Abstract. We extend BCK algebras to pseudo-BCK algebras, as MV algebras and BL algebras were extended to pseudo-MY algebras and pseudo-BL algebras, respectively. We make the connection with pseudo-MV algebras and with pseudo-BL algebras.

BL algebras were introduced by Ha Âjek as algebraic structures for his Basic Logic, starting from continuous t-norms on 0; 1. MV algebras, product algebras and Go Èdel algebras are particular cases of BL algebras. On the other hand, the pseudo-MV algebras extend the MV-algebras in the same way in which the arbitrary l-groups extend the abelian l-groups. We have generalized the BL algebras and pseudo-MV algebras, introducing the pseudo-BL algebras. In this paper we introduce weak-BL algebras and weak-pseudo-BL algebras. We also introduce non-commutative t-norms (we call them pseudo-t-norms) and use them in constructing pseudo-BL algebras and weak-pseudo-BL algebras. IntroductionWehave started from the following situation:(1) We know BL algebras, algebraic structures for which the principal examples come from: ± the real interval [0, 1] with the structure given by a continuous t-norm and ± abelian l-groups.(2) The non-commutative case (pseudo-MV algebras and pseudo-BL algebras) was developed starting from arbitrary l-groups.The natural problem was: can be de®ned a concept of pseudo-t-norm (by weaking the axioms of t-norms) on [0, 1] or, more general, on bounded chains, bounded lattices, in order to obtain new examples of pseudo-BL algebras?The present paper tries to answer to this problem in the following way:(1) We de®ne the notion of pseudo-t-norm, by throwing away the axiom of commutativity; (2) On arbitrary l-groups, we re®nd the already known examples;(3) On [0, 1], the condition of continuity is replaced by the weaker condition of left continuity in both variables. Then, [0, 1] is endowed with a weaker structure, that of weak-pseudo-BL algebra. This structure gives birth to the new concept of weak-BL algebra, in the commutative case.In conclusion, we de®ne a concept of pseudo-t-norm, that leads to pseudo-BL algebras on arbitrary l-groups; on [0, 1], we can not do this, without exiting from commutative case, the adequate structure being that of weakpseudo-BL algebra. In this paper we present a general up-today picture of the algebra of non-commutative logic. t-Norms and U-operatorsFirst, we shall recall the de®nitions of t-norms (t-conorms) and of their associated U-operators de®ned on the real interval [0, 1]. De®nition 2.1 (Cf. [32]) (a) A binary operation T on the real interval [0, 1] is a t-norm iff: (t0) it is commutative, (t1) it is associative, (t2) it is non-decreasing (isotone) in the ®rst argument (i.e. if x y, then Tx; z T y; z, for Focus All our gratitude to Radko Mesiar for his patient reading of the manuscript. We all three are new in the ®eld of t-norms, therefore his very kind suggestions were very useful to improve the paper, as for example: to complete the bibliography with [13, 14, 18, 41, 42], to modify the proof of Theorem 5.15; he noted that our Examples 8.3(1) is a left-continuous modi®cation of Example 1.12 from [42] and our further examples of pseudo-t-norms are ordinal sums in the sense of Clifford (Theorem 3.42 from [42] or [41]); he also noted that then relevant residual operators h...

We introduce Iséki algebras as a special class of BCK algebras and we prove they are categorically equivalent with BL algebras. Other connections are established.

Starting from quasi-Wajsberg algebras (which are generalizations of Wajsberg algebras), whose regular sets are Wajsberg algebras, we introduce a theory of quasi-algebras versus, in parallel, a theory of regular algebras. We introduce the quasi-RM, quasi-RML, quasi-BCI, (commutative, positive implicative, quasi-implicative, with product) quasi-BCK, quasi-Hilbert and quasi-Boolean algebras as generalizations of RM, RML, BCI, (commutative, positive implicative, implicative, with product) BCK, Hilbert and Boolean algebras respectively.In Part I, the first part of the theory of quasi-algebras -versus the first part of a theory of regular algebras -is presented. We introduce the quasi-RM and the quasi-RML algebras and we present two equivalent definitions of quasi-BCI and of quasi-BCK algebras.

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