A "classification" of congruence primal arithmetical algebras

Abstract: We classify essential algebras whose irredundant non-refinable covers consist of primal algebras. The proof is obtained by constructing one to one correspondence between such algebras and partial orders on finite sets. Further, we prove that for a finite algebra, it has an irredundant non-refinable cover consists of primal algebras if and only if it is the both congruence primal and arithmetical. Finally, we obtain combinatorial description of congruence primal arithmetical algebras.