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

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 ? RTR ? RTR is asymmetric,
where is the universal relation and RT is the converse relation of [1]
  • is semiconnex I  ? R ? RTR ? RT ? I R is antisymmetric,
where I  is the complementary relation of the identity relation and RT is the converse relation of [1]


  • The edge relation[6] of a tournament graph 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 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


  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)[permanent dead 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 if a graph edge leads from vertex to vertex
  7. ^ 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.
  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 and are impossible, so follows from the semiconnex property.

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