Transformation semigroups preserving a binary relation

We investigate the semigroups of full and partial transformations of a set X which preserve a binary relation σ defined on X. We consider in detail the case where σ is an order or a quasi-order relation. There are conditions of regularity of such semigroups. We introduce two definitions of preservation of σ for the semigroup of binary relations. It is proved that subsets of B(X) preserving σ are semigroups in each case. We give the condition of regularity of B _σ (X) in the case where σ(X) is a quasi-order.