Implication Algebra Research Articles

Non-commutative logical algebras and algebraic quantales

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.

Multiary α-Resolution Principle for a Lattice-Valued Logic

This paper focuses on resolution-based automated reasoning theory in a lattice-valued logic system with truth values that are defined in a lattice-valued logical algebraic structure-lattice implication algebras (LIAs) - which essentially aims to extend the classical logic to handle automated deduction under an uncertain environment. Concretely, we investigate a generalization of the known conjunctive normal form (CNF) in the classical logic representation that we call the generalized conjunctive normal form (GCNF), which aims to characterize the constants and implication connectives that are essentially different from the ones in classical logic. We then extend the established resolution principle at a certain truth-value level α (called α-resolution) in this lattice-valued logic to a more general form, i.e., from binary α -resolution to multiary α-resolution. The extension to multiary α-resolution starts from lattice-valued propositional logic LP(X), while its theorems of both soundness and completeness are proved. Multiary α-resolution principle is then further established in the corresponding lattice-valued first-order logic LF(X), along with its soundness theorem, lifting lemma, and completeness theorem. Meanwhile, an important result that multiary α-resolution principle in LF(X) can be equivalently transformed into that in LP(X) to some extent is obtained. All these works will theoretically support the establishment of an automated reasoning algorithm and its implementation with further applications into automated deduction and decision-making problems under uncertainty.

An axiomatizable logical foundation for lattice-ordered qualitative linguistic approach for reasoning with words

It is commonsense that people do express, think, reason, and make decision directly using linguistic terms in natural language rather than using numerical quantification. How to establish the formalized approach imitating the common way of human being’s to manipulate directly linguistic terms without numerical approximation to provide the rational decision is still an open research area. Investigations on the algebraic structure of linguistic term set for varied decision making applications (especially in social science) still lack a formalism for development of strict linguistic valued logic system as a theoretical foundation and its approximate reasoning scheme in practice. To attain this goal we characterize and construct a typical structure of commonly used linguistic term sets in natural language by a lattice-ordered algebra structure – lattice implication algebra (LIA), where Łukasiewicz implication algebra, as a special case of LIA, plays a substantial role. By using Łukasiewicz logic’s axiomatizability in terms of Pavelka type fuzzy logic, we propose a new axiomatizable lattice ordered qualitative linguistic truth-valued logic system based on LIA to place an important foundation for further establishing formal linguistic truth-valued logic based approximate reasoning and decision making with applications. This proposed logic system has a distinct feature of handling comparable or incomparable linguistic terms directly without numerical quantification, will be especially beneficial for perception-based decision making processes. It attempts to enhance the quantitative theory of decision science with qualitative, algebraic and logic-oriented approaches to achieve reasoning with words.

Quantum B-algebras

Abstract The concept of quantale was created in 1984 to develop a framework for non-commutative spaces and quantum mechanics with a view toward non-commutative logic. The logic of quantales and its algebraic semantics manifests itself in a class of partially ordered algebras with a pair of implicational operations recently introduced as quantum B-algebras. Implicational algebras like pseudo-effect algebras, generalized BL- or MV-algebras, partially ordered groups, pseudo-BCK algebras, residuated posets, cone algebras, etc., are quantum B-algebras, and every quantum B-algebra can be recovered from its spectrum which is a quantale. By a two-fold application of the functor “spectrum”, it is shown that quantum B-algebras have a completion which is again a quantale. Every quantale Q is a quantum B-algebra, and its spectrum is a bigger quantale which repairs the deficiency of the inverse residuals of Q. The connected components of a quantum B-algebra are shown to be a group, a fact that applies to normal quantum B-algebras arising in algebraic number theory, as well as to pseudo-BCI algebras and quantum BL-algebras. The logic of quantum B-algebras is shown to be complete.

Soft Lattice Implication Subalgebras

In this paper, we introduce the concepts of soft lattice implication subalgebras, endow a parameter set as a lattice implication algebra, and further discuss its equivalent characterization. Then, new operations of soft lattice implication subalgebras are introduced, under which two soft lattice implication subalgebras is also a soft lattice implication subalgebra. Finally, the concepts of image and preimage of a soft lattice implication subalgebra and their properties are presented.

IFI-ideals of lattice implication algebras

Abstract The notion of IFI–ideal is introduced in lattice implication algebras. Firstly, the equivalent conditions of IF–ideals and IFI–ideals are given in lattice implication algebras. Then the proposition of IFI–ideal is investigated in lattice implication algebras. Next, the relations between IFI–ideal and IF–ideal, between IFI–ideal and IFI–filter, between IFI–ideal and fuzzy impilcative ideals, between IFI–ideal and implicative ideals are discussed in lattice implication algebras. Moreover, the extension theorem of IFI–ideals is obtained, and Ψ(L) which is composed of all IFI–ideals constitutes a closure system. Finally, we prove that ∀α ∈ [0;1], A = (µ0,α ; ) is an IFI–ideal of lattice implication algebra L if and only if L is a lattice H implication algebra.

On an algebra of linguistic truth-valued intuitionistic lattice-valued logic

In this paper, we propose a kind of linguistic truth-valued intuitionistic fuzzy lattice based on the point view of intuitionistic fuzzy set and linguistic truth-valued lattice implication algebra. As an algebra fundament of linguistic truth-valued intuitionistic fuzzy logic, some properties of linguistic truth-valued intuitionistic fuzzy algebra are discussed. The results show that linguistic truth-valued intuitionistic fuzzy lattice is a residual lattice, but it is not MTL-algebra, R0-algebra, BL-algebra, MV-algebra and quasi lattice implication algebra.

Determination of 3-Ary -Resolution in Lattice-valued Propositional Logic LP(X)

Abstract One of key issues for α-n(t) ary resolution automated reasoning based on lattice-valued logic with truth-value in a lattice implication algebra is to investigate the α-n(t) ary resolution of some generalized literals. In this article, the determination of α-resolution of any 3-ary generalized literals which include the implication operators not more than 2 in LP(X). It not only lay the foundation for practical implementation of automated reasoning algorithm in LP(X), but also provides the strong support for α-n(t) ary resolution automated reasoning approaches.

On the algebraic structure of binary lattice-valued fuzzy relations

From a general algebraic point of view, this paper aims at providing an algebraic analysis for binary lattice-valued relations based on lattice implication algebras--a kind of lattice-valued propositional logical algebra. By abstracting away from the concrete lattice-valued relations and the operations on them, such as composition and converse, the notion of lattice-valued relation algebra is introduced, LRA for short. The reduct of an LRA is a lattice implication algebra. Such an algebra generalizes Boolean relation algebras by general distributive lattices and can provide a fundamental algebraic theory for establishing lattice-valued first-order logic. Some important results are generalized from the classical case. The notion of cylindric filter is introduced and the generated cylindric filters are characterized.

Implication operators on the set of ∨-irreducible element in the linguistic truth-valued intuitionistic fuzzy lattice

We construct a kind of linguistic truth-valued intuitionistic fuzzy lattice based on linguistic truth-valued lattice implication algebras to deal with linguistic truth values. We get some properties of implication operators on the set of ∨-irreducible elements. And furthermore the implication operators on the linguistic truth-valued intuitionistic fuzzy lattice are discussed. The proposed system can better express both comparable and incomparable information. Also it can deal with both positive and negative evidences which are represented by linguistic truth values at the same time during the information processing system.

Syntax theory of finite lattice-valued propositional logic

In this paper, we establish the graded syntax theory of lattice-valued propositional logic based on finite lattice implication algebras, define the notions of syntactic consequence operation and formal proof, and develop a kind of graded finite lattice-valued propositional calculus. By generalizing classic provable equivalence relation, we present a kind of generalized provable equivalence relation, and establish the corresponding quotient algebra. Finally, we establish the generalized deduction theorem by syntactic consequence operation, and establish the completeness in Pavelka’s sense based on finite chains.

Lattice-valued modal propositional logic based on M-lattice implication algebras

The modal lattice implication algebra (i. e., M-lattice implication algebra) is introduced and its properties are investigated. The modal lattice-valued propositional logical system is introduced by considering the M-lattice implication algebra as the valuation field, and the syntax and semantic of the logical system are discussed, respectively.

Ideal resolution principle for lattice-valued first-order logic based on lattice implication algebra

As a continuate work, ideal-based resolution principle for lattice-valued first-order logic system LF(X) is proposed, which is an extension of α-resolution principle in lattice-valued logic system based on lattice implication algebra. In this principle, the resolution level is an ideal of lattice implication algebra, instead of an element in truth-value field. Moreover, the soundness theorem is given. In the light of lifting lemma, the completeness theorem is established. This can provide a new tool for automated reasoning.

Associativity, Commutativity and Symmetry in Residuated Structures

In this paper we develop the study of extended-order algebras, recently introduced by C. Guido and P. Toto, which are implicative algebras that generalize all the widely considered integral residuated structures. Particular care is devoted to the requirement of completeness that can be obtained by the MacNeille completion process. Associativity, commutativity and symmetry assumptions are characterized and their role is discussed toward the structure of the algebra and of its completion. As an application, further operations corresponding to the logical connectives of conjunction negation and disjunction are considered and their properties are investigated, either assuming or excluding the additional conditions of associativity, commutativity and symmetry. An overlook is also devoted to the relationship with other similar structures already considered such as implication algebras (in particular Heyting algebras), BCK algebras, quantales, residuated lattices and closed categories.

Implicative Algebras

In this paper we introduce the concept of implicative algebras which is an equivalent definition of lattice implication algebra of Xu (1993) and further we prove that it is a regular Autometrized Algebra. Further we remark that the binary operation → on lattice implicative algebra can never be associative.Keywords: Implicative algebra, lattice implication algebras, Autometrized Algebras, regular Autometrized Algebras

Extended-order algebras and fuzzy implicators

In this work we reconsider the notion of implicator in a complete lattice L and discuss its properties, taking into account the viewpoint of the implication operation of classes of (weak) extended-order algebras, introduced by C. Guido and P. Toto and included in the class of implicative algebras considered by E. Rasiowa. In fact, such an implication, that is an extension of an order relation, can be viewed as an implicator in L, whose properties depend on those characterizing the structure of the algebra. We also propose in a (weak) right-distributive complete extended-order algebra $$(L,\rightarrow,\top)$$ with adjoint product ? a relative implication as a further implicator beyond $${\rightarrow}{.}$$ The relative implication allows an extension of the inclusion relation between L-sets A and B, different from the subsethood degree, that consists in seeing to which extent A is included in its conjunction with B. Moreover, we introduce in $$(L,\rightarrow,\top)$$ a further binary operation that we call conditional conjunction that can be read as "a" and "b, given a", which motivates the term we have chosen to denote it. This operation, strictly related to the divisibility condition of BL-algebras, satisfies most conditions usually asked to a conjunction and it is well behaved with the adjoint product ? and the meet operation ?.

α-ordered linear resolution method for lattice-valued logic system based on lattice implication algebra

An α-ordered linear resolution method for lattice-valued logic based on lattice implication algebra is proposed to provide an efficient resolution approach for resolution-based mechanical theorem proving. Firstly, some essential concepts for α-ordered linear resolution are given. The α-ordered linear resolution method for lattice-valued propositional logic system LP(X) based on lattice implication algebra is presented. Both soundness and weak completeness theorems are established. Secondly, the lifting lemma is established from LP(X) to LF(X), which is then used for obtaining the weak completeness theorem of α-ordered linear resolution method in LF(X) based on lattice implication algebra. Finally, an α-ordered linear resolution automated reasoning algorithm for lattice-valued logic system based on lattice implication algebra is designed.

α-Generalized lock resolution method in linguistic truth-valued lattice-valued logic

Abstract This paper focuses on effcient non-clausal resolution-based automated reasoning methods and algorithms for a lattice-ordered linguistic truth-valued logic, which corresponds to extensions of α-lock resolution. Firstly, α-generalized lock resolution is proposed for lattice-valued propositional logic and first order logic, respectively, along with their concepts, soundness and completeness. Then, α-generalized lock resolution for first order linguistic truth-valued lattice-valued logic LV(n×2) F(X) is equivalently transformed into that for propositional logic Ln P(X), which can greatly reduce the complexity of the resolution procedure. Finally, α-generalized linear semi-lock resolution is discussed, and its general algorithm is also contrived. This work provides more efficient and natural resolution automated reasoning scheme in linguistic truth-valued logic based on lattice implication algebra with the aim at establishing formal tools for symbolic natural language processing.

Logic of Quantum Mechanics for Information Technology Field

Quantum mechanics is a branch of physics for a mathematical description of the particle wave, and it is applied to information technology such as quantum computer, quantum information, quantum network and quantum cryptography, etc. In 1936, Garrett Birkhoff and John von Neumann introduced the logic of quantum mechanics (quantum logic) in order to investigate projections on a Hilbert space. As another type of quantum logic, orthomodular implication algebra was introduced by Chajda et al. This algebra has the logical implication as a binary operation. In pure mathematics, there are many algebras such as Hilbert algebras, implicative models, implication algebras and dual BCK-algebras (DBCK-algebras), which have the logical implication as a binary operation. In this paper, we introduce the definitions and some properties of those algebras and clarify the relations between those algebras. Also, we define the implicative poset which is a generalization of orthomodular implication algebras and DBCK-algebras, and research properties of this algebraic structure.

Filteristic Soft Lattice Implication Algebras

Applying the idea of soft set theory to lattice implication algebras, the novel concept of (implicative) filteristic soft lattice implication algebras which related to (implicative) filter(for short, (IF -)F -soft lattice implication algebras) are introduced. Basic properties of (IF -)F -soft lattice implication algebras are derived. Two kinds of fuzzy filters (i.e.(∈,∈ ∨qk)((∈,∈ ∨ qk))-fuzzy (implicative) filter) of L are introduced, which are generalizations of fuzzy (implicative) filters. Some characterizations for a soft set to be a (IF -)F -soft lattice implication algebra are provided. Analogously, this idea can be used in other types of filteristic lattice implication algebras (such as fantastic (positive implicative) filteristic soft lattice implication algebras). Keywords—Soft set; (implicative) filteristic lattice implication algebras; fuzzy (implicative) filters; ((∈,∈ ∨qk)) (∈,∈ ∨ qk)-fuzzy (implicative) filters.