A Variety Containing EMV-Algebras and Pierce Sheaves

TL;DR

This paper introduces the least variety containing EMV-algebras by defining wEMV-algebras, extending EMV-algebras with a new operation, and explores their structure through Pierce sheaves.

Contribution

It defines wEMV-algebras as the minimal variety containing EMV-algebras and extends EMV-algebras with a new operation to embed them into wEMV-algebras.

Findings

wEMV is the least variety containing EMV-algebras

Every wEMV-algebra can be embedded into an associated wEMV-algebra

Pierce sheaves of proper EMV-algebras are studied

Abstract

According to \cite{Dvz}, we know that the class of all EMV-algebras, $\mathsf{EMV}$, is not a variety, since it is not closed under the subalgebra operator. The main aim of this work is to find the least variety containing $\mathsf{EMV}$. For this reason, we introduced the variety $\mathsf{wEMV}$ of wEMV-algebras of type $(2,2,2,2,0)$ induced by some identities. We show that, adding a derived binary operation $\ominus$ to each EMV-algebra $(M;\vee,\wedge,\oplus,0)$, we extend its language, so that $(M;\vee,\wedge,\oplus,\ominus,0)$, called an associated wEMV-algebra, belongs to $\mathsf{wEMV}$. Then using the congruence relations induced by the prime ideals of a wEMV-algebra, we prove that each wEMV-algebra can be embedded into an associated wEMV-algebra. We show that $\mathsf{wEMV}$ is the least subvariety of the variety of wEMV-algebras containing $\mathsf{EMV}$. Finally, we study Pierce sheaves of proper EMV-algebras.