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.
  is connex  is asymmetric,
 where  is the universal relation and  is the converse relation of .
  is semiconnex  is antisymmetric,
 where  is the complementary relation of the identity relation  and  is the converse relation of .

Properties 
 The edge relation  of a tournament graph  is always a semiconnex relation on the set of 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  cannot be antitransitive, provided  has at least 4 elements. On a 3-element set }, e.g. the relation } has both properties.
 If  is a semiconnex relation on , then all, or all but one, elements of  are in the range of . Similarly, all, or all but one, elements of  are in the domain of .