Connex relation
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Look up connexity in Wiktionary, the free dictionary. |
In mathematics, a homogeneous relation is called a connex relation,[1] or a relation having the property of connexity, if it relates all pairs of elements in some way. More formally, the homogeneous relation R over a set X is connex when
Every pair of elements is either in R or in the converse relation RT.
A homogeneous relation is called a semiconnex relation,[1] or a relation having the property of semiconnexity, if it relates all pairs of distinct elements in some way. More formally, the homogeneous relation R over a set X is semiconnex when
Several authors define only the semiconnex property, and call it connex rather than semiconnex.[2][3][4]
The connex properties originated from order theory: if a partial order is also a connex relation, then it is a total order. Therefore, in older sources, a connex relation was said to have the totality property;[citation needed] however, this terminology is disadvantageous as it may lead to confusion with, e.g., the unrelated notion of right-totality, also known as surjectivity. Some authors call the connex property of a relation completeness.[citation needed]
Characterizations
Let R be a homogeneous relation.
- R is connex ↔ U ⊆ R ∪ RT ↔ R ⊆ RT ↔ R is asymmetric,
- where U is the universal relation and RT is the converse relation of R.[1]
- R is semiconnex ↔ I ⊆ R ∪ RT ↔ R ⊆ RT ∪ I ↔ R is antisymmetric,
- where I is the complementary relation of the identity relation I and RT is the converse relation of R.[1]
Properties
- The edge relation[5] E of a tournament graph G is always a semiconnex relation on the set of G's vertices.
- A connex relation cannot be symmetric, except for the universal relation.
- A relation is connex if, and only if, it is semiconnex and reflexive.[6]
- A semiconnex relation on a set X cannot be antitransitive, provided X has at least 4 elements.[7] On a 3-element set {a, b, c}, e.g. the relation {(a, b), (b, c), (c, a)} has both properties.
- If R is a semiconnex relation on X, then all, or all but one, elements of X are in the range of R.[8] Similarly, all, or all but one, elements of X are in the domain of R.
References
- ↑ 1.0 1.1 1.2 1.3 Schmidt & Ströhlein 1993, p. 12.
- ↑ Bram van Heuveln. "Sets, Relations, Functions". Troy, NY. http://www.cogsci.rpi.edu/~heuveb/teaching/Logic/CompLogic/Web/Handouts/SetsRelationsFunctions.pdf. Retrieved 2018-05-28. Page 4.
- ↑ Carl Pollard. "Relations and Functions". Ohio State University. http://www.ling.ohio-state.edu/~pollard/680/chapters/relations.pdf. Retrieved 2018-05-28. Page 7.
- ↑ Felix Brandt; Markus Brill; Paul Harrenstein (2016). "Tournament Solutions". in Felix Brandt. Handbook of Computational Social Choice. Cambridge University Press. ISBN 978-1-107-06043-2. Page 59, footnote 1.
- ↑ defined formally by vEw if a graph edge leads from vertice v to vertice w
- ↑ For the only if direction, both properties follow trivially. — For the if direction: when x≠y, then xRy ∨ yRx follows from the semiconnex property; when x=y, even xRy follows from reflexivity.
- ↑ Jochen Burghardt (Jun 2018). Simple Laws about Nonprominent Properties of Binary Relations (Technical Report). Lemma 8.2, p.8.
- ↑ If x, y∈X\ran(R), then xRy and yRx are impossible, so x=y follows from the semiconnex property.
- Schmidt, Gunther; Ströhlein, Thomas (1993). Relations and Graphs: Discrete Mathematics for Computer Scientists. Berlin: Springer-Verlag. ISBN 978-3-642-77970-1. https://books.google.fr/books?id=ZgarCAAAQBAJ.
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