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

\forall x,\ y\in X,\,\,x\ R\ y\ \lor \ y\ R\ x

Binary relations

	Symmetric	Antisymmetric	Connex	Well-founded	Has joins	Has meets
Equivalence relation	✓	✗	✗	✗	✗	✗
Preorder (Quasiorder)	✗	✗	✗	✗	✗	✗
Partial order	✗	✓	✗	✗	✗	✗
Total preorder	✗	✗	✓	✗	✗	✗
Total order	✗	✓	✓	✗	✗	✗
Prewellordering	✗	✗	✓	✓	✗	✗
Well-quasi-ordering	✗	✗	✗	✓	✗	✗
Well-ordering	✗	✓	✓	✓	✗	✗
Lattice	✗	✓	✗	✗	✓	✓
Join-semilattice	✗	✓	✗	✗	✓	✗
Meet-semilattice	✗	✓	✗	✗	✗	✓

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.

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

\forall x,\ y\in X,\,\,x\neq y\rightarrow x\ R\ y\ \lor \ y\ R\ x

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 $\leftrightarrow U \subseteq R \cup R T \leftrightarrow R \subseteq R T \leftrightarrow R$ is asymmetric,

where

U

is the universal relation and

R T

is the converse relation of

R

.[1]

$R$ is semiconnex $\leftrightarrow I \subseteq R \cup R T \leftrightarrow R \subseteq R T \cup I \leftrightarrow R$ is antisymmetric,

where

I

is the complementary relation of the identity relation

I

and

R T

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 $G$ '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 ${a, b, c$ }, e.g. the relation ${(a, b), (b, c), (c, a)$ } 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

Schmidt & Ströhlein 1993, p. 12.
Bram van Heuveln. "Sets, Relations, Functions" (PDF). Troy, NY. Retrieved 2018-05-28. Page 4.
Carl Pollard. "Relations and Functions" (PDF). Ohio State University. Retrieved 2018-05-28. 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 22 Jan 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.

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[FOOTNOTESchmidtStröhlein199312-1] Schmidt & Ströhlein 1993, p. 12.

[2] Bram van Heuveln. "Sets, Relations, Functions" (PDF). Troy, NY. Retrieved 2018-05-28. Page 4.

[3] Carl Pollard. "Relations and Functions" (PDF). Ohio State University. Retrieved 2018-05-28. 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 22 Jan 2019. Page 59, footnote 1.

[5] rmally 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.