Weak Pseudo EMV-algebras. I: Basic Properties.

We define weak pseudo EMV-algebras which are a non-commutative generalization of weak EMV-algebras, pseudo MV-algebras, and generalized Boolean algebras, respectively. In contrast to pseudo EMV-algebras, the class of wPEMV-algebras is a variety. We present basic properties and examples of wPEMV-algebras. The main aim is to show when a wPEMV-algebra can be embedded into a wPEMV-algebra N with top element, called a representing wPEMV-algebra, as a maximal and normal ideal of N. The paper is divided into two parts. Part I studies wPEMV-algebras from the point of semiclans, generalized pseudo effect algebras and integral GMV-algebras. We describe congruences via normal ideals, and we show when a wPEMV-algebra possesses a representing one. Part II. It studies representable wPEMV-algebras and it shows an equational base for them. Left and right unitizing automorphisms enable us to construct representing wPEMV-algebras. We present the Basic Representation Theorem. Finally, we study subvarieties of cancellative wPEMV-algebras, perfect wPEMV-algebras, weakly commutative wPEMV-algebras, and normal-valued wPEMV-algebras, respectively.

Authors

• Anatolij Dvurecenskij
• Omid Zahiri

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