Classes of Pseudo-BCK algebras -Part I

In this paper we study particular classes of pseudo-BCK algebras, bounded or not bounded, as pseudo-BCK(pP) algebras, pseudo-BCK(pP) lattices, pseudo-Hájek(pP) algebras and pseudo-Wajsberg algebras. We introduce new classes of pseudo-BCK(pP) lattices, we establish hierarchies and we give some examples. We work with left-defined algebras and we work with → and ; as primitive operations, not with the pseudo-t-norm. The paper has two parts. In the first part, we introduce a methodology for the simultaneous work with several algebras of logicand not only of logicand consequently, since we already have pseudo-BCK(pP) algebras and pseudo-Wajsberg algebras, categorically isomorphic with porims (partially ordered residuated integral monoids) and pseudo-MV algebras, respectively, we introduce pseudo-BCK(pP) lattices, pseudo-Hájek(pP) algebras, weakpseudo-Hájek(pP) algebras and divisible pseudo-BCK(pP) lattices as particular classes of pseudo-BCK algebras, categorically isomorphic with non-commutative residuated lattices, pseudo-BL algebras, weak-pseudoBL algebras (pseudo-MTL algebras) and divisible non-commutative residuated lattices, respectively. We analyse some bounded algebras connected with logic, but corresponding not-bounded algebras too (when there is only a greatest element, 1); therefore, we define the generalized-pseudo-Hájek(pP) algebras, the generalized-pseudoWajsberg algebras, the weak-generalized-pseudo-Hájek(pP) algebras (generalized-pseudo-MTL algebras) etc. We divide the properties of (generalized-) pseudo-Hájek(pP) algebras into three groups: those coming from the fact that they are pseudo-BCK(pP) algebras, those coming from the fact that they are lattices (pseudo-BCK(pP) lattices) and those coming from pseudo-divisibility, (pdiv), and pseudo-pre-linearity, (pprel), conditions; we present these properties, old and new ones. We find equivalent conditions with (pdiv) and (pprel) and, since a pseudo-Hájek(pP) algebra satisfies also (pC∨) condition, we show its connection with (pdiv) and (pprel) conditions: