left and right euclidean;
\forall (x, y: A),\; x \nsim y \;\wedge\; y \nsim x \;\Rightarrow\; x = y .
Using excluded middle, it is equivalent to say that every two elements are related in some order or equal:
\forall (x, y: A),\; x \sim y \;\vee\; y \sim x \;\vee\; x = y .
On the other hand, there is a stronger notion that may be used in constructive mathematics, if is already equipped with a tight apartness . In that case, we say that is strongly connected if any two distinct elements are related in one order or the other:
\forall (x, y: A),\; x \# y \;\Rightarrow\; x \sim y \;\vee\; y \sim x .
Since is connected itself, every strongly connected relation is connected; the converse holds with excluded middle (through which every set has a unique tight apartness).