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In the paper, pre-Hilbert algebras are defined as a generalization of Hilbert algebras (namely, a Hilbert algebra is just a pre-Hilbert algebra satisfying the property of antisymmetry). Pre-Hilbert algebras have been inspired by Henkin's Positive Implicative Logic. Their properties and characterizations are investigated. Some important results and examples are given. Moreover, positive implicative pre-Hilbert algebras are introduced and studied, their connections with some algebras of logic are presented. The hierarchies existing between the classes of algebras considered here are shown. [ABSTRACT FROM AUTHOR]

GE algebras (generalized exchange algebras) and pre-Hilbert algebras are a generalization of well-known Hilbert algebras. In the paper, connections between these algebras are studied. In particular, it is proven that pi-BE algebras are equivalent with GE algebras satisfying the exchange property. Some characterizations of transitive GE algebras and exchange pre-Hilbert algebras are given. It is shown that the intersection of classes of GE algebras and pre-Hilbert algebras is the class of transitive GE algebras. Moreover, GE, BE and pre-Hilbert algebras with the antisymmetry property are investigated. It is proven that transitive GE algebras satisfying the property of antisymmetry coincide with Hilbert algebras. Finally, positive implicative GE and pre-Hilbert algebras are considered, their connections with some algebras of logic are presented. In addition, the hiearchies existing between the classes of algebras studied here are shown. [ABSTRACT FROM AUTHOR]

The notions of a soft general algebra and a soft subalgebra are introduced and studied. The operations on them such as a restricted intersection, an extended intersection, a restricted union, a ∧-intersection, a ∨-union and a cartesian product are established.

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 investigate the property of commutativity for various generalizations of pseudo-BCK algebras (pseudo-M, pseudo-RM, pseudo-RML, pseudo-aRML** algebras and many others). We give an axiom system for commutative pseudo-aRML** algebras and show that every such algebra (A,→,⤳, 1) is a join-semilattice with respect to the associated join operation V given by x V y = (x → y) ⤳ y. We define the commutative deductive systems of pseudo-M algebras and prove that a pseudo-aRML** algebra with the additional condition (pD) is commutative if and only if each of its deductive systems is commutative. We introduce the notion of BB-deductive system and then we construct the quotient algebra A/D of a pseudo-RM algebra A via a closed BBdeductive system D of A. Finally, we show that a BB-deductive system D of a pseudo-RML algebra A with (pD) is commutative if and only if A/D is a commutative pseudo-aRML** algebra. [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]

We show that UP-algebras are the same as Komori's BCC-algebras and that KU-algebras are reversed BCK-algebras. Moreover, we prove that pseudo-KU algebras are in fact reversed pseudo-BCK algebras. [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]

The class of all RM algebras contains BCK, BCI, BCH, BZ, RME, pre- BZ algebras and many others. In this paper, we introduce the notion of translation deductive system in RM algebras and investigate its elementary properties. We prove that every closed translation deductive system is a congruence kernel. Moreover, we give the construction of quotient algebra A/D of a RM algebra A via a translation deductive system D of A. Finally, we consider commutative RM, RME and pre-BZ algebras and show that commutative RME algebras coincide with commutative pre-BZ algebras and with commutative BCK algebras. We also show that there is a bijection between congruences and closed translation deductive systems in commutative RM algebras. [ABSTRACT FROM AUTHOR]

The notions of a soft general algebra and a soft subalgebra are introduced and studied. The operations on them such as a restricted intersection, an extended intersection, a restricted union, a ∧-intersection, a ∨-union and a cartesian product are established. [ABSTRACT FROM AUTHOR]