Classes of Pseudo-BCK algebras – Part I
n 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 logic- and not only of logic- 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, weakpseudo- 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 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 analyze 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. InPart II,we introduce the ordinal sum of two bounded pseudo-BCK(pP) lattices and prove in what conditions we get pseudo-Hajek(pP) algebras or other structures, more general, or more particular. We introduce new generalizations of generalized-pseudo-Hajek(pP) algebras, named pseudo-a, pseudo-b, pseudo-g , pseudo-e , pseudo-d, pseudo-ab, . . . , as pseudo-BCK(pP) lattices verifying one, two, three, four or five of the above conditions (pC ), (pCv), (pC ), (pC ) and (pCx). By adding (pDN) and (pWNM) conditions to these algebras, when they are bounded, we get more classes. We establish hierarchies between all these new classes and the old classes pointed out in Part I and we give some examples.