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Abstract

The notion of a RM algebra, introduced recently, is a generalization of many other algebras of logic. The class of RM algebras contains (weak-)BCC algebras, BCH algebras, BCI algebras, BCK algebras and many others. A RM algebra is an algebra A = (A; →, 1) of type (2, 0) satisfying the identities: xx = 1 and 1 → x = x. In this paper we study the set of maximal elements of a RM algebra, branches of a RM algebra and moreover translation deductive systems of a RM algebra giving so called the Representation Theorem for RM algebras.

Abstract

In this paper, we introduce some new classes of algebras related to UP-algebras and semigroups, called a left UP-semigroup, a right UP-semigroup, a fully UP-semigroup, a left-left UP-semigroup, a right-left UP-semigroup, a left-right UP-semigroup, a right-right UP-semigroup, a fully-left UP-semigroup, a fully-right UP-semigroup, a left-fully UP-semigroup, a right-fully UP-semigroup, a fully-fully UP-semigroup, and find their examples.

Abstract

In this paper we introduce the notion of a left zeroid and a right zeroid of Γ -semirings. We prove that, a left zeroid of a simple Γ-semiring M is regular if and only if M is a regular Γ -semiring.

Abstract

Injective pseudo-BCI algebras are studied. There is shown that the only injective pseudo-BCI algebra is the trivial one.

Abstract

In this paper, we introduced the concept of a soft hoop and we investigated some of their properties. Then, we established different types of intersections and unions of the family of soft hoops. We defined two operations ⊙ and → on the set of all soft hoops and we proved that with these operations, it is a hoop and also is a Heyting algebra. Finally we introduced a congruence relation on the set of all soft hoops and we investigated the quotient of it.

Summary

We show that the set of all partial predicates over a set D together with the disjunction, conjunction, and negation operations, defined in accordance with the truth tables of S.C. Kleene’s strong logic of indeterminacy [], forms a Kleene algebra. A Kleene algebra is a De Morgan algebra [] (also called quasi-Boolean algebra) which satisfies the condition x¬:xy¬ :y (sometimes called the normality axiom). We use the formalization of De Morgan algebras from [].

The term “Kleene algebra” was introduced by A. Monteiro and D. Brignole in []. A similar notion of a “normal i-lattice” had been previously studied by J.A. Kalman []. More details about the origin of this notion and its relation to other notions can be found in [, , , ]. It should be noted that there is a different widely known class of algebras, also called Kleene algebras [, ], which generalize the algebra of regular expressions, however, the term “Kleene algebra” used in this paper does not refer to them.

Algebras of partial predicates naturally arise in computability theory in the study on partial recursive predicates. They were studied in connection with non-classical logics [, , , , , ]. A partial predicate also corresponds to the notion of a partial set [] on a given domain, which represents a (partial) property which for any given element of this domain may hold, not hold, or neither hold nor not hold. The field of all partial sets on a given domain is an algebra with generalized operations of union, intersection, complement, and three constants (0, 1, n which is the fixed point of complement) which can be generalized to an equational class of algebras called DMF-algebras (De Morgan algebras with a single fixed point of involution) []. In [] partial sets and DMF-algebras were considered as a basis for unification of set-theoretic and linguistic approaches to probability.

Partial predicates over classes of mathematical models of data were used for formalizing semantics of computer programs in the composition-nominative approach to program formalization [, , , ], for formalizing extensions of the Floyd-Hoare logic [, ] which allow reasoning about properties of programs in the case of partial pre- and postconditions [, , , ], for formalizing dynamical models with partial behaviors in the context of the mathematical systems theory [, , , , ].

Abstract

In this paper we define strong ideals and horizontal ideals in pseudo-BCH-algebras and investigate the properties and characterizations of them.

Abstract

In this paper, we introduce the notion of topological UP-algebras and several types of subsets of topological UP-algebras, and prove the generalization of these subsets. We also introduce the notions of quotient topological spaces of topological UP-algebras and topological UP-homomorphisms. Furthermore, we study the relation between topological UP-algebras, Hausdor spaces, discrete spaces, and quotient topological spaces, and prove some properties of topological UP-algebras.

Abstract

In this paper, we construct the fundamental theorem of UP-homomorphisms in UP-algebras. We also give an application of the theorem to the first, second, third and fourth UP-isomorphism theorems in UP-algebras.

Abstract

In this paper, the concepts of n-fold implicative ideals and n-fold obstinate ideals in BL-algebras are introduced. With respect to this concepts, some related results are given. In particular, it is proved that an ideal is an n-fold implicative ideal if and only if is an n-fold Boolean ideal. Also, it is shown that a BL-algebra is an n-fold integral BL-algebra if and only if trivial ideal {0} is an n-fold obstinate ideal. Moreover, the relation between n-fold obstinate ideals and n-fold (integral) obstinate filters in BL-algebras are studied by using the set of complement elements. Finally, it is proved that ideal I of BL-algebra L is an n-fold obstinate ideal if and only if LT is an n-fold obstinate BL-algebra.