Connex relation

In mathematics, a homogeneous relation is called a connex relation, 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

x, yX, xRyyRx.

Every pair of elements is either in R or in the converse relation RT.

A homogeneous relation is called a semiconnex relation, 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

x, yX, xyxRyyRx.

Several authors define only the semiconnex property, and call it connex rather than semiconnex.

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. Some authors call the connex property of a relation completeness.[citation needed]

Characterizations

Let R be a homogeneous relation.

• R is connex ↔ URRTRRTR is asymmetric,
where U is the universal relation and RT is the converse relation of R.
• R is semiconnex ↔ I RRTRRTIR is antisymmetric,
where I  is the complementary relation of the identity relation I and RT is the converse relation of R.

Properties

• The edge relation 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.
• A semiconnex relation on a set X cannot be antitransitive, provided X has at least 4 elements. 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. Similarly, all, or all but one, elements of X are in the domain of R.