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 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)

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)

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.

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.

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.

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)

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.

The present paper introduces and studies the variety [MATHEMATICAL SCRIPT CAPITAL W]ℋn of n-linear weakly Heyting algebras. It corresponds to the algebraic semantic of the strict implication fragment of the normal modal logic K with a generalization of the axiom that defines the linear intuitionistic logic or Dummett logic. Special attention is given to the variety [MATHEMATICAL SCRIPT CAPITAL W]ℋ2 that generalizes the linear Heyting algebras studied in [10] and [12], and the linear Basic algebras introduced in [2].

In this paper we shall introduce the variety FWHA of frontal weak Heyting algebras as a generalization of the frontal Heyting algebras introduced by Leo Esakia in [ 10 ]. A frontal operator in a weak Heyting algebra A is an expansive operator τ preserving finite meets which also satisfies the equation $${\tau(a) \leq b \vee (b \rightarrow a)}$$, for all $${a, b \in A}$$. These operators were studied from an algebraic, logical and topological point of view (...) by Leo Esakia in [ 10 ]. We will study frontal operators in weak Heyting algebras and we will consider two examples of them. We will give a Priestley duality for the category of frontal weak Heyting algebras in terms of relational spaces $${\langle X, \leq, T, R \rangle}$$ where $${\langle X, \leq, T \rangle}$$ is a WH -space [ 6 ], and R is an additional binary relation used to interpret the modal operator. We will also study the WH -algebras with successor and the WH -algebras with gamma. For these varieties we will give two topological dualities. The first one is based on the representation given for the frontal weak Heyting algebras. The second one is based on certain particular classes of WH -spaces. (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 ?

We use the left self‐distributive axiom to introduce and study a special class of weak Heyting algebras, called self‐distributive weak Heyting algebras (SDWH‐algebras). We present some useful properties of SDWH‐algebras and obtain some equivalent conditions of them. A characteristic of SDWH‐algebras of orders 3 and 4 is given. Finally, we study the relation between the variety of SDWH‐algebras and some of the known subvarieties of weak Heyting algebras such as the variety of Heyting algebras, the variety of (...) basic algebras, the variety of subresiduated lattices, the variety of reflexive WH‐algebras (RWH‐algebras), and the variety of transitive WH‐algebras (TWH‐algebras). (shrink)

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)

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.

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.

Modifying the descriptive and theoretical generalizations of Relativized Minimality, we argue that a significant subset of weak island violations arise when an extracted phrase should scope over some intervener but is unable to. Harmless interveners seem harmless because they can support an alternative reading. This paper focuses on why certain wh-phrases are poor wide scope takers, and offers an algebraic perspective on scope interaction. Each scopal element SE is associated with certain operations (e.g., not with complements). When a wh-phrase (...) scopes over some SE, the operations associated with that SE are performed in its denotation domain. The requisite operations may or may not be available in a domain, however. We present an empirical analysis of a variety of wh-phrases. It is argued that the wh-phrases that escape all weak islands (i.e., can scope over any intervener) are those that range over individuals, the reason being that all Boolean operations are defined for their domain. Collectives, manners, amounts, numbers, etc. all denote in domains with fewer operations and are thus selectively sensitive to scopal interveners—a “semantic relativized minimality effect”. (shrink)

We show that the variety of n -dimensional weakly higher order cylindric algebras, introduced in Németi [9], [8], is finitely axiomatizable when n > 2. Our result implies that in certain non-well-founded set theories the finitization problem of algebraic logic admits a positive solution; and it shows that this variety is a good candidate for being the cylindric algebra theoretic counterpart of Tarski’s quasi-projective relation algebras.

Built on the foundations laid by Peirce, Schröder, and others in the 19th century, the modern development of relation algebras started with the work of Tarski and his colleagues [21, 22]. They showed that relation algebras can capture strong first‐order theories like ZFC, and so their equational theory is undecidable. The less expressive class WA of weakly associative relation algebras was introduced by Maddux [7]. Németi [16] showed that WA's have a decidable universal theory. There has been extensive research on (...) increasing the expressive power of WA by adding new operations [1, 4, 11, 13, 20]. Extensions of this kind usually also have decidable universal theories. Here we give an example – extending WA's with set‐theoretic projection elements – where this is not the case. These “logical” connectives are set‐theoretic counterparts of the axiomatic quasi‐projections that have been investigated in the representation theory of relation algebras [22, 6, 19]. We prove that the quasi‐equational theory of the extended class PWA is not recursively enumerable. By adding the difference operator D one can turn WA and PWA to discriminator classes where each universal formula is equivalent to some equation. Hence our result implies that the projections turn the decidable equational theory of “WA + D ” to non‐recursively enumerable (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim). (shrink)

In this paper we investigate some basic semantic and syntactic conditions characterizing the equivalence connective. In particular we define three basic classes of algebras: the class of weak equivalential algebras, the class of equivalential algebras and the class of regular equivalential algebras.Weak equivalential algebras can be used to study purely equivalential fragments of relevant logics and strict equivalential fragments of some modal logics. Equivalential algebras are suitable to study purely equivalential fragment of BCI and BCK logic. A subclass (...) of the class of regular equivalential algebras is suitable to study equivalential fragments of ukasiewicz logics. Some subvarieties of the class of regular equivalential algebras provide natural semantics for equivalential fragments of the intuitionistic prepositional logic and various intermediate logics. (shrink)

In this paper we shall introduce the variety WQS of weak-quasi-Stone algebras as a generalization of the variety QS of quasi-Stone algebras introduced in [9]. We shall apply the Priestley duality developed in [4] for the variety N of ¬-lattices to give a duality for WQS. We prove that a weak-quasi-Stone algebra is characterized by a property of the set of its regular elements, as well by mean of some principal lattice congruences. We will also determine the (...) simple and subdirectly irreducible algebras. (shrink)

For (finitary) deductive systems, we formulate a signature‐independent abstraction of the weak excluded middle law (WEML), which strengthens the existing general notion of an inconsistency lemma (IL). Of special interest is the case where a quasivariety algebraizes a deductive system ⊢. We prove that, in this case, if ⊢ has a WEML (in the general sense) then every relatively subdirectly irreducible member of has a greatest proper ‐congruence; the converse holds if ⊢ has an inconsistency lemma. The result extends, (...) in a suitable form, to all protoalgebraic logics. A super‐intuitionistic logic possesses a WEML iff it extends. We characterize the IL and the WEML for normal modal logics and for relevance logics. A normal extension of has a global consequence relation with a WEML iff it extends, while every axiomatic extension of with an IL has a WEML. (shrink)

We propose a definition of weak o-minimality for structures expanding a Boolean algebra. We study this notion, in particular we show that there exist weakly o-minimal non o-minimal examples in this setting.

We show that there is a restriction, or modification of the finite-variable fragments of First Order Logic in which a weak form of Craig's Interpolation Theorem holds but a strong form of this theorem does not hold. Translating these results into Algebraic Logic we obtain a finitely axiomatizable subvariety of finite dimensional Representable Cylindric Algebras that has the Strong Amalgamation Property but does not have the Superamalgamation Property. This settles a conjecture of Pigozzi [12].

Paraconsistent Weak Kleene logic is the 3-valued logic with two designated values defined through the weak Kleene tables. This paper is a first attempt to investigate PWK within the perspective and methods of abstract algebraic logic. We give a Hilbert-style system for PWK and prove a normal form theorem. We examine some algebraic structures for PWK, called involutive bisemilattices, showing that they are distributive as bisemilattices and that they form a variety, \, generated by the 3-element algebra (...) WK; we also prove that every involutive bisemilattice is representable as the Płonka sum over a direct system of Boolean algebras. We then study PWK from the viewpoint of AAL. We show that \ is not the equivalent algebraic semantics of any algebraisable logic and that PWK is neither protoalgebraic nor selfextensional, not assertional, but it is truth-equational. We fully characterise the deductive filters of PWK on members of \ and the reduced matrix models of PWK. Finally, we investigate PWK with the methods of second-order AAL—we describe the class \ of PWK-algebras, algebra reducts of basic full generalised matrix models of PWK, showing that they coincide with the quasivariety generated by WK—which differs from \—and explicitly providing a quasiequational basis for it. (shrink)

In this note we prove that the abstract property "weakly subdirectly indecomposable" does not characterize the class IWs α of weak cylindric set algebras. However, we give another (similar) abstract property characterizing IWs α . The original property does characterize the directed unions of members of $\mathrm{IWs}_alpha \operatorname{iff} \alpha$ is countable. Free algebras will be shown to satisfy the original property.

We investigate the theories of linear algebra, which were originally defined to study the question of whether commutativity of matrix inverses has polysize Frege proofs. We give sentences separating quantified versions of these theories, and define a fragment in which we can interpret a weak theory V 1 of bounded arithmetic and carry out polynomial time reasoning about matrices - for example, we can formalize the Gaussian elimination algorithm. We show that, even if we restrict our language, proves (...) the commutativity of inverses. (shrink)

In this paper we introduced various classes of weakly associative relation algebras with polyadic composition operations. Among them is the class RWA of representable weakly associative relation algebras with polyadic composition operations. Algebras of this class are relativized representable relation algebras augmented with an infinite set of operations of increasing arity which are generalizations of the binary relative composition. We show that RWA is a canonical variety whose equational theory is decidable.

We give a complete characterization of Boolean algebras admitting weak elimination of imaginaries in terms of elementary invariants.

In this paper we shall introduce the variety WQS of weak-quasi-Stone algebras as a generalization of the variety QS of quasi-Stone algebras introduced in [9]. We shall apply the Priestley duality developed in [4] for the variety N of ¬-lattices to give a duality for WQS. We prove that a weak-quasi-Stone algebra is characterized by a property of the set of its regular elements, as well by mean of some principal lattice congruences. We will also determine the (...) simple and subdirectly irreducible algebras. (shrink)

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.

It is known that for all finite n ≥ 5, there are relation algebras with n-dimensional relational bases but no weak representations. We prove that conversely, there are finite weakly representable relation algebras with no n-dimensional relational bases. In symbols: neither of the classes RA n and wRRA contains the other.

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)