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 on a set X is connex when for all ${\displaystyle x,y\in X,}$

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

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 ${\displaystyle x\neq y}$ or, equivalently, when for all ${\displaystyle x,y\in X,}$

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

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;[] 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.[5]

## Characterizations

Let ${\displaystyle R}$ be a homogeneous relation.

• ${\displaystyle R}$ is connex U ? R ? RTR ? RTR is asymmetric,
where ${\displaystyle U}$ is the universal relation and RT is the converse relation of ${\displaystyle R.}$[1]
• ${\displaystyle R}$ is semiconnex I  ? R ? RTR ? RT ? I R is antisymmetric,
where I  is the complementary relation of the identity relation ${\displaystyle I}$ and RT is the converse relation of ${\displaystyle R.}$[1]

## Properties

• The edge relation[6]${\displaystyle E}$ of a tournament graph ${\displaystyle G}$ 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.[7]
• A semiconnex relation on a set ${\displaystyle X}$ cannot be antitransitive, provided ${\displaystyle X}$ has at least 4 elements.[8] On a 3-element set {a, b, c}, e.g. the relation {(a, b), (b, c), (c, a)} has both properties.
• If ${\displaystyle R}$ is a semiconnex relation on ${\displaystyle X,}$ then all, or all but one, elements of ${\displaystyle X}$ are in the range of ${\displaystyle R.}$[9] Similarly, all, or all but one, elements of ${\displaystyle X}$ are in the domain of ${\displaystyle R.}$

## References

1. ^ a b c d Schmidt & Ströhlein 1993, p. 12.
2. ^ Bram van Heuveln. "Sets, Relations, Functions" (PDF). Troy, NY. Retrieved . CS1 maint: discouraged parameter (link) Page 4.
3. ^ Carl Pollard. "Relations and Functions" (PDF). Ohio State University. Retrieved . CS1 maint: discouraged parameter (link) 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 2019. Page 59, footnote 1.
5. ^ Jehle, Geoffrey; Reny, Philip (2011). Advanced Microeconomic Theory. Edinburgh Gate, Harlow, Essex, CM20 2JE, England: Prentice Hall, Financial Times. p. 5. ISBN 978-0-273-73191-7.CS1 maint: location (link)
6. ^ defined formally by ${\displaystyle vEw}$ if a graph edge leads from vertex ${\displaystyle v}$ to vertex ${\displaystyle w}$
7. ^ For the only if direction, both properties follow trivially. — For the if direction: when ${\displaystyle x\neq y,}$ then xRy ? yRx follows from the semiconnex property; when ${\displaystyle x=y,}$ even ${\displaystyle xRy}$ follows from reflexivity.
8. ^ Jochen Burghardt (Jun 2018). Simple Laws about Nonprominent Properties of Binary Relations (Technical Report). arXiv:1806.05036. Bibcode:2018arXiv180605036B. Lemma 8.2, p.8.
9. ^ If x, y?X\ran(R), then ${\displaystyle xRy}$ and ${\displaystyle yRx}$ are impossible, so ${\displaystyle x=y}$ follows from the semiconnex property.