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GE algebras (generalized exchange algebras), transitive GE algebras (tGE algebras, for short) and aGE algebras (that is, GE algebras verifying the antisymmetry) are a generalization of Hilbert algebras. Here some properties and characterizations of these algebras are investigated. Connections between GE algebras and other classes of algebras of logic are studied. The implicative and positive implicative properties are discussed. It is shown that the class of positive implicative GE algebras (resp. the class of implicative aGE algebras) coincides with the class of generalized Tarski algebras (resp. the class of Tarski algebras). It is proved that for any aGE algebra the property of implicativity is equivalent to the commutative property. Moreover, several examples to illustrate the results are given. Finally, the interrelationships between some classes of implicative and positive implicative algebras are presented. [ABSTRACT FROM AUTHOR]

We consider some generalizations of BCK algebras such as for example RM, tRM, aRM (= BH) algebras and others. We investigate modal operators on such algebras. In particular, we show that the composition of two modal operators on an aRM algebra verifying the transitivity property is also a modal operator if and only if they commute. It is also proved that if two modal operators on an aRM algebra have the same image, then they coincide. Moreover, we introduce and study modal RM algebras. We define modal upper sets of RM algebras and we investigate some properties of these sets. We construct quotient modal RM algebras and we prove that if φ is a homomorphism of a modal RM algebra A onto a modal aRM algebra B, then A/Ker(φ) is isomorphic to B. Furthermore, we establish connections between deductive systems and congruences for modal RM algebras. [ABSTRACT FROM AUTHOR]

We consider some generalizations of BCK algebras (RML, BE, aBE, BE** and aBE** algebras). We investigate the property of implicativity for these algebras. We prove that for any implicative BE** algebra the commutativity property is equivalent to the property of antisymmetry and show that implicative aBE** algebras are commutative BCK algebras. We also show that the class of implicative BE** algebras is a variety. [ABSTRACT FROM AUTHOR]

In this paper we introduce and study the monadic RM algebras. As particular cases we get monadic *RM algebras, monadic RM** algebras, monadic BE algebras, monadic BCI algebras, monadic BCK algebras, etc. The monadic deductive systems and R-congruence relations of a monadic RM algebra are defined and their properties are investigated. For the monadic *RM** algebras, it is proved that there is a one-to-one order-preserving correspondence between the R-congruence relations and the monadic closed t-deductive systems. Finally, we prove that if A is a monadic pre-BBBZ algebra, then the lattice of monadic deductive systems of A is isomorphic to the lattice of the deductive systems of (A∃∀,→, 1), where A∃∀ is the set of all fixed elements of A. [ABSTRACT FROM AUTHOR]

Some generalizations of BCK algebras (such as, from example, RM, BH, pre-BBBCC and pre-BCK algebras) are considered. It is proved that the set of all deductive systems of a pre-BBBCC algebra is an algebraic distributive lattice. Irreducible deductive systems and prime deductive systems of RM algebras are introduced and studied. In particular, the homomorphic properties of these deductive systems are provided. Finally, subdirect products of BH algebras by using prime deductive systems are investigated. [ABSTRACT FROM AUTHOR]

We consider various generalizations of BCK algebras (RML, aRML, *RML, aRML**, pre-BCC, pre-BCK, weak BCK* algebras and many others). We investigate the property of implicativity for these algebras. We prove that every implicative *RML** algebra is a pre-BCC algebra and also show that implicative *aRML** algebras are Tarski algebras. Moreover, we give several examples of implicative finite algebras. Finally, we review some classes of implicative RML algebras and prove that they are varieties. [ABSTRACT FROM AUTHOR]

A simplified axiomatization of pseudo-BCK algebras is given. In addition, we obtain a system of axioms defining bounded commutative pseudo-BCK algebras and show that these axioms are independent (that is, none follows from the others). [ABSTRACT FROM AUTHOR]