Classes of pseudo-BCK algebras – Part II
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-Hajek(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 non-commutative logic-and not only-and 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-Hajek(pP) algebras, weak-pseudo-Hajek(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-pseudo-BL algebras (pseudo-MTL algebras) and divisible non-commutative residuated lattices, respectively. We analyse some bounded algebras connected with non-commutative logic, but corresponding not-bounded algebras too (when there is only a greatest element, 1); therefore, we define the generalized-pseudo-Hajek(pP) algebras, the generalized-pseudo-Wajsberg algebras, the weak-generalized-pseudo-Hajek(pP) algebras (generalized-pseudo-MTL algebras), etc. We divide the properties of (generalized) pseudo-Hajek(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-Hajek(pP) algebra satisfies also (pCv) condition, we show its connection with (pdiv) and (pprel) conditions:
(pprel) ⇔ (pCv) + (pC) ⇔ (pC^) + (pC◊) and (pdiv) ⇔ (pC⇒) + (pC^) + (pCx);
we make the above decompositions in the general case of not-bounded pseudo-BCK(pP) lattices. We analize the influence of (pDN) (pseudo-double negation) condition on these decompositions and we study the additional (pWNM) (pseudo-weak nilpotent minimum) condition in the bounded case.
In part II, we generalize good pseudo-BL algebras, by introducing the notion of good pseudo-BCK algebras. We introduce and study the ordinal sum of two bounded pseudo-BCK algebras and prove in what conditions we get pseudo-Hajek(pP) algebras or other structures, more general, or more particular. We prove that (pdiv) ⇔ (pCX ) and improve some results from Part I concerning the decompositions of (pprel) and (pdiv). We introduce new generalizations of generalized-pseudo-Hajek(pP) algebras, named pseudo-α algebras, pseudo-β algebras, pseudo-g algebras,pseudo-αβ algebras, etc. By adding (pDN), (good) and (pWNM) conditions to these algebras, when they are bounded, we get more classes. We establish hierarchies between these new classes and the old classes pointed out in Part I and we give some examples in the bounded case.