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.