hcju/proofwikips huggingface dataset:
Definition:Underlying Set Functor
Let $\mathbf{Set}$ be the category of sets. Let $\mathbf C$ be a metacategory with: :A collection of sets as objects; :A collection of mappings as morphisms. The '''underlying set functor''' $\left\vert{\cdot}\right\vert: \mathbf C \to \mathbf{Set}$ is defined in the following cases. {{expand|more to come}} === Category of Monoids === {{:Definition:Underlying Set Functor/Category of Monoids}}
Definition:Underlying Set Functor/Category of Monoids
Let $\mathbf{Mon}$ be the category of monoids. The '''underlying set functor''' $\left\vert{\cdot}\right\vert : \mathbf{Mon} \to \mathbf{Set}$ is the functor defined by: {{begin-axiom}} {{axiom|lc= Object functor:        |m = \left\vert{\left({M, \circ}\right)}\right\vert := M }} {{axiom|lc= Morphism functor:        |m = \left\vert{f}\right\vert := f }} {{end-axiom}}
Definition:Free Monoid
Let $S$ be a set. A '''free monoid over $S$''' is a monoid $M$ together with a mapping $i: S \to M$, subject to: :For all monoids $N$, for all mappings $f: S \to N$, there is a unique monoid homomorphism $\bar f: M \to N$, such that: ::$\bar f \circ i = f$ This condition is called the '''universal (mapping) property''' or '''UMP''' of the '''free monoid over $S$'''.
Definition:Cantor Normal Form
Let $x$ be an ordinal. The '''Cantor normal form''' of $x$ is an ordinal summation: :$x = \omega^{a_1} n_1 + \dots + \omega^{a_k} n_k$ where: :$k \in \N$ is a natural number :$\omega$ is the minimal infinite successor set :$\langle a_i \rangle$ is a strictly decreasing finite sequence of ordinals. :$\langle n_i \rangle$ is a finite sequence of finite ordinals In summation notation: :$x = \displaystyle \sum_{i \mathop = 1}^k \omega^{a_i} n_i$ {{explain|It still needs to be explained why, when used in pages that link to this, that the summation does not include the object $\omega$ in it, just some ordinal $x$ instead. It is unclear exactly what this definition means, because $\omega$, as currently defined on this website, is the Definition:Minimal Infinite Successor Set. Thus this definition appears to be saying: "Every ordinal (which of course has to include finite ones) can be expressed as finite sums of infinite ordinals." How can a finite number (an ordinal is a number, right?) be expressed as the sum of infinite numbers?}}
Definition:Free Category
Let $G$ be a digraph. The '''free category''' on $G$, denoted $\map {\mathbf C} G$, is the category with: {{DefineCategory | ob = The vertices of $G$ | mor = The walks in $G$ | comp = concatenation of walks | id = $\operatorname{id}_v$ is the empty walk at $v$ }}
Definition:Graph Functor
Let $\mathbf{Set}$ and $\mathbf{Rel}$ be the category of sets and the category of relations, respectively. The '''graph functor''' $G: \mathbf{Set} \to \mathbf{Rel}$ is defined by: {{begin-axiom}} {{axiom|lc= Object functor:        |t = $GX := X$ }} {{axiom|lc= Morphism functor:        |t = $Gf := G_f$, the graph of $f$ }} {{end-axiom}}
Definition:Standard Transitive Model
Let $p$ be a WFF in the language of set theory. Let $A$ be a class. Then $A$ is a '''standard transitive model''' of $p$ : :$A \ne \varnothing$ :$A$ is a transitive class and: :$A \models p$
Definition:Inverse Relation Functor
Let $\mathbf{Rel}$ be the category of relations. The '''inverse relation (contravariant) functor''' $C: \mathbf{Rel} \to \mathbf{Rel}$ is the contravariant functor defined by: {{begin-axiom}} {{axiom|lc= Object functor:        |t = $CX := X$ }} {{axiom|lc= Morphism functor:        |t = $C \mathcal R := \mathcal R^{-1}$, the inverse relation to $\mathcal R$ }} {{end-axiom}} That it is in fact a contravariant functor is shown on Inverse Relation Functor is Contravariant Functor.
Definition:Transitive Closure (Set Theory)
=== Definition 1 === {{:Definition:Transitive Closure (Set Theory)/Definition 1}} The following is not equivalent to the above, but they are almost the same. === Definition 2 === {{:Definition:Transitive Closure (Set Theory)/Definition 2}}
Definition:Split Monomorphism
Let $\mathbf C$ be a metacategory. Let $f: C \to D$ be a morphism of $\mathbf C$. Then $f$ is said to be a '''split monomorphism''' iff for some $g: D \to C$, one has: :$g \circ f = \operatorname{id}_C$ where $\operatorname{id}_C$ is the identity morphism of $C$.
Definition:Split Epimorphism
Let $\mathbf C$ be a metacategory. Let $f: C \to D$ be a morphism of $\mathbf C$. Then $f$ is said to be a '''split epimorphism'''  for some $g: D \to C$, one has: :$f \circ g = \operatorname{id}_D$ where $\operatorname{id}_D$ is the identity morphism of $D$. That is,  $f$ has a section.
Definition:Section (Category Theory)
Let $\mathbf C$ be a metacategory. Let $f: C \to D$ be a morphism of $\mathbf C$. A '''section of $f$''' is a morphism $g: D \to C$ such that: :$f \circ g = \operatorname{id}_D$
Definition:Retraction
Let $\mathbf C$ be a metacategory. Let $f: C \to D$ be a morphism of $\mathbf C$. A '''retraction of $f$''' is a morphism $g: D \to C$ such that: :$g \circ f = \operatorname{id}_C$ === Retract === {{:Definition:Retract}}
Definition:Retract
Let $g: D \to C$ be a retraction of $f$. Then $D$ is said to be a '''retract''' of $C$.
Definition:Projective Object
Let $\mathbf C$ be a metacategory. Let $P \in \mathbf C_0$ be an object of $\mathbf C$. Then $P$ is said to be '''projective''' iff: :For all epimorphisms $e: E \twoheadrightarrow X$ and morphisms $f: P \to X$, there exists $\bar f: P \to E$ such that: ::$\begin{xy} <0em,0em>*+{P} = "P", <4em,0em>*+{X} = "X", <4em,4em>*+{E} = "E", "P";"E" **@{-} ?>*@{>} ?*!/_.6em/{\bar f}, "P";"X" **@{-} ?>*@{>} ?*!/^.6em/{f}, "E";"X" **@{-} ?>*@2{>} ?<>(.7)*{\vee} ?*!/_.6em/{e}, \end{xy}$ :is a commutative diagram, i.e. such that $f = e \circ \bar f$. In this situation, $f$ is said to '''lift across $e$'''.
Definition:Transitive with Respect to a Relation
Let $A$ be a class. Let $\RR$ be a relation on $A$. Let $S$ be a set. Then $S$ is '''transitive with respect to $\RR$''' : :$\forall x \in A: \forall y \in S: \paren {x \mathrel \RR y \implies x \in S}$
Definition:Initial Object
Let $\mathbf C$ be a metacategory. An '''initial object''' of $\mathbf C$ is an object $0 \in \mathbf C_0$ of $\mathbf C$ such that: :For all $C \in \mathbf C_0$, there is a unique morphism $0 \to C$.
Definition:Terminal Object
Let $\mathbf C$ be a metacategory. A '''terminal object''' of $\mathbf C$ is an object $1 \in \mathbf C_0$ of $\mathbf C$ such that: :For all $C \in \mathbf C_0$, there is a unique morphism $C \to 1$.
Definition:Orbit (Group Theory)/Length
The '''length''' of the orbit $\Orb x$ of $x$ is the number of elements of $X$ it contains: :$\size {\Orb x}$
Definition:Orbit (Group Theory)/Set of Orbits
The quotient set $X / \mathcal R_G$ is called the '''set of orbits of $X$ under the action of $G$'''.
Definition:Relational Closure
Let $A$ be a class. Let $\RR \subseteq A \times A$ be a relation. Let $x$ be a small class that is a subset of $A$. The '''relational closure''' of $x$ is the smallest small class containing $x$ that is also $\RR$-transitive.
Definition:Closed Relation
Let $A$ be a class. Let $\mathcal R$ be an arbitrary relation. Then $\mathcal R$ is '''closed''' with respect to $A$ iff $A$ is transitive with respect to $\mathcal R^{-1}$.
Definition:Supertransitive Class
Let $A$ be a transitive class. Then $A$ is said to be a '''supertransitive class''' : :$\forall x \in A: \powerset x \subseteq A$ That is, if $A$ contains the power set of all of its elements. {{explain|This phrasing is a bit awkward; element and subset are bound to be confused}}
Definition:Von Neumann Hierarchy
Let $U$ denote the universal class. The '''von Neumann hierarchy''' is a mapping $V: \On \to U$ on the ordinals, defined via transfinite recursion: :$\map V x = \begin{cases} \O & : x = 0 \\ & \\ \powerset {\map V n} & : x = n^+ \\ & \\ \displaystyle \bigcup_{y \mathop \in x} \map V y & : x \in \operatorname {Lim} \\ \end{cases}$ where: :$\powerset x$ denotes the power set of $x$ :$\operatorname {Lim}$ denotes the set of limit ordinals.
Definition:Boolean Ring
Let $\struct {R, +, \circ}$ be a ring. Then $R$ is called a '''Boolean ring'''  $R$ is an idempotent ring with unity.
Definition:Well-Founded Set
Let $S$ be a small class. Let $V \left({ x }\right)$ denote the von Neumann hierarchy. Then $S$ is a '''well-founded set'''  there is some ordinal $x$ such that $S \in V\left({x}\right)$.
Definition:Rank (Set Theory)
Let $A$ be a set. Let $V$ denote the von Neumann hierarchy. Then the '''rank''' of $A$ is the smallest ordinal $x$ such that $A \in V \left({x+1}\right)$, given that $x$ exists.
Definition:Prime Number/Negative Prime
A '''negative prime''' is an integer of the form $-p$ where $p$ is a (positive) prime number.
Definition:Composite Number/Negative Composite
A negative integer $n$ is composite  $\left|{n}\right|$ is composite.
Definition:Prime Triplet
A '''prime triplet''' is a set of three (positive) prime numbers $\left\{{n, n+2, n+4}\right\}$.
Definition:Power (Algebra)/Integer
Let $x \in \R$ be a real number. Let $n \in \Z$ be an integer. The expression $x^n$ is called '''$x$ to the power of $n$'''. $x^n$ is defined recursively as: :$x^n = \begin{cases} 1 & : n = 0 \\ & \\ x \times x^{n - 1} & : n > 0 \\ & \\ \dfrac {x^{n + 1} } x & : n < 0 \end{cases}$ where $\dfrac{x^{n + 1} } x$ denotes quotient.
Definition:Power (Algebra)/Even Power
Let $x \in \R$ be a real number. Let $n \in \Z$ be an even integer. Then $x^n$ is called an '''even power of $x$'''.
Definition:Power (Algebra)/Odd Power
Let $x \in \R$ be a real number. Let $n \in \Z$ be an odd integer. Then $x^n$ is called an '''odd power of $x$'''
Definition:Power (Algebra)/Rational Number
Let $x \in \R$ be a real number such that $x > 0$. Let $m \in \Z$ be an integer. Let $y = \sqrt [m] x$ be the $m$th root of $x$. Then we can write $y = x^{1/m}$ which means the same thing as $y = \sqrt [m] x$. Thus we can define the power to a positive rational number: Let $r = \dfrac p q \in \Q$ be a positive rational number where $p \in \Z_{\ge 0}, q \in \Z_{> 0}$. Then $x^r$ is defined as: :$x^r = x^{p/q} = \paren {\sqrt [q] x}^p = \sqrt [q] {\paren {x^p} }$. When $r = \dfrac {-p} q \in \Q: r < 0$ we define: :$x^r = x^{-p/q} = \dfrac 1 {x^{p/q}}$ analogously for the negative integer definition.
Definition:Power (Algebra)/Real Number
Let $x \in \R_{>0}$ be a (strictly) positive real number. Let $r \in \R$ be a real number. === Definition 1 === {{:Definition:Power (Algebra)/Real Number/Definition 1}} === Definition 2 === {{:Definition:Power (Algebra)/Real Number/Definition 2}} === Definition 3 === {{:Definition:Power (Algebra)/Real Number/Definition 3}}
Definition:Power (Algebra)/Complex Number
Let $z, k \in \C$ be any complex numbers. '''$z$ to the power of $k$''' is defined as the multifunction: :$z^k := e^{k \ln \paren z}$ where $e^z$ is the exponential function and $\ln$ is the  natural logarithm multifunction.
Definition:Power (Algebra)/Power of Zero
Let $r \in \R$ be a real number. (This includes the situation where $r \in \Z$ or $r \in \Q$.) When $x=0$, $x^r$ is defined as follows: :$0^r = \begin{cases} 1 & : r = 0 \\ 0 & : r > 0 \\ \text{Undefined} & : r < 0 \\ \end{cases}$ This takes account of the awkward case $0^0$: it is "generally accepted" that $0^0 = 1$ as this convention agrees with certain general results which would otherwise need a special case.
Definition:Power (Algebra)/Exponent
In the expression $x^r$, the number $r$ is known as the '''exponent''' of $x$, particularly for $r \in \R$.
Definition:Two (Boolean Algebra)
Denote with $\top$ the canonical tautology. Denote with $\bot$ the canonical contradiction. Define $\mathbf 2 := \set {\bot, \top}$, read '''two'''. When endowed with the logical operations $\lor$, $\land$ and $\neg$, $\mathbf 2$ becomes a Boolean algebra. These operations have the following Cayley tables: :$\begin{array}{c|cc}  \lor & \bot & \top \\ \hline  \bot & \bot & \top \\  \top & \top & \top \end{array} \qquad \begin{array}{c|cc}  \land & \bot & \top \\ \hline  \bot  & \bot & \bot \\  \top  & \bot & \top \end{array} \qquad \begin{array}{c|cc}       & \bot & \top \\ \hline  \neg & \top & \bot \end{array}$ That $\mathbf 2$ thus becomes a Boolean algebra is shown on Two is Boolean Algebra.
Definition:Category of Boolean Algebras
The '''category of Boolean algebras''', denoted $\mathbf{BA}$, is the metacategory with: {{DefineCategory |ob  = Boolean algebras |mor =  Boolean homomorphisms }}
Definition:Operation/Unary Operation
A '''unary operation''' is the special case of an operation where the operation has exactly one operand. Thus, a '''unary operation''' on a set $S$ is a mapping whose domain and codomain are both $S$.
Definition:Constant (Category Theory)
Let $\mathbf C$ be a metacategory, and let $1$ be a terminal object of $\mathbf C$. A '''constant of $\mathbf C$''' is a morphism $f: 1 \to C$ of $\mathbf C$ which has $1$ as its domain.
Definition:Variable Element
Let $\mathbf C$ be a metacategory, and let $C$ be an object of $\mathbf C$. A '''variable element''' of $C$ is a morphism $f: B \to C$ with codomain $C$.
Definition:Enough Constants
Let $\mathbf C$ be a metacategory. Then $\mathbf C$ is said to have '''enough constants''' iff: :For all morphisms $f,g : C \to D$ with $f \ne g$, there is a constant $c: 1 \to C$ such that $f \circ c \ne g \circ c$
Definition:Product (Category Theory)
=== Binary Product === {{:Definition:Product (Category Theory)/Binary Product}} === General Definition === {{:Definition:Product (Category Theory)/General Definition}}
Definition:Cardinal Number
Let $S$ be a set. The '''cardinal number''' of $S$ is defined as follows: :$\card S = \displaystyle \bigcap \set {x \in \On : x \sim S}$ where $\On$ is the class of all ordinals. Compare cardinality.
Definition:Aleph Mapping
Let $\mathcal N'$ denote the class of all infinite cardinal numbers. Then $\aleph$ (that is: '''aleph''') is defined as the unique order isomorphism between the two ordered structures $\left({\operatorname{On}, \in}\right)$ and $\left({\mathcal N', \in}\right)$ where $\operatorname{On}$ denotes the class of ordinal numbers. {{explain|A link is needed to a proof that such a unique order isomorphism exists.}}
Definition:Class of Cardinals
The '''class of cardinals''' is the class consisting of all cardinal numbers: :$\NN = \set {x \in \On: \exists y: x = \size y}$ where $\size y$ denotes the cardinal number corresponding to $y$.
Definition:Infinite Cardinal Class
The '''infinite cardinal class''' is the class of all infinite cardinal numbers. That is: :$\mathcal N’ = \mathcal N \setminus \omega$
Definition:Definiendum
An object being defined in a definition is called a '''definiendum'''.
Definition:Definiens
The part of a definition that explains the meaning of the definiendum is the '''definiens'''.
Definition:Group Axioms/Right
A '''group''' is an algebraic structure $\struct {G, \circ}$ which satisfies the following four conditions: {{begin-axiom}} {{axiom | n = \text G 0         | lc= Closure Axiom         | q = \forall a, b \in G         | m = a \circ b \in G }} {{axiom | n = \text G 1         | lc= Associativity Axiom         | q = \forall a, b, c \in G         | m = a \circ \paren {b \circ c} = \paren {a \circ b} \circ c }} {{axiom | n = \text G_{\text R} 2         | lc= Right Identity Axiom         | q = \exists e \in G: \forall a \in G         | m = a \circ e = a }} {{axiom | n = \text G_{\text R} 3         | lc= Right Inverse Axiom         | q = \forall a \in G: \exists b \in G         | m = a \circ b = e }} {{end-axiom}}
Definition:Group Axioms/Left
A '''group''' is an algebraic structure $\struct {G, \circ}$ which satisfies the following four conditions: {{begin-axiom}} {{axiom | n = \text G 0         | lc= Closure Axiom         | q = \forall a, b \in G         | m = a \circ b \in G }} {{axiom | n = \text G 1         | lc= Associativity Axiom         | q = \forall a, b, c \in G         | m = a \circ \paren {b \circ c} = \paren {a \circ b} \circ c }} {{axiom | n = \text G_{\text L} 2         | lc= Left Identity Axiom         | q = \exists e \in G: \forall a \in G         | m = e \circ a = a }} {{axiom | n = \text G_{\text L} 3         | lc= Left Inverse Axiom         | q = \forall a \in G: \exists b \in G         | m = b \circ a = e }} {{end-axiom}}
Definition:Product of Morphisms
Let $\mathbf C$ be a metacategory. Let $A, A'$ and $B, B'$ be pairs of objects admitting binary products: ::$\begin{xy}\xymatrix@R-1em@C+1em@L+3px{  A &  A \times A'   \ar[l]_*+{p_1}   \ar[r]^*+{p_2} &  A' \\  B &  B \times B'   \ar[l]_*+{q_1}   \ar[r]^*+{q_2} &  B' }\end{xy}$ Let $f: A \to B$ and $f': A' \to B'$ be morphisms. The '''product morphism of $f$ and $f'$''', denoted $f \times f'$, is the unique morphism making the following diagram commute: ::$\begin{xy}\xymatrix@+1em@L+3px{  A   \ar[d]_*+{f} &  A \times A'   \ar[l]_*+{p_1}   \ar[r]^*+{p_2}   \ar@{-->}[d]^*+{\hskip{1.3em} f \times f'} &  A'   \ar[d]^*+{f'} \\  B &  B \times B'   \ar[l]^*+{q_1}   \ar[r]_*+{q_2} &  B' }\end{xy}$ Thus we see that $f \times f'$ is the morphism $\left\langle{fp_1, f'p_2}\right\rangle$.
Definition:Fixed Element of Permutation/Moved
$x$ '''moved by $\pi$''' : :$\map \pi x \ne x$
Definition:Fixed Element of Permutation/Set of Fixed Elements
The set of elements of $S$ which are fixed by $\pi$ can be denoted $\Fix \pi$.
Definition:Permutation on n Letters/Set of Permutations
The '''set of permutations of $\N_n$''' is denoted $S_n$.
Definition:Product Functor
Let $\mathbf C$ be a metacategory with binary products. Let $\mathbf C \times \mathbf C$ be the product category of $\mathbf C$ with itself. The '''product functor''' on $\mathbf C$ is the functor $\times: \mathbf C \times \mathbf C \to \mathbf C$ defined by: {{begin-axiom}} {{axiom|lc= Object functor:        |m = \times \left({C, D}\right) := C \times D }} {{axiom|lc= Morphism functor:        |m = \times \left({f, f'}\right) := f \times f' }} {{end-axiom}} where $C \times D$ is a binary product of $C$ and $D$, and $f \times f'$ is the product of $f$ and $f'$. That it is in fact a functor is shown on Product Functor is Functor.
Definition:One (Ordinal)
The ordinal $1$ is defined as: :$1 = 0^+$ where $0^+$ denotes the successor of the ordinal $0$.
Definition:Two (Ordinal)
The ordinal $2$ is defined as: :$2 = 1^+$ where $1^+$ denotes the successor of the ordinal $1$.
Definition:Composable Morphisms
Let $\mathbf C$ be a metacategory. Let $f, g \in \mathbf C_1$ be morphisms of $\mathbf C$. Then $f$ is said to be '''composable with $g$''' iff: :$\operatorname{cod} f = \operatorname{dom} g$ that is, iff the codomain of $f$ is the domain of $g$. When the order of composition is to be made more explicit, one says that '''$\left({g, f}\right)$ is a composable pair'''. The collection of all such '''composable pairs''' in $\mathbf C$ is denoted $\mathbf C_2$.
Definition:Category with Products/Binary
Let $\mathbf C$ be a metacategory. Then $\mathbf C$ is said to '''have binary products''' or to be a '''(meta)category with binary products''' iff: :For all objects $C, D \in \mathbf C_0$, there is a binary product $C \times D$ for $C$ and $D$.
Definition:Cofinal Relation on Ordinals
Let $x$ and $y$ be ordinals. Then $y$ is said to be '''cofinal''' with respect to $x$  there exists a mapping $f: y \to x$ such that: :$(1): \quad y \le x$ :$(2): \quad f$ is strictly increasing. :$(3): \quad$ For all $a \in x$, there is some $b \in y$ such that $f \left({b}\right) \ge a$.
Definition:Regular Cardinal
Let $\kappa$ be an infinite cardinal. Then $\kappa$ is a '''regular cardinal''' : :$\operatorname{cf} \left({\kappa}\right) = \kappa$ That is,  the cofinality of $\kappa$ is equal to itself.
Definition:Singular Cardinal
Let $\kappa$ be an infinite cardinal. Then $\kappa$ is a '''singular cardinal'''  $\operatorname{cf} \left({\kappa}\right) < \kappa$. That is, the cofinality of $\kappa$ is less than itself.
Definition:Weakly Inaccessible Cardinal
An infinite cardinal $\aleph_\kappa$ is '''weakly inaccessible''' : :$(1): \quad \kappa$ is a limit ordinal. :$(2): \quad \aleph_\kappa$ is a regular cardinal.
Definition:Strongly Inaccessible Cardinal
An infinite cardinal $\aleph_\kappa$ is called a '''strongly inaccessible cardinal''' : :$(1): \quad \aleph_\kappa$ is a weakly inaccessible cardinal :$(2): \quad \forall x \in \kappa: \left|{\mathcal P \left({x}\right)}\right| \in \kappa$
Definition:Greatest Common Divisor/Integers/General Definition
Let $S = \set {a_1, a_2, \ldots, a_n} \subseteq \Z$ such that $\exists x \in S: x \ne 0$ (that is, at least one element of $S$ is non-zero). Then the '''greatest common divisor''' of $S$: :$\gcd \paren S = \gcd \set {a_1, a_2, \ldots, a_n}$ is defined as the largest $d \in \Z_{>0}$ such that: :$\forall x \in S: d \divides x$
Definition:Inverse Morphism
A morphism $g: Y \to X$ is said to be an '''inverse (morphism)''' for $f$ : :$g \circ f = I_X$ :$f \circ g = I_Y$ where $I_X$ denotes the identity morphism on $X$.
Definition:Population
The '''population''' of a statistical study is everything in the universe of discourse that is relevant to the study. It includes all objects, measures, and observations under discussion. === Finite Population === {{:Definition:Population/Finite}} === Infinite Population === {{:Definition:Population/Infinite}}
Definition:Sample
A '''sample''' is a non-empty, finite, proper subset of a population.
Definition:Individual
An '''individual''' is an object in a population.
Definition:Object (Category Theory)
Let $\mathbf C$ be a metacategory. An '''object''' of $\mathbf C$ is an object which is considered to be atomic from a category theoretic perspective. It is a conceptual device introduced mainly to make the discussion of morphisms more convenient. '''Objects''' in a general metacategory are usually denoted with capital letters like $A,B,C,X,Y,Z$. The collection of '''objects''' of $\mathbf C$ is denoted $\mathbf C_0$. That '''objects''' don't play an important role in category theory is apparent from the fact that the notion of a metacategory can be described while avoiding to mention '''objects''' altogether. {{wtd|That is a big claim which needs a page to back it up; this is demonstrated eg. in 'Categories for the Working Mathematician' by MacLane}} Nonetheless the notion of '''object''' is one of the two basic concepts of metacategories and as such of category theory.
Definition:Maximal Subgroup
Let $G$ be a group. Let $M \le G$ be a proper subgroup of $G$. Then $M$ is a '''maximal subgroup of $G$''' : :For every subgroup $H$ of $G$, $M \subseteq H \subseteq G$ means $M = H$ or $H = G$. That is, if there is no subgroup of $G$, except $M$ and $G$ itself, which contains $M$.
Definition:Join of Subgroups/General Definition
Let $H_1, H_2, \ldots, H_n$ be subgroups of $G$. Then the '''join''' of $H_1, H_2, \ldots, H_n$ is defined as: :$\displaystyle \bigvee_{k \mathop = 1}^n H_k := \left \langle {\bigcup_{k \mathop = 1}^n H_k}\right \rangle$ or: :$\displaystyle \bigvee_{k \mathop = 1}^n H_k := \bigcap \left\{{T: T \text { is a subgroup of } G: \bigcup_{k \mathop = 1}^n H_k \subseteq T}\right\}$
Definition:Simple Random Sample
Let $P$ be a population. Let $S \subsetneq P$ be a sample. Then $S$ is a '''simple random sample''' (of size $n$) iff it fulfils the following criteria: * The process used to select the individuals of $S$ from $P$ was random; * Every individual in $P$ had an equal chance of being selected to be in $S$; * Every $n$-combination of $P$ had an equal chance of being constructed as a potential $S$.
Definition:Stratified Random Sample
Let $P$ be a population. Let $S \subsetneq P$ be a sample. Then $S$ is a '''stratified random sample''' : :there exists a finite expansion $P = P_1 \mid P_2 \mid \cdots \mid P_k$ of $P$ such that $S$ is the union of $k$ simple random samples, one taken from each $P_i$.
Definition:Convenience Sample
Let $P$ be a population in the physical universe. Let $S \subsetneq P$ be a sample. Suppose that data about some individuals in $P$ are more easily accessible than data about others. Then $S$ is a '''convenience sample'''  the primary criterion for the construction of $S$ is the higher accessibility of data of certain individuals in $P$ over others. That is, the objects in $S$ are the ones conveniently at hand.
Definition:Unambiguous
A statement is '''unambiguous''' if it is not ambiguous.