $\cl$)'''  $x$ is a fixed point of $\cl$: :$\map \cl x = x$
Definition:Closed Element/Definition 2
The element $x$ is a '''closed element of $S$ (with respect to $\cl$)'''  $x$ is in the image of $\cl$: :$x \in \Img \cl$
Definition:Univalent Relation
Let $\RR$ be a relation on a set $S$. Then $\RR$ is '''univalent''' : :$\RR \circ \RR^{-1} \subseteq \Delta_S$ That is, $\RR$ composed with its inverse $\RR^{-1}$ is a subset of the diagonal relation on $S$.
Definition:Dual Relation/Complement of Inverse
Let $\RR \subseteq S \times T$ be a binary relation. Then the '''dual''' of $\RR$ is denoted $\RR^d$ and is defined as: :$\RR^d := \overline {\paren {\RR^{-1} } }$ where: :$\RR^{-1}$ denotes the inverse of $\RR$ :$\overline {\paren {\RR^{-1} } }$ denotes the complement of the inverse of $\RR$.
Definition:Dual Relation/Inverse of Complement
Let $\RR \subseteq S \times T$ be a binary relation. Then the '''dual''' of $\RR$ is denoted $\RR^d$ and is defined as: :$\RR^d := \paren {\overline \RR}^{-1}$ where: :$\overline \RR$ denotes the complement of $\RR$ :$\paren {\overline \RR}^{-1}$ denotes the inverse of the complement of $\RR$.
Definition:Dual Relation
=== Inverse of Complement === {{:Definition:Dual Relation/Inverse of Complement}} === Complement of Inverse === {{:Definition:Dual Relation/Complement of Inverse}}
Definition:Trivial Gradation
Let $\left({R, +, \circ}\right)$ be a ring. Let $\left({M, \cdot, e}\right)$ be a monoid. Define a gradation on $R$ as follows: :$\displaystyle R_e = R$ :$\displaystyle \forall m \in M \setminus \left\{{e}\right\} : R_m = \mathbf 0$ where $\mathbf 0$ is the zero ring. This is called the '''trivial $M$-grading''' on $R$.
Definition:Irreducible (Representation Theory)
=== Linear Representation === {{:Definition:Irreducible (Representation Theory)/Linear Representation}} === G-Module === {{:Definition:Irreducible (Representation Theory)/G-Module}}
Definition:Transcendental (Abstract Algebra)/Ring
Let $\struct {R, +, \circ}$ be a commutative ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $\struct {D, +, \circ}$ be an integral subdomain of $R$. Let $x \in R$. Then $x$ is '''transcendental over $D$''' : :$\displaystyle \forall n \in \Z_{\ge 0}: \sum_{k \mathop = 0}^n a_k \circ x^k = 0_R \implies \forall k: 0 \le k \le n: a_k = 0_R$ That is, $x$ is '''transcendental over $D$'''  the only way to express $0_R$ as a polynomial in $x$ over $D$ is by the null polynomial.
Definition:Transcendental (Abstract Algebra)/Field Extension/Element
Let $E / F$ be a field extension. Let $\alpha \in E$. Then $\alpha$ is '''transcendental over $F$''' : : $\nexists f \left({x}\right) \in F \left[{x}\right] \setminus \left\{{0}\right\}: f \left({\alpha}\right) = 0$ where $f \left({x}\right)$ denotes a polynomial in $x$ over $F$.
Definition:Transcendental (Abstract Algebra)/Field Extension
A field extension $E / F$ is said to be '''transcendental''' : :$\exists \alpha \in E: \alpha$ is transcendental over $F$ That is, a field extension is '''transcendental'''  it contains at least one transcendental element.
Definition:Monomial of Free Commutative Monoid/Multiplication
The set of monomials over $\family {X_j: j \in J}$ has '''multiplication''' $\circ$ defined by: :$\ds \paren {\prod_{j \mathop \in J} X_j^{k_j} } \circ \paren {\prod_{j \mathop \in J} X_j^{k_j'} } = \paren {\prod_{j \mathop \in J} X_j^{k_j + k_j'} }$ which using multiindex addition notation reads: :$\mathbf X^k \circ \mathbf X^{k'} = \mathbf X^{k + k'}$
Definition:Monomial of Free Commutative Monoid/Degree
The '''degree''' of a monomial is defined as: :$\ds \sum_{j \mathop \in J} k_j$ that is, the modulus of the corresponding multiindex.
Definition:Degree of Polynomial/Null Polynomial
The null polynomial $0_R \in S \left[{X}\right]$ does ''not'' have a degree.
Definition:Degree of Polynomial/Zero
A polynomial $f \in S \sqbrk x$ in $x$ over $S$ is of '''degree zero'''  $x$ is a non-zero element of $S$, that is, a constant polynomial.
Definition:Multiplication of Polynomials
Let $\struct {R, +, \circ}$ be a ring. Let $\struct {S, +, \circ}$ be a subring of $R$. Let $x \in R$. Let: :$\displaystyle f = \sum_{j \mathop = 0}^n a_j x^j$ :$\displaystyle g = \sum_{k \mathop = 0}^n b_k x^k$ be polynomials in $x$ over $S$ such that $a_n \ne 0$ and $b_m \ne 0$. {{questionable|this does not need to be defined; it's just the product of ring elements}} The '''product of $f$ and $g$''' is defined as: :$\displaystyle f g := \sum_{l \mathop = 0}^{m + n} c_l x^l$ where: :$\displaystyle \forall l \in \set {0, 1, \ldots, m + n}:  c_l = \sum_{\substack {j \mathop + k \mathop = l \\ j, k \mathop \in \Z}} a_j b_k$ === Polynomial Forms === {{:Definition:Multiplication of Polynomials/Polynomial Forms}} === Polynomials as Sequences === {{:Definition:Multiplication of Polynomials/Sequence}}
Definition:Ring of Polynomials in Ring Element
The subring of $R$ consisting of all the polynomials in $x$ over $D$ is called the '''ring of polynomials over $D$''' and is denoted $D \sqbrk x$.
Definition:Order Embedding/Definition 1
$\phi$ is an '''order embedding of $S$ into $T$''' : :$\forall x, y \in S: x \preceq_1 y \iff \map \phi x \preceq_2 \map \phi y$