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 on a set X is connex when for all x and y in X,

${\displaystyle x\ R\ y\quad {\text{or}}\quad y\ R\ x.}$

A homogeneous relation is called a semiconnex relation,[1] or a relation having the property of semiconnexity, if the same property holds for all pairs of distinct elements xy, or, equivalently, when for all x and y in X,

${\displaystyle x\ R\ y\quad {\text{or}}\quad y\ R\ x\quad {\text{or}}\quad x=y.}$

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 URRTRRTR is asymmetric,
where U is the universal relation and RT is the converse relation of R.[1]
• R is semiconnex I RRTRRTIR 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. ^ a b c d Schmidt & Ströhlein 1993, p. 12.
2. ^ Bram van Heuveln. "Sets, Relations, Functions" (PDF). Troy, NY. Retrieved 2018-05-28. Page 4.
3. ^ Carl Pollard. "Relations and Functions" (PDF). Ohio State University. Retrieved 2018-05-28. 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 December 2017. Retrieved 22 Jan 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 xy, then xRyyRx 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, yX\ran(R), then xRy and yRx are impossible, so x=y follows from the semiconnex property.