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 over a set is connex when
Every pair of elements is either in or in the converse relation.
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 over a set is semiconnex when
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, also known as surjectivity. Some authors call the connex property of a relation completeness.

Characterizations

Let be a homogeneous relation.