On regularity of semigroups of maps that preserve a binary relation

We consider semigroups of full transformations, partial maps, and multi-value maps from a set $X$ to $X$, where every map preserves a binary relation $\sigma$ defined on $X$. We suggest several definitions for the preservation of $\sigma$ for partial maps and for multi-value maps. We review results about regularity of the semigroups mentioned above in the cases where $\sigma$ is a partial order, a quasi-order, an equivalence, or one of some special kinds of binary relations. Also we consider the question about regularity of the semigroup of full transformations that preserve a partial order and an equivalence simultaneously.