Connex Relation

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## Characterizations

## Properties

## References

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Connex Relation

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A "✓" indicates that the column property is required in the row definition. For example, the definition of an equivalence relation requires it to be symmetric. All definitions tacitly require transitivity and reflexivity. |

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

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 or, equivalently, when for all

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]}

Let be a homogeneous relation.

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

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

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

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

- The
*edge*relation^{[6]}of a tournament graph 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.
^{[7]} - A semiconnex relation on a set cannot be antitransitive, provided 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 is a semiconnex relation on then all, or all but one, elements of are in the range of
^{[9]}Similarly, all, or all but one, elements of are in the domain of

- ^
^{a}^{b}^{c}^{d}Schmidt & Ströhlein 1993, p. 12. **^**Bram van Heuveln. "Sets, Relations, Functions" (PDF). Troy, NY. Retrieved . CS1 maint: discouraged parameter (link)^{[permanent dead link]}Page 4.**^**Carl Pollard. "Relations and Functions" (PDF). Ohio State University. Retrieved . CS1 maint: discouraged parameter (link) 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 December 2017. Retrieved 2019. Page 59, footnote 1.**^**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)**^**defined formally by if a graph edge leads from vertex to vertex**^**For the*only if*direction, both properties follow trivially. — For the*if*direction: when then*xRy*?*yRx*follows from the semiconnex property; when even 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 and are impossible, so 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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