The class of UMP algebras arises in several classification problems in the context of derived categories of finite-dimensional algebras. In this paper we define the class of UMP algebras and develop algebraic combinatorics tools in order to present a characterization of this class of algebras which are \textit{locally monomial} (see Definition 15) and special multiserial algebras. Among other things, we use this characterization to prove that, in the case of monomial quadratic algebras, the class of UMP algebras coincides with the class of special multiserial algebras.