This paper introduces the novel concept of Neutro-BCK-algebra. In Neutro-BCK-algebra, the outcome of any given two elements under an underlying operation (neutro-sophication procedure) has three cases, such as: appurtenance, non-appurtenance, or indeterminate.

In this paper we prove that the equational class generated by bounded BCK-algebras is the variety generated by the class of finite simple bounded BCK-algebras. To obtain these results we prove that every simple algebra in the equational class generated by bounded BCK-algebras is also a relatively simple bounded BCK-algebra. Moreover, we show that every simple bounded BCK-algebra can be embedded into a simple integral commutative bounded residuated lattice. We extend our main results to some richer subreducts (...) of the class of integral commutative bounded residuated lattices and to the involutive case. (shrink)

In this paper we prove that the equational class generated by bounded BCK-algebras is the variety generated by the class of finite simple bounded BCK-algebras. To obtain these results we prove that every simple algebra in the equational class generated by bounded BCK-algebras is also a relatively simple bounded BCK-algebra. Moreover, we show that every simple bounded BCK-algebra can be embedded into a simple integral commutative bounded residuated lattice. We extend our main results to some richer subreducts (...) of the class of integral commutative bounded residuated lattices and to the involutive case. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim). (shrink)

In 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.

Let |$\textbf{A}$| be a |$BCK$|-algebra and |$f:A^{k}\rightarrow A$| a function. The main goal of this article is to give a necessary and sufficient condition for |$f$| to be compatible with respect to every relative congruence of |$\textbf{A}$|⁠. We extend this result in some related algebras, as e.g. in pocrims.

We discuss the interrelations between BCK-algebras and posets with difference. Applications are given to bounded commutative BCK-algebras, difference posets, MV-algebras, quantum MV-algebras and orthoalgebras.

Pseudo-BCK-algebras are a non-commutative generalization of well-known BCK-algebras. The paper describes a situation when a linearly ordered pseudo-BCK-algebra is an ordinal sum of linearly ordered cone algebras. In addition, we present two identities giving such a possibility of the decomposition and axiomatize the residuation subreducts of representable pseudo-hoops and pseudo-BL-algebras.

The notions of a C-energetic subset and permeable C-value in BCK-algebras are introduced, and related properties are investigated. Conditions for an element t in [0, 1] to be an permeable C-value are provided. Also conditions for a subset to be a C-energetic subset are discussed. We decompose BCK-algebra by a partition which consists of a C-energetic subset and a commutative ideal.

The notion of positive implicative soju ideal in BCK-algebra is introduced, and several properties are investigated. Relations between soju ideal and positive implicative soju ideal are considered, and characterizations of positive implicative soju ideal are established. Finally, extension property for positive implicative soju ideal is constructed.

In this paper, we define the notion of PC-lattice, as a generalization of finite positive implicative BCK-algebras with condition 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 is a distributive BCK-algebra. (shrink)

Formally, a description of weak BCK-algebras can be obtained by replacing the first BCK axiom \ - \le z - y}\) by its weakening \. It is known that every weak BCK-algebra is completely determined by the structure of its initial segments. We consider weak BCK-algebras with De Morgan complemented, orthocomplemented and orthomodular sections, as well as those where sections satisfy a certain compatibility condition, and characterize each of these classes of algebras by an equation or quasi-equation. (...) For instance, those weak BCK-algebras in which all initial segments are De Morgan complemented are just commutative weak BCK-algebras. (shrink)

This main aim of this paper is to prove that in a direct commutative BCK-algebra an ideal I is nitely generated if and only if I is a principal ideal. This result generalizes the result obtained by E. Y. Deeba in [2]. We also give an answer to the question posed by E. Y. Deeba in [1]: for what class of BCK-algebras is every Noetherian algebra a principal ideal algebra ?

In previous work the author has introduced a lambda calculus SLR with modal and linear types which serves as an extension of Bellantoni–Cook's function algebra BC to higher types. It is a step towards a functional programming language in which all programs run in polynomial time. In this paper we develop a semantics of SLR using BCK -algebras consisting of certain polynomial-time algorithms. It will follow from this semantics that safe recursion with arbitrary result type built up from N (...) and ⊸ as well as recursion over trees and other data structures remains within polynomial time. In its original formulation SLR supported only natural numbers and recursion on notation with first-order functional result type. (shrink)

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). (shrink)

The notions of a commutative (∈, ∈)-neutrosophic ideal and a commutative falling neutrosophic ideal are introduced, and several properties are investigated. Characterizations of a commutative (∈, ∈)-neutrosophic ideal are obtained. Relations between commutative (∈, ∈)-neutrosophic ideal and (∈, ∈)-neutrosophic ideal are discussed. Conditions for an (∈, ∈)-neutrosophic ideal to be a commutative (∈, ∈)-neutrosophic ideal are established. Relations between commutative (∈, ∈)-neutrosophic ideal, falling neutrosophic ideal and commutative falling neutrosophic ideal are considered. Conditions for a falling neutrosophic ideal to be (...) commutative are provided. (shrink)

Using the semantic embedding technique the theorem announced by the title is proved.

In this paper a definition of n-valued system in the context of the algebraizable logics is proposed. We define and study the variety V3, showing that it is definitionally equivalent to the equivalent quasivariety semantics for the “Three-valued BCK-logic”. As a consequence we find an axiomatic definition of the above system.

In the paper we obtain a new characterization of the BCK-algebras which are subdirect product of BCK-chains. We give an axiomatic algebraizable extension of the BCK-calculus, by means of a recursively enumerable set of axioms, such that its equivalent algebraic semantics is definitionally equivalent to the quasivariety of BCK-algebras generated by the BCK-chains. We propose the concept of "linearization of a system" and we give some examples.

A new structure, called equality algebras, will be introduced. It has two connectives, a meet operation and an equivalence, and a constant. A closure operator will be defined in the class of equality algebras, and we call the closed algebras equivalential. We show that equivalential equality algebras are term equivalent with BCK-algebras with meet. As a by-product, we obtain a quite general result, which is analogous to a result of Kabziński and Wroński: we provide an (...) equational characterization for the equivalential fragment of BCK-algebras with meet. (shrink)

Given i, j, k ∈ {1,2,3,4}, the notion of -length neutrosophic subalgebras in BCK=BCI-algebras is introduced, and their properties are investigated. Characterizations of length neutrosophic subalgebras are discussed by using level sets of interval neutrosophic sets. Conditions for level sets of interval neutrosophic sets to be subalgebras are provided.

A BCK-algebra is an algebra in which the terms are generated by a set of variables, 1, and an arrow. We mean by aBCK-identity an equation valid in all BCK-algebras. In this paper using a syntactic method we show that for two termss andt, if neithers=1 nort=1 is a BCK-identity, ands=t is a BCK-identity, then the rightmost variables of the two terms are identical.This theorem was conjectured firstly in [5], and then in [3]. As a corollary of this theorem, (...) we derive that the BCK-algebras do not form a variety, which was originally proved algebraically by Wroski ([4]). (shrink)

Characterizations of an (∈, ∈)-neutrosophic ideal are considered. Any ideal in a BCK/BCI-algebra will be realized as level neutrosophic ideals of some (∈, ∈)-neutrosophic ideal. The relation between (∈, ∈)-neutrosophic ideal and (∈, ∈)-neutrosophic subalgebra in a BCK-algebra is discussed. Conditions for an (∈, ∈)-neutrosophic subalgebra to be a (∈, ∈)-neutrosophic ideal are provided. Using a collection of ideals in a BCK/BCI-algebra, an (∈, ∈)-neutrosophic ideal is established. Equivalence relations on the family of all (∈, ∈)-neutrosophic ideals are introduced, and (...) related properties are investigated. (shrink)

Based on the hesitant fuzzy set theory which is introduced by Torra in the paper [12], the notions of Inf-hesitant fuzzy subalgebras, Inf-hesitant fuzzy ideals and Inf-hesitant fuzzy p-ideals in BCK/BCI-algebras are introduced, and their relations and properties are investigated. Characterizations of an Inf-hesitant fuzzy subalgebras, an Inf-hesitant fuzzy ideals and an Inf-hesitant fuzzy p-ideal are considered. Using the notion of BCK-parts, an Inf-hesitant fuzzy ideal is constructed. Conditions for an Inf-hesitant fuzzy ideal to be an Inf-hesitant fuzzy p-ideal (...) are discussed. Using the notion of Inf-hesitant fuzzy ideals, a characterization of a p-semisimple BCI-algebra is provided. Extension properties for an Inf-hesitant fuzzy p-ideal is established. (shrink)

This paper introduces the concept of single-valued neutrosophic hyper \(BCK\)-subalgebras as a generalization and alternative of hyper \(BCK\)-algebras and on any given nonempty set constructs at least one single-valued neutrosophic hyper \(BCK\)-subalgebra and one a single-valued neutrosophic hyper \(BCK\)-ideal. In this study level subsets play the main role in the connection between singlevalued neutrosophic hyper \(BCK\)-subalgebras and hyper \(BCK\)-subalgebras and the connection between single-valued neutrosophic hyper \(BCK\)-ideals and hyper \(BCK\)-ideals. The congruence and (strongly) regular equivalence relations are the important (...) tools for connecting hyperstructures and structures, so the major contribution of this study is to apply and introduce a (strongly) regular relation on hyper \(BCK\)-algebras and to investigate their categorical properties (quasi commutative diagram) via single-valued neutrosophic hyper \(BCK\)-ideals. Indeed, by using the single-valued neutrosophic hyper \(BCK\)-ideals, we define a congruence relation on (weak commutative) hyper \(BCK\)-algebras that under some conditions is strongly regular and the quotient of any (single-valued neutrosophic)hyper \(BCK\)-(sub)algebra via this relation is a (single-valued neutrosophic)(hyper \(BCK\)-subalgebra) \(BCK\)-(sub)algebra. (shrink)

In this article, we apply the notion of interval neutrosophic sets to ideal theory in BCK/BCI-algebras.

As a generation of fuzzy set, the notion of complex fuzzy set which is an innovative concept is introduced by Ramot, Milo, Friedman and Kandel. The purpose of this article is to apply complex fuzzy set to BCK/BCI-algebras. The notions of a complex subalgebra and a complex left reduced ideal in a BCK/BCI- algebra are introduced, and related properties are investigated. Characterizations of a complex subalgebra are provided, and the homomorphic image of a complex subalgebra and a complex left (...) reduced ideal. (shrink)

We show that every BCK-algebra can be embeded into a WMV - algebra and prove the interpolation and amalgamation properties for WMV - algebras. In proofs Gentzen-method is used. For information on BCK- algebras the reader is referred to K. Iseki, S. Tanaka [2] while for Gentzen- method to G. Takeuti [7]. I would like to thank Prof. A. Wronski for raising the problem.

The classical Glivenko theorem asserts that a propositional formula admits a classical proof if and only if its double negation admits an intuitionistic proof. By a natural expansion of the BCK-logic with negation we understand an algebraizable logic whose language is an expansion of the language of BCK-logic with negation by a family of connectives implicitly defined by equations and compatible with BCK-congruences. Many of the logics in the current literature are natural expansions of BCK-logic with negation. The validity of (...) the analogous of Glivenko theorem in these logics is equivalent to the validity of a simple one-variable formula in the language of BCK-logic with negation. (shrink)

In this paper we characterize the MV-algebras containing as subalgebras Post algebras of finitely many orders. For this we study cyclic elements in MV-algebras which are the generators of the fundamental chain of the Post algebras.

In the paper some varieties E s n of BCK-algebras with condition defined by W. H. Cornish [1] are considered. A characterization of simple algebras in E s 2 is given and some properties of simple algebras in E s 3 are indicated.

Quantum B-algebras, the partially ordered implicational algebras arising as subreducts of quantales, are introduced axiomatically. It is shown that they provide a unified semantic for non-commutative algebraic logic. Specifically, they cover the vast majority of implicational algebras like BCK-algebras, residuated lattices, partially ordered groups, BL- and MV-algebras, effect algebras, and their non-commutative extensions. The opposite of the category of quantum B-algebras is shown to be equivalent to the category of logical quantales, in the (...) way that every quantum B-algebra admits a natural embedding into a logical quantale, the enveloping quantale. Partially defined products of algebras related to effect algebras are handled efficiently in this way. The unit group of the enveloping quantale of a quantum B-algebra X is shown to be always contained in X, which gives a functorial subgroup X× of X. Similar subfunctors are obtained for the non-commutative extensions of BCK-algebras and effect algebras. The results of Galatos, Jónsson, and Tsinakis on the splitting of generalized BL-algebras into a semidirect product of a partially ordered group operating on an integral residuated poset are extended to a characterization of twisted semidirect products of a po-group by a quantum B-algebra. (shrink)

More general form of -neutrosophic ideal is introduced, and their properties are investigated. Relations between -neutrosophic ideal and )-neutrosophic ideal are discussed. Characterizations of )-neutrosophic ideal are discussed, and conditions for a neutrosophic set to be an )-neutrosophic ideal are displayed.

A new structure, called pseudo equality algebras, will be introduced. It has a constant and three connectives: a meet operation and two equivalences. A closure operator will be introduced in the class of pseudo equality algebras; we call the closed algebras equivalential. We show that equivalential pseudo equality algebras are term equivalent with pseudo BCK-meet-semilattices. As a by-product we obtain a general result, which is analogous to a result of Kabziński and Wroński: we provide an equational (...) characterization for the equivalence operations of pseudo BCK-meet-semilattices. Our result treats a much more general algebraic structure, namely, pseudo BCK-meet-semilattice instead of Heyting algebras, on the other hand, we also need to use the meet operation. Finally, we prove that the variety of pseudo equality algebras is a subtractive, 1-regular, arithmetical variety. (shrink)

GE algebras (generalized exchange algebras), transitive GE algebras (tGE algebras, for short) and aGE algebras (that is, GE algebrasverifying 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. (shrink)

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 (...) class='Hi'>algebras considered here are shown. (shrink)

Matthew Spinks [35] introduces implicative BCSK-algebras, expanding implicative BCK-algebras with an additional binary operation. Subdirectly irreducible implicative BCSK-algebras can be viewed as flat posets with two operations coinciding only in the 1- and 2-element cases, each, in the latter case, giving the two-valued implication truth-function. We introduce the resulting logic (for the general case) in terms of matrix methodology in §1, showing how to reformulate the matrix semantics as a Kripke-style possible worlds semantics, thereby displaying the distinction (...) between the two implications in the more familiar language of modal logic. In §§2 and 3 we study, from this perspective, the fragments obtained by taking the two implications separately, and – after a digression (in §4) on the intuitionistic analogue of the material in §3 – consider them together in §5, closing with a discussion in §6 of issues in the theory of logical rules. Some material is treated in three appendices to prevent §§1–6 from becoming overly distended. (shrink)

Łukasiewicz implication algebras are {→,1}-subreducts of Wajsberg algebras (MV-algebras). They are the algebraic counterpart of Super-Łukasiewicz Implicational logics investigated in Komori, Nogoya Math J 72:127–133, 1978. The aim of this paper is to study the direct decomposability of free Łukasiewicz implication algebras. We show that freely generated algebras are directly indecomposable. We also study the direct decomposability in free algebras of all its proper subvarieties and show that infinitely freely generated algebras are indecomposable, (...) while finitely free generated algebras can be only decomposed into a direct product of two factors, one of which is the two-element implication algebra. (shrink)

Recently, a new algebraic structure called pseudo-equality algebra has been defined by Jenei and Kóródi as a generalization of the equality algebra previously introduced by Jenei. As a main result, it was proved that the pseudo-equality algebras are term equivalent with pseudo-BCK meet-semilattices. We found a gap in the proof of this result and we present a counterexample and a correct version of the theorem. The correct version of the corresponding result for equality algebras is also given.

The following logics are the most noteworthy from the perspective of the calculus of combinators: the Hilbert’s positive implicational logic , the Church’s weak theory of implication , the BCK-logic, and the BCI-logic. Their significance is due to a certain correspondence between combinators and implicational formulas . The first three logics mentioned have been immensely investigated but it was not so in case of the remaining one. The BCI-logics was mentioned by A. N. Prior in the second edition of his (...) Formal Logic of 1962 where it was credited to C. A. Meredith and dated in 1956 . A. (shrink)

We show that the elimination rule for the multiplicative (or intensional) conjunction $\wedge$ is admissible in many important multiplicative substructural logics. These include LL m (the multiplicative fragment of Linear Logic) and RMI m (the system obtained from LL m by adding the contraction axiom and its converse, the mingle axiom.) An exception is R m (the intensional fragment of the relevance logic R, which is LL m together with the contraction axiom). Let SLL m and SR m be, respectively, (...) the systems which are obtained from LL m and R m by adding this rule as a new rule of inference. The set of theorems of SR m is a proper extension of that of R m , but a proper subset of the set of theorems of RMI m . Hence it still has the variable-sharing property. SR m has also the interesting property that classical logic has a strong translation into it. We next introduce general algebraic structures, called strong multiplicative structures, and prove strong soundness and completeness of SLL m relative to them. We show that in the framework of these structures, the addition of the weakening axiom to SLL m corresponds to the condition that there will be exactly one designated element, while the addition of the contraction axiom corresponds to the condition that there will be exactly one nondesignated element (in the first case we get the system BCK m , in the second - the system SR m ). Various other systems in which multiplicative conjunction functions as a true conjunction are studied, together with their algebraic counterparts. (shrink)

The quasivariety of BCK-algebras is widely known and investigated class of algebras. It is a natural semantic for the BCK-logic but there are also others quasivarieties of algebras with the above property and there are even some varieties among them. The aim of this note is to bring to the reader’s a attention the lattice they form. In what follows we shall only consider classes of algebras of type.

The purpose of this paper is to connect the proof theory and the model theory of a family of prepositional logics weaker than Heyting's. This family includes systems analogous to the Lambek calculus of syntactic categories, systems of relevant logic, systems related to BCK algebras, and, finally, Johansson's and Heyting's logic. First, sequent-systems are given for these logics, and cut-elimination results are proved. In these sequent-systems the rules for the logical operations are never changed: all changes are made in (...) the structural rules. Next, Hilbert-style formulations are given for these logics, and algebraic completeness results are demonstrated with respect to residuated lattice-ordered groupoids. Finally, model structures related to relevant model structures (of Urquhart, Fine, Routley, Meyer, and Maksimova) are given for our logics. These model structures are based on groupoids parallel to the sequent-systems. This paper lays the ground for a kind of correspondence theory for axioms of logics with implication weaker than Heyting's, a correspondence theory analogous to the correspondence theory for modal axioms of normal modal logics.Below is the sequel to the first part of the paper, which appeared in the previous issue of this journal (vol. 47 (1988), pp. 353–386). The first part contained sections on sequent-systems and Hilbert-formulations, and here is the third section on groupoid models. This second part is meant to be read in conjunction with the first part. (shrink)

We recapitulate (Section 1) some basic details of the system of implicative BCSK logic, which has two primitive binary implicational connectives, and which can be viewed as a certain fragment of the modal logic S5. From this modal perspective we review (Section 2) some results according to which the pure sublogic in either of these connectives (i.e., each considered without the other) is an exact replica of the material implication fragment of classical propositional logic. In Sections 3 and 5 we (...) show that for the pure logic of one of these implicational connectives two-in general distinct-consequence relations (global and local) definable in the Kripke semantics for modal logic turn out to coincide, though this is not so for the pure logic of the other connective, and that there is an intimate relation between formulas constructed by means of the former connective and the local consequence relation. (Corollary 5.8. This, as we show in an Appendix, is connected to the fact that the 'propositional operations' associated with both of our implicational connectives are close to being what R. Quackenbush has called pattern functions.) Between these discussions Section 4 examines some of the replacement-of-equivalents properties of the two connectives, relative to these consequence relations, and Section 6 closes with some observations about the metaphor of identical twins as applied to such pairs of connectives. (shrink)

In [9] it is proved the categorical isomorphism of two varieties: bounded commutative BCK-algebras and MV -algebras. The class of MV -algebras is the algebraic counterpart of the infinite valued propositional calculus L of Lukasiewicz . The main objective of the present paper is to study that isomorphism from the perspective of logic. The B-C-K logic is algebraizable and the quasivariety of BCKalgebras is the equivalent algebraic semantics for that logic . We call commutative B-C-K logic, briefly (...) cBCK, to the extension of B-C-K logic associated to the variety of commutative BCK–algebras. Moreover, we present the extension Boc of cBCK obtained by adding the axiom of “boundness”. We prove that the deductive system Boc is equivalent to L. We observe that cBCK admits two interesting extensions: the logic Boc, treated in this paper, which is equivalent to the system L of Lukasiewicz, and the logic Co that is naturally associated to the system Balo of `-groups . This constructions establish a link between L and Balo , that would be a logical approach to the categorical relationship between MV–algebras and `-groups. (shrink)

A syntactic derivation of Cornish identity from the axioms of HBCK is presented which amounts to a syntactic proof of Wronski's conjecture that naturally ordered BCK-algebras form a variety.