$\mathbf U$-semigroups of relations

By a $\mathbf u$-semigroup we mean an algebra $(A,\cdot,\mathbf u)$ where $(A,\cdot)$ is a semigroup and $\mathbf u\in A$. A set of binary relations closed under the relation product and containing the universal relation $\mathbf U$ forms a $\mathbf U$-semigroup of relations which can be considered as partially ordered by inclusion. In the paper, an axiomatic description of the classes of $\mathbf U$-semigroups and partially ordered $\mathbf U$-semigroups of relations, and a basis of identities and quasiidentities of the variety and quasivariety generated by these classes are obtained.