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We prove that almost all the properties of quantum-MV algebras are verified by orthomodular algebras, the new algebras introduced in a previous paper. We put a special insight on transitive antisymmetric orthomodular (taOM) algebras, generalizations of MV algebras. We make the connection with IMTL and NM algebras. [ABSTRACT FROM AUTHOR]

Hilbert algebras are particular cases of BCK algebras, while BCK algebras are particular cases of BCI algebras. In Part I, starting with a list of properties of BCK algebras and with the algebras BCH, BCC, BZ, BE, pre-BCK, old generalizations of BCI algebras or of BCK algebras, we introduce new generalizations of BCI or of BCK algebras and, consequently, new generalizations of Hilbert algebras. Namely, we found thirty one new distinct generalizations of BCI or of BCK algebras and twenty new distinct generalizations of Hilbert algebras. We show the hierarchies existing between all these algebras, old or new ones. In Part II, we present 83 examples of proper finite algebras, at least one example of each of the 59 (= 7 (old) + 1 (Hilbert) + 31 + 20) old or new algebras. [ABSTRACT FROM AUTHOR]

The l-implicative-group is a term equivalent definition of the group coming from algebras of logic. In this paper, we study the representability of l-implicative-groups and of associated algebras of logic. First, we find equivalent conditions for an l-implicativegroup to be representable. Then, we prove that representability at l-implicative-group level is inherited by the algebras obtained by restricting the l-implicative-group operations to the negative, positive cones. [ABSTRACT FROM AUTHOR]

In this paper, Part III, we shall illustrate by classes of finite examples the close connections between MV algebras, by one side, and BL algebras and divisible bounded residuated lattices, on the other side; we present classes of examples of proper MV algebras, BL algebras and divisible bounded residuated lattices, satisfying or not satisfying condition Weak Nilpotent Minimum (WNM). We use equivalent definitions of these algebras, as particular cases of reversed left-BCK algebras, as established in Part I. [ABSTRACT FROM AUTHOR]

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 non-commutative logic- and not only and consequently, since we already have pseudo-BCK(pP) algebras and pseudo-Wajsberg algebras, cate gorically isomorphic with porims (partially ordered residuated integral monoids) and pseudo-MV algebras, respectively, we introduce pseudo-BCK(pP) lattices, pseudo-Hájek(pP) algebras, weak-pseudo-Hájek(pP) algebras and divisible pseudo-BCK(pP) lattices as particular classes of pseudo-BCK algebras, cate gorically isomorphic withnon-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-Hájek(pP) algebras, the generalized-pseudo-Wajsberg 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, (pdi v), 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: (pprel) ⇔ (pC∨) + (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-Hájek(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-Hájek(pP) algebras, named pseudo-α algebras, pseudo-β algebras, pseudo-γ 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. [ABSTRACT FROM AUTHOR]

We introduce pseudo-Iséki algebras as a special class of pseudo-BCK algebras and we prove they are categorically equivalent with pseudo-BL algebras. Other connections are established. [ABSTRACT FROM AUTHOR]

A foreword to the "Journal of Multiple-Valued Logic & Soft Computing" is presented.