hcju/proofwikips huggingface dataset:
Definition:Local Sub-Basis
Let $\left({X, \tau}\right)$ be a topological space. Let $x \in X$. Then a '''local sub-basis''' of $x$ relative to $\tau$ is a neighborhood sub-basis of $x$ consisting of open sets.
Definition:Neighborhood Sub-Basis
Let $\struct {S, \tau}$ be a topological space. Let $x \in S$. Let $\BB$ be a set of neighborhoods of $x$. Then $\BB$ is a '''neighborhood sub-basis of $x$ relative to $\tau$''' : :for each neighborhood $N$ of $x$, there exists a finite subset $K$ of $\BB$ such that $\bigcap K \subseteq N$.
Definition:Star Convex Set/Star Center
The point $a \in A$ is called a '''star center of $A$'''.
Definition:Sublattice
Let $\left({L,\wedge_L,\vee_L,\preceq_L}\right)$ be a lattice. Let $S$ be a subset of $L$ and let $\wedge_S$, $\vee_S$, and $\preceq_S$ be the restrictions to $S$ of $\wedge_L$, $\vee_L$, and $\preceq_L$, respectively. Then $\left({S,\wedge_S,\vee_S,\preceq_S}\right)$ is a '''sublattice''' of $\left({L,\wedge_L,\vee_L,\preceq_L}\right)$ iff $S$ is closed under  $\wedge_S$ and $\vee_S$. If in addition $L$ is a bounded lattice and its top and bottom elements are in $S$, then $S$ is called a '''$0,1$-sublattice''' of $L$. If in addition $L$ is a complete lattice and for each subset $T$ of $S$, $\sup_L T,\inf_L T \in S$, then $S$ is called a '''complete sublattice''' of $L$.
Definition:Generalized Ordered Space
Let $\struct {S, \preceq}$ be a totally ordered set. Let $\tau$ be a topology for $S$. === Definition 1 === {{:Definition:Generalized Ordered Space/Definition 1}} === Definition 2 === {{:Definition:Generalized Ordered Space/Definition 2}} === Definition 3 === {{:Definition:Generalized Ordered Space/Definition 3}}
Definition:Generalized Ordered Space/Definition 2
$\left({S, \preceq, \tau}\right)$ is a '''generalized ordered space''' : :$(1): \quad$ there exists a linearly ordered space $\left({S', \preceq', \tau'}\right)$ :$(2): \quad$ there exists a mapping $\phi: S \to S'$ such that $\phi$ is both an order embedding and a topological embedding.
Definition:Generalized Ordered Space/Definition 1
$\left({S, \preceq, \tau}\right)$ is a '''generalized ordered space''' : :$(1): \quad \left({S, \tau}\right)$ is a Hausdorff space :$(2): \quad$ there exists a basis for $\left({S, \tau}\right)$ whose elements are convex in $S$.
Definition:Argumentum ad Hominem
An '''argumentum ad hominem''' is a logical argument that concludes the falsehood of a statement because it is made by a particular person or group of people. It can be argued that it is reasonable to be suspicious of an argument if most of the assertions made by said person or group regarding the topic under discussion are false. However, such reasoning is still fallacious.
Definition:Argumentum ad Baculum
An '''argumentum ad baculum''' is a logical argument that, rather than prove or present evidence for a claim, threatens any who dare argue with the person or group making the claim. The presence of such a threat is often a good reason to avoid ''stating'' an opposing view, but manifestly does not support the truth or falsity of the claim under consideration.
Definition:Conjunction/General Definition
Let $p_1, p_2, \ldots, p_n$ be statements. The '''conjunction''' of $p_1, p_2, \ldots, p_n$ is defined as: :$\displaystyle \bigwedge_{i \mathop = 1}^n \ p_i = \begin{cases} p_1 & : n = 1 \\ & \\ \displaystyle \left({\bigwedge_{i \mathop = 1}^{n-1} \ p_i}\right) \land p_n & : n > 1 \end{cases}$ That is: :$\displaystyle \bigwedge_{i \mathop = 1}^n \ p_i = p_1 \land p_2 \land \cdots \land p_{n-1} \land p_n$ In terms of the set $P = \left\{{p_1, \ldots, p_n}\right\}$, this can also be rendered: :$\displaystyle \bigwedge P$ and is referred to as the '''conjunction of $P$'''.
Definition:Disjunction/General Definition
Let $p_1, p_2, \ldots, p_n$ be statements. The '''disjunction''' of $p_1, p_2, \ldots, p_n$ is defined as: :$\displaystyle \bigvee_{i \mathop = 1}^n \ p_i = \begin{cases} p_1 & : n = 1 \\ & \\ \displaystyle \paren {\bigvee_{i \mathop = 1}^{n - 1} \ p_i} \lor p_n & : n > 1 \end{cases}$ That is: :$\displaystyle \bigvee_{i \mathop = 1}^n \ p_i = p_1 \lor p_2 \lor \cdots \lor p_{n - 1} \lor p_n$ In terms of the set $P = \set {p_1, \ldots, p_n}$ this can also be rendered: :$\displaystyle \bigvee P$ and is referred to as the '''disjunction of $P$'''.
Definition:Countable Set/Countably Infinite
{{:Definition:Countable Set/Countably Infinite/Definition 1}}
Definition:Logical Not/Boolean Interpretation
The truth value of $\neg \mathbf A$ under a boolean interpretation $v$ is given by: :$\map v {\neg \mathbf A} = \begin{cases} \T & : \map v {\mathbf A} = \F \\ \F & : \map v {\mathbf A} = \T \end{cases}$
Definition:Collation
A '''collation''' is a structured alignment with certain placeholders that underpins the construction of formal languages. These placeholders may be replaced by elements of an alphabet $\mathcal A$ under consideration. A '''collation in $\mathcal A$''' is one where all placeholders are replaced by symbols from $\mathcal A$. For example, if we take $\square$ to denote a placeholder, then $\square\square\square\square\square$ represents the collation "a word of length $5$". We can see that then the word "sheep" is an instance of the '''collation''' "a word of length $5$" '''in the English alphabet''', as is "axiom". Typical examples of '''collations''' encountered in mathematics are words or structured graphics like labeled trees. {{transclude:Definition:Collation/Collation System   |section = def   |title = Collation System   |header = 3   |link = true }} {{transclude:Definition:Collation/Unique Readability|   |section = def   |title = Unique Readability   |header = 3   |link = true }}
Definition:Conditional/Boolean Interpretation
The truth value of $\mathbf A \implies \mathbf B$ under a boolean interpretation $v$ is given by: :$\map v {\mathbf A \implies \mathbf B} = \begin{cases} \T & : \map v {\mathbf A} = \F \text{ or } \map v {\mathbf B} = \T \\ \F & : \text{otherwise} \end{cases}$ and the truth value of $\mathbf A \impliedby \mathbf B$ under a boolean interpretation $v$ is given by: :$\map v {\mathbf A \impliedby \mathbf B} = \begin{cases} \T & : \map v {\mathbf A} = \T \text{ or } \map v {\mathbf B} = \F \\ \F & : \text{otherwise} \end{cases}$
Definition:Conjunction/Boolean Interpretation
The truth value of $\mathbf A \land \mathbf B$ under a boolean interpretation $v$ is given by: :$\map v {\mathbf A \land \mathbf B} = \begin{cases} \T & : \map v {\mathbf A} = \map v {\mathbf B} = \T \\ \F & : \text{otherwise} \end{cases}$
Definition:Disjunction/Boolean Interpretation
The truth value of $\mathbf A \lor \mathbf B$ under a boolean interpretation $v$ is given by: :$\map v {\mathbf A \lor \mathbf B} = \begin{cases} \F & : \map v {\mathbf A} = \map v {\mathbf B} = \F \\ \T & : \text{otherwise} \end{cases}$
Definition:Biconditional/Boolean Interpretation
The truth value of $\mathbf A \iff \mathbf B$ under a boolean interpretation $v$ is given by: :$\map v {\mathbf A \iff \mathbf B} = \begin{cases} \T & : \map v {\mathbf A} = \map v {\mathbf B} \\ \F & : \text{otherwise} \end{cases}$
Definition:Exclusive Or/Boolean Interpretation
The truth value of $\mathbf A \oplus \mathbf B$ under a boolean interpretation $v$ is given by: :$v \left({\mathbf A \oplus \mathbf B}\right) = \begin{cases} F & : v \left({\mathbf A}\right) = v \left({\mathbf B}\right) \\ T & : \text{otherwise} \end{cases}$
Definition:Order Automorphism
Let $\left({S, \preceq}\right)$ be an ordered set. Let $\phi: S \to S$ be an order isomorphism from $\left({S, \preceq}\right)$ to $\left({S, \preceq}\right)$. Then $\phi$ is an '''(order) automorphism'''.
Definition:Logical NAND/Boolean Interpretation
The truth value of $\mathbf A \uparrow \mathbf B$ under a boolean interpretation $v$ is given by: :$v \left({\mathbf A \uparrow \mathbf B}\right) = \begin{cases} T & : v \left({\mathbf A}\right) = F \text{ or } v \left({\mathbf B}\right) = F\\ F & : \text{otherwise} \end{cases}$
Definition:Logical NOR/Boolean Interpretation
The truth value of $\mathbf A \downarrow \mathbf B$ under a boolean interpretation $v$ is given by: :$v \left({\mathbf A \downarrow \mathbf B}\right) = \begin{cases} T & : v \left({\mathbf A}\right) = v \left({\mathbf B}\right) = F\\ F & : \text{otherwise} \end{cases}$
Definition:Dual Isomorphism (Order Theory)
Let $\left({S, \preceq_S}\right)$ and $\left({T, \preceq_T}\right)$ be ordered sets. Let $\phi:S \to T$ be a bijection. Then $\phi$ is a '''dual isomorphism''' between $\left({S, \preceq_S}\right)$ and $\left({T, \preceq_T}\right)$ iff $\phi$ and $\phi^{-1}$ are decreasing mappings. If there is a dual isomorphism between $\left({S, \preceq_S}\right)$ and $\left({T, \preceq_T}\right)$, then $\left({S, \preceq_S}\right)$ is '''dual''' to $\left({T, \preceq_T}\right)$. Equivalently, $\left({S, \preceq_S}\right)$ is '''dual''' to $\left({T, \preceq_T}\right)$ iff $S$ with the dual ordering is isomorphic to $T$.
Definition:Self-Dual (Order Theory)
Let $\left({S, \preceq}\right)$ be an ordered set. If $S$ is dual to $S$, then $S$ is '''self-dual'''.
Definition:Labeled Tree
A '''labeled tree''' is a finite rooted tree in which each node has attached to it a '''label'''. These '''labels''' may be any symbol or other object that one can think of, but usually consist of one letter or symbol. {{expand|A picture like [http://mathworld.wolfram.com/images/eps-gif/PrueferCode_700.gif this], exemplifying the below}} The parent function is usually depicted by placing the ancestor above the child and connecting them with a line. The various children of a node may also be considered to be in a particular order. This is usually depicted by placing them in a left-to-right order. Because of the intuitive clarity of these relations, '''labeled trees''' may be considered collations as basic as strings are. However, because of the space limitations that most printed media have to take into account, it is usual that one quickly resorts to representing trees by strings in one way or another.
Definition:Upper Set
Let $\left({S, \preceq}\right)$ be an ordered set. Let $U \subseteq S$.
Definition:Lower Set
Let $\left({S, \preceq}\right)$ be an ordered set. Let $L \subseteq S$. Then $L$ is a '''lower set''' in $S$ : : For all $l \in L$ and $s \in S$: if $s \preceq l$ then $s \in L$. That is, $L$ is a '''lower set'''  it contains its own lower closure.
Definition:Generalized Ordered Space/Definition 3
$\left({S, \preceq, \tau}\right)$ is a '''generalized ordered space''' : :$(1): \quad \left({S, \tau}\right)$ is a Hausdorff space :$(2): \quad$ there exists a sub-basis for $\left({S, \tau}\right)$ each of whose elements is an upper set or lower set in $S$.
Definition:Singular Statement/Individuating Description
An '''individuating description''' is a predicate whose purpose is to uniquely identify a particular object.
Definition:In-Order Traversal of Labeled Tree
Let $T$ be a binary labeled tree. '''In-order traversal''' of $T$ is an algorithm designed to obtain a string representation of $T$. The steps are as follows: === Variant 1 === {{:Definition:In-Order Traversal of Labeled Tree/Variant 1}} === Variant 2 === {{:Definition:In-Order Traversal of Labeled Tree/Variant 2}}
Definition:In-Order Traversal of Labeled Tree/Variant 1
$\mathtt{Inorder} (T):$ * $n \gets t$, where $t$ is the root node of $T$. * If $n$ is a leaf node, output the label of $n$, and stop. * Let $T_1$ and $T_2$ be the left and right subtrees of $T$. * If $n$ has only one child, skip this step. Output $\mathtt{Inorder} (T_1)$. * Output the label of $n$. * Output $\mathtt{Inorder} (T_2)$. * Stop.
Definition:In-Order Traversal of Labeled Tree/Variant 2
$\mathtt{Inorder} (T):$ * $n \gets t$, where $t$ is the root node of $T$. * If $n$ is a leaf node, output the label of $n$, and stop. * Let $T_1$ and $T_2$ be the left and right subtrees of $T$. * Output a left bracket $($. * If $n$ has only one child, skip this step. Output $\mathtt{Inorder} (T_1)$. * Output the label of $n$. * Output $\mathtt{Inorder} (T_2)$. * Output a right bracket $)$. * Stop.
Definition:Pre-Order Traversal of Labeled Tree
Let $T$ be a binary labeled tree. '''Pre-order traversal''' of $T$ is an algorithm designed to obtain a string representation of $T$. The steps are as follows: $\mathtt{Preorder} (T):$ * $n \gets t$, where $t$ is the root node of $T$. * Output the label of $n$. * If $n$ is a leaf node, stop. * Let $T_1$ and $T_2$ be the left and right subtrees of $T$. * If $n$ has only one child, skip this step. Output $\mathtt{Preorder} (T_1)$. * Output $\mathtt{Preorder} (T_2)$. * Stop. The resulting string will be in Polish notation.
Definition:Polish Notation
'''Polish notation''' is a method of writing expressions without the need for parentheses. The main method in writing an expression in '''Polish notation''' is to output an operator $\Box$, directly followed by the output of all its arguments $p, q, \ldots$, yielding: :$\Box p q \ldots$ If we had another, operator $\diamond$ taking only one argument, then also: :$\Box {\diamond} p {\diamond} q \ldots$ is in '''Polish notation'''. Note that for the applicability of '''Polish notation''', it is vital to be able to efficiently distinguish an argument from an operator, or a sequence of arguments. === Formal Definition === {{:Definition:Polish Notation/Formal Definition}} === Reverse Polish Notation === {{:Definition:Polish Notation/Reverse Polish Notation}} {{expand|Could benefit from being defined in BNF or otherwise formally, as an exercise in formal languages}}
Definition:Many-to-One Relation/Defined/Element
Let $f \subseteq S \times T$ be a many-to-one relation. <section notitle="true" name="tc-defn"> Let $s \in S$. Then $f$ is '''defined at $s$'''  $s \in \Dom f$, the domain of $f$. </section>
Definition:Many-to-One Relation/Defined/Set
Let $f \subseteq S \times T$ be a many-to-one relation. <section notitle="true" name="tc-defn"> Let $R \subseteq S$. Then $f$ is '''defined on $R$'''  it is defined at all $r \in R$. Equivalently,  $R \subseteq \Dom f$, the domain of $f$. </section>
Definition:Logical Not/Notational Variants
Various symbols are encountered that denote the concept of the logical not: {| border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; |- ! Symbol  ! Origin  ! Known as |- | $\neg p$ |   |   |- | $\mathsf{NOT}\ p$ |   |   |- | $\sim p$ or $\tilde p$ | {{BookReference|Principia Mathematica|1910|Alfred North Whitehead|author2 = Bertrand Russell}} | '''tilde''' or '''curl''' |- | $- p$ |  |   |- | $\bar p$ |   | '''Bar $p$''' |- | $p'$ |   | '''$p$ prime''' or '''$p$ complement''' |- | $! p$ |   | '''Bang $p$''' |- | $\operatorname{N} p$ | {{AuthorRef|Jan Łukasiewicz|Łukasiewicz}}'s Polish notation |  |}
Definition:Deductive Method
'''The deductive method''' is a general term for the process of building a logical argument from a (usually) limited number of premises. Thus from a small number of statements which have been accepted as true, and a small number of rules of deduction, a theoretically unlimited number of further statements are determined to be true as a result.
Definition:Conjunction/Notational Variants
Various symbols are encountered that denote the concept of logical conjunction: {| border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; |- ! Symbol  ! Origin  ! Known as |- | $p \land q$ |   | '''wedge''' |- | $p\ \mathsf{AND} \ q$ |   |   |- | $p \ . \ q$ | {{BookReference|Principia Mathematica|1910|Alfred North Whitehead|author2 = Bertrand Russell}} | '''dot''' |- | $p \ \And \ q$ |  | '''Ampersand''' |- | $\operatorname K p q$ | {{AuthorRef|Jan Łukasiewicz|Łukasiewicz}}'s Polish notation |  |}
Definition:Conditional/Notational Variants
Various symbols are encountered that denote the concept of the conditional: {| border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; |- ! Symbol  ! Origin  ! Known as |- | $p \implies q$ |  | '''Implies''' |- | $p \to q$ |   | often used when space is limited |- | $p \supset q$ | {{BookReference|Principia Mathematica|1910|Alfred North Whitehead|author2 = Bertrand Russell}} | '''hook''' or '''horseshoe''' |- | $p \, \mathop {-\!\!\!<} q$ | {{AuthorRef|Charles Sanders Peirce}} | '''sign of illation''' |- | $\operatorname C p q$ | {{AuthorRef|Jan Łukasiewicz|Łukasiewicz}}'s Polish notation |  |} In mathematics, as opposed to works concerned purely with logic, it is usual to use "$\implies$", as then it can be ensured that it is understood to mean exactly the same thing when we use it in the "mathematical" context. There are other uses in mathematics for the other symbols.
Definition:Biconditional/Notational Variants
Various symbols are encountered that denote the concept of biconditionality: {| border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; |- ! Symbol  ! Origin  |- | $p \iff q$ |   |- | $p\ \mathsf{EQ} \ q$ |   |- | $p \equiv q$ | {{BookReference|Principia Mathematica|1910|Alfred North Whitehead|author2 = Bertrand Russell}} |- | $p = q$ | |- | $p \leftrightarrow q$ | |- | $\operatorname E p q$ | {{AuthorRef|Jan Łukasiewicz|Łukasiewicz}}'s Polish notation |} It is usual in mathematics to use $\iff$, as there are other uses for the other symbols.
Definition:Universal Quantifier/Notational Variants
Various symbols are encountered that denote the concept of universal quantifier: {| border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; |- ! Symbol  ! Origin  |- | $\forall x$ | {{AuthorRef|Gerhard Gentzen}}: ''Untersuchungen über das logische Schließen'' (1935) |- | $\paren x$ | {{BookReference|Principia Mathematica|1910|Alfred North Whitehead|author2 = Bertrand Russell}} |- | $\Pi x$ | {{AuthorRef|Jan Łukasiewicz|Łukasiewicz}}'s Polish notation |- | $\wedge x$ or $\bigwedge x$ |  |- | $\ds \operatorname{\Large {\textsf A} } \limits_{x, y, \dotsc}$ | {{BookReference|Introduction to Logic and to the Methodology of Deductive Sciences|1946|Alfred Tarski}} |}
Definition:Disjunction/Notational Variants
Various symbols are encountered that denote the concept of disjunction: {| border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; |- ! Symbol  ! Origin  ! Known as |- | $p \lor q$ | {{BookReference|Principia Mathematica|1910|Alfred North Whitehead|author2 = Bertrand Russell}} | '''vee''' or '''vel''' |- | $p\ \mathsf{OR} \ q$ |   |   |- | $p + q$ |   |   |- | $\operatorname A p q$ | {{AuthorRef|Jan Łukasiewicz|Łukasiewicz}}'s Polish notation |   |}
Definition:Existential Quantifier/Notational Variants
Various symbols are encountered that denote the concept of existential quantifier: {| border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; |- ! Symbol  ! Origin  |- | $\exists x$ | {{AuthorRef|Giuseppe Peano}}: ''Formulario Mathematico (2nd ed.)'' (1896) |- | $\Sigma x$ | {{AuthorRef|Jan Łukasiewicz|Łukasiewicz}}'s Polish notation |- | $\lor x$ or $\bigvee x$ |  |- | $\ds \operatorname {\Large {\textsf E} } \limits_{x, y \dotsc}$ | {{BookReference|Introduction to Logic and to the Methodology of Deductive Sciences|1946|Alfred Tarski}} |}
Definition:Exclusive Or/Notational Variants
Various symbols are encountered that denote the concept of exclusive or: {| border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; |- ! Symbol  ! Origin  ! Known as |- | $p \oplus q$ | | sometimes called '''o-plus''' |- | $p\ \mathsf{XOR} \ q$ | | |- | $p + q$ | | |- | $p \not \Leftrightarrow q$ | | |- | $p \not \equiv q$ | | |- | $p \ne q$ | | |- | $p \ \dot \lor \ q$ | | |- | $p \ \_ \lor \ q$ | | |}
Definition:Logical NAND/Notational Variants
Various symbols are encountered that denote the concept of logical NAND: {| border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; |- ! Symbol  ! Origin  ! Known as |- | $p \mid q$ | {{AuthorRef|Henry Maurice Sheffer|Henry Sheffer}} | Sheffer stroke |- | $p \uparrow q$ |  | Also sometimes referred to as the '''Sheffer stroke''' |- | $p \ \mathsf{NAND} \ q$ |  |   |- | $p / q$ |{{BookReference|Introduction to Symbolic Logic|1959|A.H. Basson|author2 = D.J. O'Connor}} | who refer to it as the '''stroke function''' |- | $p \bar \curlywedge q$ | {{AuthorRef|Charles Sanders Peirce}} | Modified Ampheck |} The all-uppercase rendition NAND originates from the digital electronics industry, where, because NAND is Functionally Complete, this operator has a high importance.
Definition:Logical NOR/Notational Variants
Various symbols are encountered that denote the concept of logical NOR: {| border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; |- ! Symbol  ! Origin  ! Known as |- | $p \downarrow q$ | {{AuthorRef|Willard Van Orman Quine|Willard Quine}} | Quine arrow |- | $p \ \mathsf{NOR} \ q$ |   |   |- | $p \mathop \bot q$ |  | |- | $p \curlywedge q$ | {{AuthorRef|Charles Sanders Peirce}} | Ampheck |} The all-uppercase rendition NOR originates from the digital electronics industry, where, because NOR is Functionally Complete, this operator has a high importance.
Definition:Axiom/Logic
An '''axiom''' in logic is a statement which is taken as '''self-evident'''. Note, however, that there has been disagreement for as long as there have been logicians and philosophers as to whether particular statements are true or not. For example, the Law of Excluded Middle is accepted as axiomatic by philosophers and logicians of the Aristotelian school but is denied by the intuitionist school.
Definition:Axiom/Formal Systems
Let $\mathcal L$ be a formal language. Part of defining a proof system $\mathscr P$ for $\mathcal L$ is to specify its '''axioms'''. An '''axiom of $\mathscr P$''' is a well-formed formula of $\mathcal L$ that $\mathscr P$ approves of by definition.
Definition:Axiom/Mathematics
The term '''axiom''' is used throughout the whole of mathematics to mean a statement which is accepted as true for that particular branch. Different fields of mathematics usually have different sets of statements which are considered as being '''axiomatic'''. So statements which are taken as axioms in one branch of mathematics may be theorems, or irrelevant, in others.
Definition:Strictly Precede/Definition 2
Let $\left({S, \preceq}\right)$ be an ordered set. Let $a \preceq b$ such that $a \ne b$. Then '''$a$ strictly precedes $b$'''.
Definition:Strictly Precede/Definition 1
Let $\left({S, \prec}\right)$ be a strictly ordered set. Let $a, b \in S$ and $a \prec b$. Then '''$a$ strictly precedes $b$'''.
Definition:Strictly Ordered Set
A '''strictly ordered set''' is a relational structure $\left({S, \prec}\right)$ such that the relation $\prec$ is an strict ordering.
Definition:Strictly Well-Ordered Set
{{expand|Add a definition to the effect that a '''strictly ordered set''' is a relational structure where the relation is a strict well-ordering.}} Let $\left({S, \prec}\right)$ be a strictly totally ordered set. Then $\left({S, \prec}\right)$ is a '''strictly well-ordered set'''  $\prec$ is a foundational relation.
Definition:Minimal/Relation
Let $\left({S, \mathcal R}\right)$ be a relational structure. Let $T \subseteq S$ be a subset of $S$. An element $x \in T$ is an '''$\mathcal R$-minimal element of $T$''' : :$\forall y \in T: y \not \mathrel {\mathcal R} x$
Definition:Maximal/Relation
Let $\left({S, \mathcal R}\right)$ be a relational structure. Let $T \subseteq S$ be a subset of $S$. An element $x \in T$ is an '''$\mathcal R$-maximal element of $T$''' : :$\forall y \in T: x \not \mathrel {\mathcal R} y$
Definition:Dual Order Embedding
Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be ordered sets. A '''dual order embedding''' is a mapping $\phi: S \to T$ such that: :$\forall x, y \in S: x \preceq_1 y \iff \phi \left({y}\right) \preceq_2 \phi \left({x}\right)$ That is: : if $\phi$ is an order embedding of $\left({S, \preceq_1}\right)$ into $\left({T, \succeq_2}\right)$ where $\succeq_2$ is the dual of $\preceq_2$.
Definition:Backus-Naur Form/Non-Terminal
Symbols that appear on the  of a specification in Backus-Naur form are called '''non-terminals.'''
Definition:Backus-Naur Form/Terminal
Symbols that never appear on the  of a specification in Backus-Naur form are called '''terminals.'''