Some Results on Quasi MV-Algebras and Perfect Quasi MV-Algebras

Abstract Quasi MV-algebras are a generalization of MV-algebras and they are motivated by the investigation of the structure of quantum logical gates. In the first part, we present relationships between ideals, weak ideals, congruences, and perfectness within MV-algebras and quasi MV-algebras, respectively. To achieve this goal, we provide a comprehensive characterization of congruence relations of a quasi MV-algebra $${\mathcal {A}}$$ A concerning the congruence relations of its MV-algebra of regular elements of $${\mathcal {A}}$$ A , along with specific equivalence relations concerning the complement of the set of regular elements. In the second part, we concentrate on perfect quasi MV-algebras. We present their representation by symmetric quasi $$\ell $$ ℓ -groups, a special kind of quasi $$\ell $$ ℓ -groups. Moreover, we establish a categorical equivalence of the category of perfect quasi MV-algebras, the category of n -perfect quasi MV-algebras, and the category of symmetric quasi $$\ell $$ ℓ -groups.