A "classification" of congruence primal arithmetical algebras

TL;DR

This paper classifies a specific type of algebra called congruence primal arithmetical algebras, establishing a correspondence with finite partial orders and providing a combinatorial description of these structures.

Contribution

It introduces a classification framework for these algebras, linking their structure to partial orders and characterizing them through primal and arithmetical properties.

Findings

Established a one-to-one correspondence with partial orders.

Proved the equivalence involving primal and arithmetical properties.

Provided a combinatorial description of the classified 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.