A Variety Containing EMV-Algebras and Pierce Sheaves

According to \cite{Dvz}, we know that the class of all EMV-algebras, $\mathsf{E}\mathsf{M}\mathsf{V}$, 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 $\mathsf{E}\mathsf{M}\mathsf{V}$. For this reason, we introduced the variety $\mathsf{w}\mathsf{E}\mathsf{M}\mathsf{V}$ 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 $\mathsf{w}\mathsf{E}\mathsf{M}\mathsf{V}$. 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 $\mathsf{w}\mathsf{E}\mathsf{M}\mathsf{V}$ is the least subvariety of the variety of wEMV-algebras containing $\mathsf{E}\mathsf{M}\mathsf{V}$. Finally, we study Pierce sheaves of proper EMV-algebras.