Commutative BCK-algebras Research Articles

On implicative BE algebras

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.

Spectral properties of cBCK-algebras

In this paper we study prime spectra of commutative BCK-algebras. We give a new construction for commutative BCK-algebras using rooted trees, and determine both the ideal lattice and prime ideal lattice of such algebras. We prove that the spectrum of any commutative BCK-algebra is a locally compact generalized spectral space which is compact if and only if the algebra is finitely generated as an ideal. Further, we show that if a commutative BCK-algebra is involutory, then its spectrum is a Priestley space. Finally, we consider the functorial properties of the spectrum and define a functor from the category of commutative BCK-algebras to the category of distributive lattices with zero. We give a partial answer to the question: what distributive lattices lie in the image of this functor?

ON COMMUTATIVE BE-ALGEBRAS

In this paper we investigate the relationship between BE-algebras, implica- tive algebras, and J-algebras. Moreover, we define commutative BE-algebras and state that these algebras are equivalent to the commutative dual BCK-algebras.

PC-lattices: A Class of Bounded BCK-algebras

In this paper, we define the notion of PC-lattice, as a generalization of finite positive implicative BCK-algebras with condition (S) and bounded commutative BCK-algebras. We investiate some results for Pc-lattices being a new class of BCK-lattices. Specially, we prove that any Boolean lattice is a PC-lattice and we show that if X is a PC-lattice with condition S, then X is an involutory BCK-algebra if and only if X is a commutative BCK-algebra. Finally, we prove that any PC-lattice with condition (S) is a distributive BCK-algebra.

Commutative deductive systems of pseudo-BCK-algebras

In this paper, we generalize the axiom systems given by M. Palasinski and B. Woźniakowska for commutative BCK-algebras to the case of commutative pseudo-BCK-algebras. A characterization of commutative pseudo-BCK-algebras is also given. We define the commutative deductive systems of pseudo-BCK-algebras, and we generalize some results proved by Yisheng Huang for commutative ideals of BCI-algebras to the case of commutative deductive systems of pseudo-BCK-algebras. We prove that a pseudo-BCK-algebra A is commutative if and only if all the deductive systems of A are commutative. We show that a normal deductive system H of a pseudo-BCK-algebra A is commutative if and only if A / H is a commutative pseudo-BCK-algebra. We introduce the notions of state operators and state-morphism operators on pseudo-BCK-algebras, and we apply these results on commutative deductive systems to investigate the properties of these operators.

A Structure of BCI-Algebras

Commutative BCI-algebras can be considered as semilattices whose sections are equipped with certain involutions. A similar view can be applied to commutative BCK-algebras. However, for general BCK-algebras a certain construction was settled by the author and J. Kuhr (Miskolc Math. Notes 8:11–21, 2007) showing that they can be considered as structures essentially weaker than semilattices but still with certain involutions in sections. The aim of this paper is to involve a similar approach for BCI-algebras.

The variety of all commutative BCK-algebras is generated by its finite members as a quasivariety

Abstract We prove the result announced by the title as well as some of its consequences.

Local pseudo-BCK algebras with pseudo-product

Abstract Pseudo-BCK algebras were introduced by G. Georgescu and A. Iorgulescu as a generalization of BCK algebras in order to give a corresponding structure to pseudo-MV algebras, since the bounded commutative BCK algebras correspond to MV algebras. Properties of pseudo-BCK algebras and their connections with other fuzzy structures were established by A. Iorgulescu and J. Kühr. The aim of this paper is to define and study the local pseudo-BCK algebras with pseudo-product. We will also introduce the notion of perfect pseudo-BCK algebras with pseudo-product and we will study their properties. We define the radical of a bounded pseudo-BCK algebra with pseudo-product and we prove that it is a normal deductive system. Another result consists of proving that every strongly simple pseudo-hoop is a local bounded pseudo-BCK algebra with pseudo-product.

The Localization of Commutative Bounded BCK-Algebras

In this paper we develop a theory of localization for bounded commutative BCK-algebras. We try to extend some results from the case of commutative Hilbert algebras (see [1]) to the case of commutative BCK-alge- bras.

Equational spectrum of Hilbert varieties

AbstractWe prove that an equational class of Hilbert algebras cannot be defined by a single equation. In particular Hilbert algebras and implication algebras are not one-based. Also, we use a seminal theorem of Alfred Tarski in equational logic to characterize the set of cardinalities of all finite irredundant bases of the varieties of Hilbert algebras, implication algebras and commutative BCK algebras: all these varieties can be defined by independent bases of n elements, for each n > 1.

Join-semilattices whose sections are residuated PO-monoids

We generalize the concept of an integral residuated lattice to join-semilattices with an upper bound where every principal order-filter (section) is a residuated semilattice; such a structure is called a sectionally residuated semilattice. Natural examples come from propositional logic. For instance, implication algebras (also known as Tarski algebras), which are the algebraic models of the implication fragment of the classical logic, are sectionally residuated semilattices such that every section is even a Boolean algebra. A similar situation rises in case of the Lukasiewicz multiple-valued logic where sections are bounded commutative BCK-algebras, hence MV-algebras. Likewise, every integral residuated (semi)lattice is sectionally residuated in a natural way. We show that sectionally residuated semilattices can be axiomatized as algebras (A, r, →, ⇝, 1) of type 〈3, 2, 2, 0〉 where (A, →, ⇝, 1) is a {→, ⇝, 1}-subreduct of an integral residuated lattice. We prove that every sectionally residuated lattice can be isomorphically embedded into a residuated lattice in which the ternary operation r is given by r(x, y, z) = (x · y) ∨ z. Finally, we describe mutual connections between involutive sectionally residuated semilattices and certain biresiduation algebras.

General Theory of Quasi-Commutative BCI-algebras

General Theory of Quasi-Commutative BCI-algebrasIt is known that commutative BCK-algebras form a variety, but BCK-algebras do not [4]. Therefore H. Yutani introduced the notion of quasicommutative BCK-algebras. In this article we first present the notion and general theory of quasi-commutative BCI-algebras. Then we discuss the reduction of the type of quasi-commutative BCK-algebras and some special classes of quasicommutative BCI-algebras.

On implication in MV-algebras

In [1], the authors introduced the notion of a weak implication algebra, which reflects properties of implication in MV-algebras, and demonstrated that the class of weak implication algebras is definitionally equivalent to the class of upper semilattices whose principal filters are compatible MV-algebras. It is easily seen that weak implication algebras are just duals of commutative BCK-algebras. We show here that most results of [1] are, in fact, immediate consequences of two well-known facts: (i) a bounded commutative BCK-algebra possesses a natural upper semilattice structure, (ii) the class of MV-algebras and that of bounded commutative BCK-algebras are definitionally equivalent.

Several Classes of BCK-algebras and their Properties

Several Classes of BCK-algebras and their PropertiesIn this article the general theory of Commutative BCK-algebras and BCI-algebras and several classes of BCK-algebras are given according to [2].

Embeddings of generalized effect algebras into complete effect algebras

Generalized effect algebras as posets are unbounded versions of effect algebras having bounded effect-algebraic extensions. We show that when the MacNeille completion MC(P) of a generalized effect algebra P cannot be organized into a complete effect algebra by extending the operation ⊕ onto MC(P) then still P may be densely embedded into a complete effect algebra. Namely, we show these facts for Archimedean GMV-effect algebras and block-finite prelattice generalized effect algebras. Moreover, we show that extendable commutative BCK-algebras directed upwards are equivalent to generalized MV-effect algebras.

On fuzzy ideals in BCK/BCI-algebras

In the present paper we give a procedure by which we generate a fuzzy ideal (resp. closed fuzzy ideal) by a fuzzy set in a BCI-algebra. As applications we prove: The set of all the fuzzy ideals in a BCI-algebra forms a complete lattice (called fuzzy ideal lattice). The set of all the closed fuzzy ideals in a BCI-algebra is a modular sublattice of the fuzzy ideal lattice, but it is not distributive in general. For commutative BCK-algebras we establish the prime fuzzy ideal theorem. In particular, we give some characterizations of Noether BCK/BCI-algebras by fuzzy ideals.

Negation and BCK‐algebras

AbstractIn this paper we consider twelve classical laws of negation and study their relations in the context of BCK‐algebras. A classification of the laws of negation is established and some characterizations are obtained. For example, using the concept of translation we obtain some characterizations of Hilbert algebras and commutative BCK‐algebras with minimum. As a consequence we obtain a theorem relating those algebras to Boolean algebras.

Algebras in the Positive Cone of po-Groups

We systemize a number of algebras that are especially known in the field of quantum structures and that in particular arise from the positive cones of partially ordered groups. Generalized effect algebras, generalized difference posets, cone algebras, commutative BCK-algebras with the relative cancellation property, and positive minimal clans are included in the text. All these structures are conveniently characterizable as special cases of generalized pseudoeffect algebras, which we introduced in a previous paper. We establish the exact relations between all mentioned structures, thereby adding new structures whenever necessary to make the scheme of order complete. Generalized pseudoeffect algebras were under certain conditions proved to be representable by means of a po-group. From this fact, we will easily establish representation theorems for all of the structures included in discussion.

Fuzzy prime ideals and invertible fuzzy ideals in BCK-algebras

We state the relation between fuzzy prime ideals and invertible fuzzy ideals in commutative BCK-algebras.

On Categorical Equivalences of Commutative BCK-algebras

A commutative BCK-algebra with the relative cancellation property is a commutative BCK-algebra (X;*,0) which satisfies the condition: if a ≤ x, a ≤ y and x * a = y * a, then x = y. Such BCK-algebras form a variety, and the category of these BCK-algebras is categorically equivalent to the category of Abelian l-groups whose objects are pairs (G, G 0), where G is an Abelian l-group, G 0 is a subset of the positive cone generating G + such that if u, v ∈ G 0, then 0 ∨ (u - v) ∈ G 0, and morphisms are l-group homomorphisms h: (G, G 0) → (G′,G′0) with f(G 0) ⫅ G′0. Our methods in particular cases give known categorical equivalences of Cornish for conical BCK-algebras and of Mundici for bounded commutative BCK-algebras (= MV-algebras).