UMP Monomial Algebras: Combinatorial and Homological Consequences

Abstract In this paper, we apply the techniques developed in [4] to present several consequences of studying UMP algebras and the ramifications graph of a monomial bound quiver algebra. Specifically, we prove that every weakly connected component of the ramifications graph of a UMP monomial algebra is unilaterally connected. Furthermore, using the main result characterizing UMP algebras in the monomial context, we prove that the class of UMP algebras is equivalent to the class of special multiserial algebras when the algebra is a quadratic monomial algebra. Based on this equivalence and the classification of Chen–Shen–Zhou on Gorenstein projective modules in [5], we extend their results to the class of monomial special multiserial UMP algebras, where we use the analysis of homological properties on quadratic monomial algebras given by these authors.