Definition:Endorelation
Definition
Let $\RR$ be a relation on $S \times S$.
Then $\RR$ is referred to as an endorelation on $S$.
An endorelation can be defined as the ordered triple:
- $\RR = \tuple {S, S, R}$
where $R \subseteq S \times S$.
General Definition
An $n$-ary relation $\RR$ on a cartesian space $S^n$ is an $n$-ary endorelation on $S$:
- $\RR = \struct {S, S, \ldots, S, R}$
where $R \subseteq S^n$.
Class Theoretical Definition
In the context of class theory, the definition follows the same lines:
Let $A$ be a class.
An endorelation $\RR$ on $A$ is a subclass of the Cartesian product $A \times A$.
That is, such that the domain and image of $\RR$ are both subclasses of $A$.
Also known as
The term endorelation is rarely seen. Once it is established that the domain and codomain of a given relation are the same, further comment is rarely needed.
Hence an endorelation on $S$ is also called:
- a relation in $S$
or:
- a relation on $S$
The latter term is discouraged, though, because it can also mean a left-total relation, and confusion can arise.
Some sources use the term binary relation exclusively to refer to a binary endorelation.
Some sources, for example 1974: P.M. Cohn: Algebra: Volume $\text { 1 }$, use the term relation for what is defined here as an endorelation, and a relation defined as a general ordered triple of sets: $\tuple {S, T, R \subseteq S \times T}$ is called a correspondence.
As this can cause confusion with the usage of correspondence to mean a relation which is both left-total and right-total, it is recommended that this is not used.
Examples
Arbitrary Relation
Let $V_0 = \set {a, b, c}$.
A possible endorelation on $V_0$ is:
- $R = \set {\tuple {a, a}, \tuple {b, b}, \tuple {b, c}, \tuple {c, b} }$
Properties of Arbitrary Relation: 1
Let $V = \set {u, v, w, x}$.
Let $E$ be the relation on $V$ defined as:
- $E = \set {\tuple {u, v}, \tuple {v, u}, \tuple {v, w}, \tuple {w, v} }$
Then $E$ is:
Properties of Arbitrary Relation: 2
Let $V = \set {a, b, c, d}$.
Let $R$ be the relation on $V$ defined as:
- $E = \set {\tuple {a, a}, \tuple {a, b}, \tuple {a, c}, \tuple {a, d}, \tuple {b, b}, \tuple {b, c}, \tuple {b, d}, \tuple {c, c}, \tuple {c, d}, \tuple {d, d} }$
Then $E$ is:
Properties of Arbitrary Relation: 3
Let $V = \set {a, b, c, d}$.
Let $R$ be the relation on $V$ defined as:
- $r = \set {\tuple {a, a}, \tuple {a, b}, \tuple {b, b}, \tuple {b, c}, \tuple {c, b}, \tuple {c, c} }$
Then $E$ is:
Also see
- Results about endorelations can be found here.
Sources
- 1951: Nathan Jacobson: Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts ... (previous) ... (next): Introduction $\S 3$: Equivalence relations
- 1959: E.M. Patterson: Topology (2nd ed.) ... (previous) ... (next): Chapter $\text {II}$: Topological Spaces: $\S 10$. Equivalence Relations
- 1964: W.E. Deskins: Abstract Algebra ... (previous) ... (next): $\S 1.2$: Definition $1.5$
- 1964: Steven A. Gaal: Point Set Topology ... (previous) ... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 2.1$. Relations on a set
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 10$: Equivalence Relations
- 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 0.3$
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Relations
- 1968: A.N. Kolmogorov and S.V. Fomin: Introductory Real Analysis ... (previous) ... (next): $\S 1.4$: Decomposition of a set into classes. Equivalence relations
- 1968: Ian D. Macdonald: The Theory of Groups ... (previous) ... (next): Appendix: Elementary set and number theory
- 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.19$: Some Important Properties of Relations: Definition $19.1$
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 6$. Indexed families; partitions; equivalence relations
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 7$: Relations
- 1977: Gary Chartrand: Introductory Graph Theory ... (previous) ... (next): Appendix $\text{A}.2$: Cartesian Products and Relations
- 1979: John E. Hopcroft and Jeffrey D. Ullman: Introduction to Automata Theory, Languages, and Computation ... (previous) ... (next): Chapter $1$: Preliminaries: $1.5$ Relations
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.4$: Equivalence relations
- 1993: Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.5$: Relations
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $2$: Maps and relations on sets: Definition $2.20$
- 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $1$: Pairs, Relations, and Functions
- 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 3$: Relations