New algebras and new connections between/in the algebras of logic and the monoidal algebras

Abstract

We introduce the implicative-group as a term equivalent definition of the group coming from algebras of logic; we also introduce the partially-ordered and the lattice-ordered implicative-group as term equivalent definitions of the partially-ordered and of the lattice-ordered group, respectively. Since G. Dymek made the connection between the pseudo-BCI algebras and the groups, by introducing the subclass of p-semisimple pseudo-BCI algebras and proving that they are equivalent with the groups, we conclude that the p-semisimple pseudo-BCI algebras are equivalent with the implicative-groups. We define the p-semisimple po-groups (po-implicative-groups) and prove that they are equivalent with the groups (implicative-groups, respectively). We draw the ``map of some algebras of logic and the analogous ``map of the corresponding monoidal algebras. The lattice-ordered implicative-group is the great piece which missed from the puzzle showing the connections between the algebras of logic and the monoidal algebras. We establish the connections between the lattice-ordered implicative-groups and the pseudo-Wajsberg algebras and the pseudo-H\'{a}jek(pP) algebras verifying some properties.