arXiv: Rings and Algebras | 2019
A Variety Containing EMV-Algebras and Pierce Sheaves
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.