Transformation semigroups preserving a binary relation

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