Pseudo EMV-algebras. I - Basic Properties.

We introduce pseudo EMV-algebras which are a non-commutative generalization of both MV-algebras and generalized Boolean algebras. The existence of a top element is not assumed. The paper has two parts. In the present one we study basic properties of pseudo EMV-algebras as ideals and homomorphisms. The class of all pseudo EMV-algebras is not a variety and rather a more general class, called a q-variety, but similar to a variety. We study representable pseudo EMV-algebras, normal-valued ones, and pseudo EMV-algebras whose every maximal ideal is normal. The second part shows that every pseudo EMV-algebra without top element can be embedded into a pseudo EMV-algebra with top element as a maximal and normal ideal of the latter one. We present a categorical equivalence of the category of pseudo EMV-algebras without top element with a special category of pseudo MV-algebras or with a special category of `-groups. Finally, we study states as finitely additive mappings as well as state-morphisms on pseudo EMValgebras and we present their representation as an integral over a regular Borel probability measure.

Authors

• Anatolij Dvurecenskij
• Omid Zahiri

Keywords

No keywords are indexed for this paper.