Connex Relation
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Connex Relation

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

${\displaystyle \forall x,\ y\in X,\,\,x\ R\ y\ \lor \ y\ R\ x}$

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

${\displaystyle \forall x,\ y\in X,\,\,x\neq y\rightarrow x\ R\ y\ \lor \ y\ R\ x}$

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;[] 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.[]

## Characterizations

Let R be a homogeneous relation.

• R is connex U ? R ? RTR ? RTR is asymmetric,
where U is the universal relation and RT is the converse relation of R.[1]
• R is semiconnex I  ? R ? RTR ? 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 Gs 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 , e.g. the relation 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. ^ a b c d Schmidt & Ströhlein 1993, p. 12.
2. ^ Bram van Heuveln. "Sets, Relations, Functions" (PDF). Troy, NY. Retrieved . Page 4.
3. ^ Carl Pollard. "Relations and Functions" (PDF). Ohio State University. Retrieved . Page 7.
4. ^ Felix Brandt; Markus Brill; Paul Harrenstein (2016). "Tournament Solutions" (PDF). In Felix Brandt; Vincent Conitzer; Ulle Endriss; Jérôme Lang; Ariel D. Procaccia (eds.). Handbook of Computational Social Choice. Cambridge University Press. ISBN 978-1-107-06043-2. Archived (PDF) from the original on 8 Dec 2017. Retrieved 2019. Page 59, footnote 1.
5. ^ defined formally by vEw if a graph edge leads from vertice v to vertice w
6. ^ 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.
7. ^ Jochen Burghardt (Jun 2018). Simple Laws about Nonprominent Properties of Binary Relations (Technical Report). arXiv:1806.05036. Bibcode:2018arXiv180605036B. Lemma 8.2, p.8.
8. ^ If x, y?X\ran(R), then xRy and yRx are impossible, so x=y follows from the semiconnex property.