Connex relation

In mathematics, a binary relation R on a set X is called a connex relation if it relates all pairs of elements from X in some way. More formally, R is connex when

∀x∀y (x ∈ X ∧ y ∈ X) ⇒(xRy ∨ yRx ).

A relation is called a semi-connex relation if it relates all distinct elements in some way. 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, a.k.a. surjectivity.

Formal definition

A connex relation is a homogeneous binary relation R on some set X for which either xRy or yRx holds for any pair (x,y). An equivalent statement in terms of the universal relation X×X is

where RT is the converse relation to R.

A relation R is semi-connex when x≠y and (x,y) ∉ R implies (y,x) ∈ R. If I is the identity relation, an alternative characterization of a semi-connex relation is

where the overbar indicates the complementary relation.

Several authors define only the latter property, and call it connex rather than semi-connex.[1][2][3]

Properties

  • The edge relation[4] E of a tournament graph G is always a semi-connex 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 semi-connex and reflexive.[5]
  • A semi-connex relation on a set X cannot be antitransitive, provided X has at least 4 elements.[6] 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 semi-connex relation on X, then all, or all but one, elements of X are in the range of R.[7] Similarly, all, or all but one, elements of X are in the domain of R.

References

  • Gunter Schmidt (2011) Relational Mathematics, page 62, Cambridge University Press ISBN 978-0-521-76268-7
  • Gunther Schmidt & Thomas Ströhlein (1993) Relations and Graphs Discrete Mathematics for Computer Scientists, EATCS Monographs on Theoretical Computer Science, page 32, Springer Verlag, ISBN 3-540-56254-0
  1. ↑ Bram van Heuveln. "Sets, Relations, Functions" (PDF). Troy, NY. Retrieved 2018-05-28. Page 4.
  2. ↑ Carl Pollard. "Relations and Functions" (PDF). Ohio State University. Retrieved 2018-05-28. Page 7.
  3. ↑ Felix Brandt; Markus Brill; Paul Harrenstein (2016). "Tournament Solutions" (PDF). In F. Brandt, V. Conitzer, U. Endriss, J. Lang, A. Procaccia. Handbook of Computational Social Choice. Cambridge University Press. Page 3, footnote 1.
  4. ↑ defined formally by vEw if a graph edge leads from vertice v to vertice w
  5. ↑ For the only if direction, both properties follow trivially. — For the if direction: when x≠y, then xRy ∨ yRx follows from the semi-connex property; when x=y, even xRy follows from reflexivity.
  6. ↑ Jochen Burghardt (Jun 2018). Simple Laws about Nonprominent Properties of Binary Relations (Technical Report). arXiv:1806.05036. Bibcode:2018arXiv180605036B. Lemma 8.2, p.8.
  7. ↑ If x,y∈X\ran(R), then xRy and yRx are impossible, so x=y follows from the semi-connex property.
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