# Algebra Reference¶

absolute

Applied to couplets, sets, constructs derived from them (like relations) and algebras. Such a construct is called absolute if its ground set is based on set A ($$A$$).

See for example absolute set.

• A relation that has only members that are elements of the Cartesian product $$A \times A$$ is an ‘absolute relation’. Example: $$\{a{\mapsto}1, b{\mapsto}2\}$$. (However, an absolute relation is not an absolute set; the members of the relation are couplets, not atoms.)
• The relation $$\{a{\mapsto}1, b{\mapsto}\{2\}\}$$ is not an absolute relation, as one of the right elements of the member couplets is not an atom (it is a set $$\{2\}$$).
absolute clan
An absolute clan is a clan that is an element of the second power set of the Cartesian product of set A ($$A$$) ($$P^2(A \times A)$$). Such a clan has only atoms as members of the couplets in the relations. Example: $$\{ \{2{\mapsto}1, 3{\mapsto}2\}, \{5{\mapsto}4, 9{\mapsto}7\} \}$$.
absolute couplet
An absolute couplet is a couplet that is an element of the Cartesian product of set A ($$A$$) ($$A \times A$$). Such a couplet has only atoms as members. Example: $$2{\mapsto}1$$.
absolute ground set

Our algebras normally have a ground set and an absolute ground set. The absolute ground set is the ground set with the elements of the algebra being expressed in terms of set A ($$A$$).

For example, the ground set of the algebra of relations is $$P(M \times M)$$; this allows sets and couplets as elements of the couplets that form the relations. The absolute ground set of the algebra of relations is $$P(A \times A)$$; this requires atoms to form the couplets that form the relations.

absolute multiclan
An absolute multiclan is a multiclan that has only atoms as members of the couplets in the relations. See also absolute clan.
absolute multiset
An absolute multiset is a multiset that is an element of the power set of the Cartesian product of set A ($$A$$) with set N ($$N$$) ($$P(A \times N)$$. Such a multiset has only atoms as elements. Example: $$\{1{:}3, 'a'{:}5\}$$.
absolute relation
An absolute relation is a set that is an element of the power set of the Cartesian product of set A ($$A$$) ($$P(A \times A)$$. Such a set has only couplets as elements that consist only of atoms. Example: $$\{2{\mapsto}1, 5{\mapsto}2\}$$.
absolute set
An absolute set is a set that is an element of the power set of set A ($$A$$) ($$P(A)$$. Such a set has only atoms as elements. Example: $$\{1, 2\}$$.
algebra
An algebra is a set together with one or more operations defined on it. For a more detailed definition see Algebras.
algebra of clans
The algebra of clans is the algebra on sets of relations. As a result its ground set is the second power set of couplets, which allows us to use more operations in addition to those on an algebra of sets. See Algebra of Clans for a more detailed explanation.
algebra of couplets
Given a collection of couplets we define an algebra on it by including the operations of composition and transposition. See Algebra of Couplets for a more detailed definition.
algebra of multiclans
An algebra defined on multisets of relations. This algebra is similar to the algebra of clans, but takes into account multiplicities of relations. See Algebra of Multiclans for more information.
algebra of multisets
An algebra on the set of all multisets of set M ($$M$$) with the operations of multiset union, multiset intersection, multiset addition, multiset difference, and submultiset. See Multiset Algebra for more detailed information.
algebra of relations
The algebra of relations is an algebra on a set of couplets under the operations of composition and transposition. See Algebra of Relations for a more detailed definition.
algebra of sets
Also called set algebra, is the algebra formed by taking the power set of a set and applying the operations of union and intersection. See Algebra of Sets for a more detailed definition.
atom
An atom is a datum that is not a set or a couplet. The set of all atoms is set A ($$A$$).
bijective
A relation is bijective if it is both left-functional and right-functional.
binary extension

A binary extension is an extension of a binary operation from a given algebra to an extension of the algebra that consists of sets of the elements of the original algebra:

$binaryExtn(op, S1, S2) := \{op(s1, s2)\ :\ s1 \in S1 \text{ and } s2 \in S2 \text{ where } op(s1, s2) \text{ is defined}\}$
binary intersection
A binary intersection is an intersection of two sets. See intersection for a complete definition.
binary multi-extension

A binary multi-extension is an extension of a binary operation from a given algebra to an extension of the algebra that consists of multisets of the elements of the original algebra:

$binMultiExtn(op, S1, S2) := \{op(s1, s2){:}(\dot{S1}(s1) \cdot \dot{S2}(s2))\ :\ s1 \in S1 \text{ and } s2 \in S2 \text{ where } op(s1, s2) \text{ is defined}\}$
binary operation
A binary operation is an operation with two arguments, typically with a result that belongs to the same ground set as the arguments (when the operation is a member of an algebra).
binary relation
We represent a binary relation as a set where every member is a couplet. We also call this simply a relation.
binary union
A binary union is a union of two sets. See union for more information.
Cartesian product
The Cartesian product of two sets $$X \times Y$$ is the set of all couplets where the first member of the couplet is a member of $$X$$ and the second member of the couplet is a member of $$Y$$.
clan
A clan is a set of relations.
clan diagonal
A clan diagonal is a clan with a single relation that is a diagonal.
complement

The complement of a given set is the collection of elements not in the given set. This definition depends on a choice of a larger set in which context every other set is a subset of. In particular, given an algebra of sets whose ground set is the power set $$P(U)$$ and $$X\in P(U)$$, then the complement of $$X$$ is:

$X' := \{x \in U: x \not\in X\} = U - X$
composition

The composition of the couplets $$a{\mapsto}b$$ and $$c{\mapsto}d$$ is defined as:

$\begin{split}c{\mapsto}d \circ a{\mapsto}b := \begin{cases} a{\mapsto}d & \text{if } b = c \\ \text{undefined} & \text{if } b \ne c \end{cases}\end{split}$

The operation may be extended to extended algebras using the binary extension and – if there is no danger of ambiguities – is then also called simply ‘composition’.

Specific extensions:

• Algebra of relations: $$R_2 \circ R_1 := \{c_2 \circ c_1\ :\ c_1 \in R_1,\ c_2 \in R_2\}$$

($$R_1$$ and $$R_2$$ are relations; $$c_1$$ and $$c_2$$ are couplets.)

• Algebra of clans: $$C_2 \circ C_1 := \{R_2 \circ R_1\ :\ R_1 \in C_1,\ R_2 \in C_2\}$$

($$C_1$$ and $$C_2$$ are clans; $$R_1$$ and $$R_2$$ are relations.)

• Algebra of multiclans: $$\dot{C}_2 \circ \dot{C}_1 := \{(R_2 \circ R_1){:}(\dot{C}_2(R_2) \cdot \dot{C}_1(R_1))\ :\ R_1 \in \dot{C}_1,\ R_2 \in \dot{C}_2\}$$

($$\dot{C}_1$$ and $$\dot{C}_2$$ are multiclans; $$R_1$$ and $$R_2$$ are relations.)

couplet
A couplet is an ordered pair, following the Kuratowski definition of an ordered pair defined as $$\{\{l\}, \{l, r\}\}$$. It is the mathematical object used to represent a datum or data point. We denote it by $$l{\mapsto}r$$, with $$l$$ called the left component, and $$r$$ the right component.
cross-functional union
A short name for cross-left-functional union in cases where no ambiguities are expected.
cross-intersection

The cross-intersection is an extension of intersection (or multiset intersection). Depending on the specific form of extension, it may be a binary extension (when extending to an algebra of sets) or a binary multi-extension (when extending to an algebra of multisets).

In the specific context, cross-intersection is the binary extension of intersection from an algebra of sets to an algebra of sets of sets (for example, from the algebra of relations to the algebra of clans). The cross-intersection of the sets (of sets) $$\mathbb{S}$$ and $$\mathbb{T}$$ is defined as:

$\mathbb{S} \blacktriangle \mathbb{T} = \{X \cap Y\ : X \in \mathbb{S} \text{ and } Y \in \mathbb{T}\}$

For the version of cross-intersection on the algebra of multiclans and on an algebra of multisets in general see multi-cross-intersection.

cross-left-functional union

The cross-left-functional union, (or cross-functional union), of two clans $$\mathbb{C}$$ and $$\mathbb{D}$$ is a binary extension of the left-functional union from the algebra of relations to the algebra of clans:

$\mathbb{C} \underset{f}{\blacktriangledown} \mathbb{D} = \{R \underset{f}{\cup} Q\ : R \in \mathbb{C} \text{ and } Q \in \mathbb{D}\}$
cross-right-functional union

The cross-right-functional union of two clans $$\mathbb{C}$$ and $$\mathbb{D}$$ is a binary extension of the right-functional union from the algebra of relations to the algebra of clans:

$\mathbb{C} \underset{rf}{\blacktriangledown} \mathbb{D} = \{R \underset{rf}{\cup} Q\ : R \in \mathbb{C} \text{ and } Q \in \mathbb{D}\}$
cross-substriction

The cross-substriction is a binary extension of substriction from an algebra of sets to an algebra of sets of sets (for example, from the algebra of relations to the algebra of clans). The cross-substriction of the sets (of sets) $$\mathbb{S}$$ and $$\mathbb{T}$$ is defined as:

$\mathbb{S} \blacktriangleleft \mathbb{T} = \{X \vartriangleleft Y\ : X \in \mathbb{S} \text{ and } Y \in \mathbb{T}\} = \{X : X \in \mathbb{S} \text{ and } X \subset Y \text{ for some } Y \in \mathbb{T} \}$

We also have a binary multi-extension of cross-substriction called the multi-cross-substriction.

cross-superstriction

The cross-superstriction is a binary extension of superstriction from an algebra of sets to an algebra of sets of sets (for example, from the algebra of relations to the algebra of clans). The cross-superstriction of the sets (of sets) $$\mathbb{S}$$ and $$\mathbb{T}$$ is defined as:

$\mathbb{S} \blacktriangleright \mathbb{T} = \{X \vartriangleright Y\ : X \in \mathbb{S} \text{ and } Y \in \mathbb{T}\} = \{X : X \in \mathbb{S} \text{ and } X \supset Y \text{ for some } Y \in \mathbb{T} \}$

We also have a binary multi-extension of cross-superstriction called the multi-cross-superstriction.

cross-union

The cross-union is a binary extension of union from an algebra of sets to an algebra of sets of sets (for example, from the algebra of relations to the algebra of clans). The cross-union of the sets (of sets) $$\mathbb{S}$$ and $$\mathbb{T}$$ is defined as:

$\mathbb{S} \blacktriangledown \mathbb{T} = \{X \cup Y\ : X \in \mathbb{S} \text{ and } Y \in \mathbb{T}\}$

We also have a binary multi-extension of cross-union called the multi-cross-union.

diagonal

The diagonal is an equivalence relation. The diagonal of a set $$S$$ is defined as:

$D_S = \{x{\mapsto}x \in S \times S\ : x \in S\}$
difference

The difference of two sets $$S$$ and $$T$$ is the set of elements in $$S$$ but not in $$T$$ (see also Wikipedia: Relative complement). The definition is:

$S \setminus T = \{x: x \in S\ \&\ x \notin T\}$
equivalence relation
A relation is said to be an equivalence relation if it is reflexive, symmetric and transitive.
extension
An extension of a given set or algebra, is an extension of the set or algebra either by expanding the ground set, the set of operations on the set or both. For a more detailed definition, see Extension.
finite intersection
A finite intersection is the intersection of a finite collection of sets. See intersection for a complete definition.
finite union
A finite union is the union of a finite collection of sets. See union for a more detailed explanation.
function
A function is a left-functional relation.
functional
A short name for left-functional in cases where no ambiguities are expected.
functional union
A short name for left-functional union in cases where no ambiguities are expected.
ground set
The set that contains all the elements of an algebra.
identity element
The element of the ground set of an algebra that, when used as one of the arguments of a binary operation produces the other argument of the operation as result of the operation.
intersection

An operation on sets that creates a set by collecting the elements in common to two or more individual sets into a new set. In mathematical terms, if $$\mathbb{S}$$ is a collection of sets, then the intersection of all of the sets in $$\mathbb{S}$$ is denoted:

$\bigcap \mathbb{S} = \bigcap_{T\in\mathbb{S}}T,$

and is the set $$\{x\ : \forall T \in \mathbb{S},\ x \in T\}$$. If $$\mathbb{S}$$ consists of only two sets, the intersection is called a binary intersection. If $$\mathbb{S}$$ consists of a finite collection of sets, the intersection is called a finite intersection. See also Wikipedia: Intersection.

left
A short name for left component where no ambiguity is expected.
left component
Given a couplet represented by $$l{\mapsto}r$$, the component $$l$$ is called the left component.
left set

The left set of a relation $$R$$ is the set of all left components of its members:

$left(R) = \{ l\ : l{\mapsto}r \in R \}$

The left set of a clan $$\mathbb{C}$$ is the union of all left sets of its member relations:

$left(\mathbb{C}) = \underset{R\ \in\ \mathbb{C}}{\bigcup} left(R)$
left-functional

A relation $$R$$ is said to be left-functional, or simply functional, if:

$x{\mapsto}y \in R\ \& \ x{\mapsto}z \in R \implies y = z$

A clan $$\mathbb{C}$$ is said to be (left-)functional if all its relations are functional:

$\forall R \in \mathbb{C}: R \text{ is left-functional}$
left-functional union

The left-functional union (or functional union) of two functions $$R$$ and $$Q$$ is the union of the two relations if the result is left-functional, otherwise the result is not defined:

$\begin{split}R \underset{f}{\cup} Q = \begin{cases} R \cup Q & \text{if }R \cup Q \text{ is left-functional} \\ \text{undefined} & \text{if it is not left-functional} \end{cases}\end{split}$
left-regular

A clan $$\mathbb{C}$$ is said to be left-regular (or short regular) if it is left-functional and the left sets of all its relations are the same:

\begin{split}\begin{align*} \forall R \in \mathbb{C}&: R \text{ is left-functional} \text{ and } \\ \forall R, Q \in \mathbb{C}&: left(R) = left(Q) \end{align*}\end{split}
lhs-cross-functional union
A short name for lhs-cross-left-functional union in cases where no ambiguities are expected.
lhs-cross-left-functional union

Given two clans the lhs-cross-left-functional union (or lhs-cross-functional union) takes the cross-left-functional union, then any relations in the clan on the left side of the operation that resulted in the empty set are collected by taking their union and combined with the result of the cross-left-functional union. Mathematically, if $$\mathbb{C}$$ and $$\mathbb{D}$$ are the two clans in question, then their lhs-cross-left-functional union is:

$\mathbb{C} \overrightarrow{\underset{f}\blacktriangledown} \mathbb{D} = \mathbb{C} \underset{f}{\blacktriangledown} \mathbb{D} \bigcup\{T : T\in\mathbb{C}\ \& \ T \underset{f}{\blacktriangledown} \mathbb{D} = \varnothing \}$
multi-cross-intersection

Let $$\dot{\mathbb{S}}$$ and $$\dot{\mathbb{T}}$$ be multisets of sets with $$X \in \dot{\mathbb{S}}$$ and $$Y \in \dot{\mathbb{T}}$$, then their multi-cross-intersection is defined in the same way as the cross-intersection with the multiplicities satisfying:

$\dot{\mathbb{S}} \blacktriangle \dot{\mathbb{T}}(X \cap Y) = \dot{\mathbb{S}}(X) \cdot \dot{\mathbb{T}}(Y)$

Where there is no ambiguity we will refer to the multi-cross-intersection simply as the cross-intersection.

multi-cross-substriction

Let $$\dot{\mathbb{S}}$$ and $$\dot{\mathbb{T}}$$ be multisets of sets with $$X \in \dot{\mathbb{S}}$$ and $$Y \in \dot{\mathbb{T}}$$, then their multi-cross-substriction is defined in the same way as the cross-substriction with the multiplicities satisfying:

$\dot{\mathbb{S}} \blacktriangleleft \dot{\mathbb{T}}(X \vartriangleleft Y) = \dot{\mathbb{S}}(X)$

Where there is no ambiguity we will refer to the multi-cross-substriction simply as the cross-substriction.

multi-cross-superstriction

Let $$\dot{\mathbb{S}}$$ and $$\dot{\mathbb{T}}$$ be multisets of sets with $$X \in \dot{\mathbb{S}}$$ and $$Y \in \dot{\mathbb{T}}$$, then their multi-cross-superstriction is defined in the same way as the cross-superstriction with the multiplicities satisfying:

$\dot{\mathbb{S}} \blacktriangleright \dot{\mathbb{T}}(X \vartriangleright Y) = \dot{\mathbb{S}}(X)$

Where there is no ambiguity we will refer to the multi-cross-superstriction simply as the cross-superstriction.

multi-cross-union

Let $$\dot{\mathbb{S}}$$ and $$\dot{\mathbb{T}}$$ be multisets of sets with $$X \in \dot{\mathbb{S}}$$ and $$Y \in \dot{\mathbb{T}}$$, then their multi-cross-union is defined in the same way as the cross-union with the multiplicities satisfying:

$\dot{\mathbb{S}} \blacktriangledown \dot{\mathbb{T}}(X \cup Y) = \dot{\mathbb{S}}(X) \cdot \dot{\mathbb{T}}(Y)$

Where there is no ambiguity we will refer to the multi-cross-union simply as the cross-union.

multiclan
A multiclan is a multiset of relations.
multiplicity
Given an element of a multiset, the multiplicity of that element is the number of times the element appears in the multiset. See Multiset for more information.
multiset
A multiset, also sometimes called a bag, is a generalization of the idea of a set where multiple instances of the same element are allowed. See Multiset for more information.
multiset addition

The multiset addition of two multisets $$\dot{S}$$ and $$\dot{T}$$ is defined as follows:

$\big(\dot{S} + \dot{T}\big)(x) = \dot{S}(x) + \dot{T}(x)$

for any $$x$$, where here we take $$\dot{S}(x)=0$$ for $$x \not\in \dot{S}$$, and $$\dot{T}(x)=0$$ for $$x \not\in \dot{T}$$.

multiset difference

The multiset difference of two multisets $$\dot{S}$$ and $$\dot{T}$$ is defined as follows:

$\begin{split}\big(\dot{S} \setminus \dot{T}\big)(x) = \begin{cases} \dot{S}(x) - \dot{T}(x) & \text{if } \dot{S}(x) - \dot{T}(x)>0 \\ \text{undefined} & \text{if } \dot{S}(x) - \dot{T}(x) \leq 0 \end{cases}\end{split}$

for any $$x$$, where here we take $$\dot{S}(x)=0$$ for $$x \not\in \dot{S}$$, and $$\dot{T}(x)=0$$ for $$x \not\in \dot{T}$$.

multiset intersection

We define the multiset intersection of two multisets $$\dot{S}$$ and $$\dot{T}$$ to be:

$\big(\dot{S} \cap \dot{T}\big)(x) = min \big(\dot{S}(x), \dot{T}(x)\big)$

for any $$x$$, where here we take $$\dot{S}(x)=0$$ for $$x \not\in \dot{S}$$, and $$\dot{T}(x)=0$$ for $$x \not\in \dot{T}$$.

multiset union

We define the multiset union of two multisets $$\dot{S}$$ and $$\dot{T}$$ to be:

$\big(\dot{S} \cup \dot{T}\big)(x) = max \big(\dot{S}(x), \dot{T}(x)\big)$

for any $$x$$, where here we take $$\dot{S}(x)=0$$ for $$x \not\in \dot{S}$$, and $$\dot{T}(x)=0$$ for $$x \not\in \dot{T}$$.

partition
A partition of a set is the splitting of a set into a collection of smaller subsets. Mathematically, given a set $$S$$, we create a set of subsets of $$S$$ such that the union of those sets is $$S$$, and whose pairwise intersection is the empty set (another term for this is that any two sets are disjoint).
power set
The power set of any set $$S$$, written $$P(S)$$, is the set of all subsets of $$S$$, including the empty set and $$S$$ itself. We also use the expressions ‘second power set’, ‘third power set’ and so on to mean successive application of the power set operation for the indicated number of times: ‘power set of the power set of the ... of $$S$$. (Adapted from Wikipedia: Power set.)
projection

Given a clan, call it $$C$$, and a collection of elements, call it $$lefts$$, the projection of $$C$$ onto $$lefts$$ is a new clan where all of the left components of all of the couplets of the relations of $$C$$ are in the set $$lefts$$.

To obtain the projection of $$C$$ onto $$lefts$$ mathematically we can do this as follows:

$\begin{split}D_{lefts} := \{l{\mapsto}l\ : l\in lefts \} \\ project(C,lefts) = \{ R \circ D_{lefts}\ : R\in C\}\end{split}$
reflexive

A relation $$R$$ is said to be reflexive if:

$\forall x \in left(R) \cup right(R): x{\mapsto}x \in R$

A couplet can also be reflexive if it is of the form $$x{\mapsto}x$$.

See also left set, right set.

regular
A short name for left-regular in cases where no ambiguities are expected.
relation
A relation is a set of couplets. See also binary relation.
right
A short name for right component where no ambiguity is expected.
right component
Given a couplet represented by $$l{\mapsto}r$$ the component $$r$$ is the right component.
right multiset

The right multiset of a multiclan $$\mathbb{C}$$ is the multiset addition of all right sets of its member relations:

$right(\mathbb{C}) = \underset{R\ \in\ \mathbb{C}}{+} right(R)$
right set

The right set of a relation $$R$$ is the set of all right components of its members:

$right(R) = \{ r\ : l{\mapsto}r \in R \}$

The right set of a clan $$\mathbb{C}$$ is the union of all right sets of its member relations:

$right(\mathbb{C}) = \underset{R\ \in\ \mathbb{C}}{\bigcup} right(R)$
right-functional

A relation $$R$$ is said to be right-functional if:

$y{\mapsto}x \in R\ \&\ z{\mapsto}x \in R \implies y = z$
right-functional union

The right-functional union of two right-functional relations $$R$$ and $$Q$$ is the union of the two relations if the result is right-functional, otherwise the result is not defined:

$\begin{split}R \underset{rf}{\cup} Q = \begin{cases} R \cup Q & \text{if }R \cup Q \text{ is right-functional} \\ \text{undefined} & \text{if it is not right-functional} \end{cases}\end{split}$
right-regular

A clan $$\mathbb{C}$$ is said to be right-regular if the right sets of all its relations are the same:

$\forall R, Q \in \mathbb{C}: right(R) = right(Q)$
set
A set is a collection of distinct objects. Each object of a set is called an element of the set. In particular, if $$X$$ is a set, then we denote the fact that $$x$$ is an element of $$X$$ by writing $$x\in X$$. We will apply the axioms of Zermelo-Fraenkel set theory with choice (ZFC) to the sets. See also Set Notation for more information about set notation
set A
The set $$A$$ is the set of all atoms. It is a subset of the set M ($$M$$).
set M
The set $$M$$ is the set of all elements that can be represented in a given system, including atoms, couplets and sets. (A consequence of this is that the power set of $$M$$ $$P(M)$$ cannot be represented in a given system, and therefore is not an element of $$M$$.)
set N
The set $$N$$ is the set of all positive integers.
submultiset

The submultiset relation is a binary relation of multisets. A multiset $$\dot{S}$$ is a submultiset of a multiset $$\dot{T}$$ if the following holds:

$\dot{S} \subset \dot{T} \iff \forall x \in \dot{S}, \: \dot{S}(x) \leq \dot{T}(x).$
subset

The subset relation is a binary relation of sets. A set $$S$$ is a subset of a set $$T$$ if every element of $$S$$ is also an element of $$T$$:

$S \subset T \implies \forall x\ [\ x \in S\ \implies\ x \in T\ ]$
substriction

Substriction is a partial binary operation on sets and multisets. The substriction of two sets or multisets $$S$$ and $$T$$ is defined as:

$S \vartriangleleft T = S\ \ \text{if}\ \ S \subset T$

(When extended to an algebra of sets of sets (for example, the algebra of clans), we obtain the cross-substriction, which is also sometimes called ‘substriction’.)

supermultiset

The supermultiset relation is a binary relation of multisets. A multiset $$\dot{S}$$ is a supermultiset of a multiset $$\dot{T}$$ if the following holds:

$\dot{S} \supset \dot{T} \iff \forall x \in \dot{T}, \: \dot{S}(x) \geq \dot{T}(x).$
superset

The superset relation is a binary relation of sets. A set $$S$$ is a superset of a set $$T$$ if every element of $$T$$ is also an element of $$S$$:

$S \supset T \implies \forall x\ [\ x \in T\ \implies\ x \in S\ ]$
superstriction

Superstriction is a partial binary operation on sets and multisets. The superstriction of two sets or multisets $$S$$ and $$T$$ is defined as:

$S \vartriangleright T := S\ \ \text{if}\ \ S \supset T$

(When extended to an algebra of sets of sets (for example, the algebra of clans), we obtain the cross-superstriction, which is also sometimes called ‘superstriction’.)

symmetric

A relation $$R$$ is said to be symmetric if:

$\forall x, y \in left(R) \cup right(R): x{\mapsto}y \in R \implies y{\mapsto}x \in R$

See also left set, right set.

symmetric difference

The symmetric difference of two sets $$S$$ and $$T$$ is the set of elements that are only in one of the sets. The definition is:

$S \vartriangle T = (S \cup T) \setminus (S \cap T)$
transitive

A relation $$R$$ is said to be transitive if:

$\forall x, y, z \in left(R) \cup right(R): (x{\mapsto}y \in R \ \& \ y{\mapsto}z \in R) \implies x{\mapsto}z \in R$

See also left set, right set.

transposition

Transposition is a unary operation on couplets. The transposition of a couplet $$a{\mapsto}b$$ is defined as:

$\overleftrightarrow{a{\mapsto}b} = b{\mapsto}a$

The operation may be extended to extended algebras (like the algebra of relations) using the unary extension and – if there is no danger of ambiguities – is then also called simply ‘transposition’.

In multisets and multiclans the operation is the same and the multiplicities do not change.

unary extension

The unary extension is the operation that extends a unary operation from its algebra to an extended algebra (which is an algebra of sets):

$unaryExtn(op, S) := \{op(s)\ :\ s \in S \text{ where } op(s) \text{ is defined}\}$
unary multi-extension

The unary multi-extension is the operation that extends a unary operation from its algebra to an extended algebra (which is an algebra of multisets). For this extension, the multiplicities do not change:

$unaryExtn(op, \dot{S}) := \{op(s){:}\dot{S}(s)\ :\ s \in S \text{ where } op(s) \text{ is defined}\}$
unary operation
An operation with only one argument, typically with a result that belongs to the same ground set as the argument (when the operation is a member of an algebra).
union

An operation on sets that creates a set by collecting the elements of two or more individual sets into a new set. In mathematical terms, if $$\mathbb{S}$$ is a collection of sets, then the union of all of the sets in $$\mathbb{S}$$ is denoted:

$\bigcup \mathbb{S} = \bigcup_{T\in\mathbb{S}}T,$

and is the set $$\{x\ : \exists T \in \mathbb{S},\ x \in T\}$$. If $$\mathbb{S}$$ consists of only two sets, the union is called a binary union. If $$\mathbb{S}$$ consists of a finite collection of sets, the union is called a finite union. See also Wikipedia: Union.

Zermelo-Fraenkel set theory with choice (ZFC)
A system of axioms on sets that is the standard form of set theory and the foundation of much of modern mathematics. See also Wikipedia: Zermelo-Fraenkel set theory with choice (ZFC).