Connex Relation

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## Characterizations

## Properties

## References

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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

Every pair of elements is either in *R* or in the converse relation *R*^{T}.

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;^{[]} 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*.^{[]}

Let *R* be a homogeneous relation.

*R*is connex*U*?*R*?*R*^{T}*R*?*R*^{T}*R*is asymmetric,

- where
*U*is the universal relation and*R*^{T}is the converse relation of*R*.^{[1]}

*R*is semiconnex*I*?*R*?*R*^{T}*R*?*R*^{T}?*I**R*is antisymmetric,

- where
*I*is the complementary relation of the identity relation*I*and*R*^{T}is the converse relation of*R*.^{[1]}

- 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 , 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*.

- ^
^{a}^{b}^{c}^{d}Schmidt & Ströhlein 1993, p. 12. **^**Bram van Heuveln. "Sets, Relations, Functions" (PDF). Troy, NY. Retrieved . Page 4.**^**Carl Pollard. "Relations and Functions" (PDF). Ohio State University. Retrieved . Page 7.**^**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.**^**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). arXiv:1806.05036. Bibcode:2018arXiv180605036B. 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.

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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