Composition of matrix products and categorical equivalence

First, we prove two finite algebras are categorically equivalent if and only if the matrix products of their irredundant non-refinable covers are isomorphic. Second, we characterize families of irreducible algebras such that there exists an algebra whose neighbourhoods in an irredundant non-refinable cover are isomorphic to the respective irreducible algebra in the given family. Finally, we exhibit two facts by constructing examples. The first one is that there is a family of irreducible algebras such that there are many algebraic structures whose neighbourhoods in an irredundant non-refinable cover are isomorphic to the respective irreducible algebra in the given family. The second example is an algebra such that the matrix product of an irredundant non-refinable cover is bigger than the given algebra.