On some Properties of quasi-MV √ Algebras and quasi-MV Algebras. Part IV

In the present paper, which is a sequel to [20, 4, 12], we investigate further the structure theory of quasi-MV algebras and √ quasi-MV algebras. In particular: we provide a new representation of arbitrary √ qMV algebras in terms of √ qMV algebras arising out of their MV* term subreducts of regular elements; we investigate in greater detail the structure of the lattice of √ qMV varieties, proving that it is uncountable, providing equational bases for some of its members, as well as analysing a number of slices of special interest; we show that the variety of √ qMV algebras has the amalgamation property; we provide an axiomatisation of the 1-assertional logic of √ qMV algebras; lastly, we reconsider the correspondence between Cartesian √ qMV algebras and a category of Abelian lattice-ordered groups with operators first addressed in [10]

On some properties of quasi-MV algebras and square root quasi-MV algebras, IV

In the present paper, which is a sequel to [20, 4, 12], we investigate further the structure theory of quasiMV algebras and √0quasi-MV algebras. In particular: we provide a new representation of arbitrary √0qMV algebras in terms of √0qMV algebras arising out of their MV* term subreducts of regular elements; we investigate in greater detail the structure of the lattice of √0qMV varieties, proving that it is uncountable, providing equational bases for some of its members, as well as analysing a number of slices of special interest; we show that the variety of √0qMV algebras has the amalgamation property; we provide an axiomatisation of the 1-assertional logic of √0qMV algebras; lastly, we reconsider the correspondence between Cartesian √0qMV algebras and a category of Abelian lattice-ordered groups with operators first addressed in [10]

Quasi-Algebras versus Regular Algebras - Part I

Starting from quasi-Wajsberg algebras (which are generalizations of Wajsberg algebras), whose regular sets are Wajsberg algebras, we introduce a theory of quasi-algebras versus, in parallel, a theory of regular algebras. We introduce the quasi-RM, quasi-RML, quasi-BCI, (commutative, positive implicative, quasi-implicative, with product) quasi-BCK, quasi-Hilbert and quasi-Boolean algebras as generalizations of RM, RML, BCI, (commutative, positive implicative, implicative, with product) BCK, Hilbert and Boolean algebras respectively. In Part I, the first part of the theory of quasi-algebras - versus the first part of a theory of regular algebras - is presented. We introduce the quasi-RM and the quasi-RML algebras and we present two equivalent definitions of quasi-BCI and of quasi-BCK algebras