States, state operators and quasi-pseudo-MV algebras

Abstract

Quasi-pseudo-MV algebras (quasi-pMV algebras, for short) arising from quantum computational logics are the generalizations of both quasi-MV algebras and pseudo-MV algebras. In this paper, we introduce the notions of states, state-morphisms, state operators and state-morphism operators to quasi-pMV algebras. First, we present the related properties of states on quasi-pMV algebras and show that states and Bosbach states coincide on any quasi-pMV algebra. And then we investigate the relationship between state-morphisms and the normal and maximal ideals of quasi-pMV algebras. We prove state-morphisms and extremal states are equivalent. The existence of states on quasi-pMV algebras is also discussed. Finally, state operators and state-morphism operators are introduced to quasi-pMV algebras, and the corresponding structures are called state quasi-pMV algebras and state-morphism quasi-pMV algebras, respectively. We investigate the related properties of ideals under state operators and state-morphism operators. Meanwhile, we show that there is a bijective correspondence between normal $$\sigma $$ -ideals and ideal congruences on state quasi-pMV algebras.