Connex Relation
Get Connex Relation essential facts below. View Videos or join the Connex Relation discussion. Add Connex Relation to your PopFlock.com topic list for future reference or share this resource on social media.
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 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

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.

  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
 



 



 
Music Scenes