Implication Algebra Research Articles

Complete and atomic Tarski algebras

Tarski algebras, also known as implication algebras or semi-boolean algebras, are the $$\left\{ \rightarrow \right\} $$ -subreducts of Boolean algebras. In this paper we shall introduce and study the complete and atomic Tarski algebras. We shall prove a duality between the complete and atomic Tarski algebras and the class of covering Tarski sets, i.e., structures $$\left $$ , where X is a non-empty set and $${\mathcal {K}}$$ is non-empty family of subsets of X such that $$\bigcup {\mathcal {K}}=X$$ . This duality is a generalization of the known duality between sets and complete and atomic Boolean algebras. We shall also analize the case of complete and atomic Tarski algebras endowed with a complete modal operator, and we will prove a duality for these algebras.

Dual quantum B-algebras

Quantum B-algebras as implicational subreducts of quantales were introduced by Rump and Yang. They cover the majority of implicational algebras and provide a unified semantics for a wide class of algebraic logics. Some concepts for quantales survive in the framework of quantum B-algebras. In this paper, we first introduce the concept of dual quantum B-algebras (Girard quantum B-algebras). Next, we prove that every dual quantum B-algebra is a residuated poset and that complete dual quantum B-algebras and dual quantales are equivalent to each other. Further, we consider the construction of Girard quantum B-algebras from dual quantum B-algebras.

Q-Filters of Quantum B-Algebras and Basic Implication Algebras

The concept of quantum B-algebra was introduced by Rump and Yang, that is, unified algebraic semantics for various noncommutative fuzzy logics, quantum logics, and implication logics. In this paper, a new notion of q-filter in quantum B-algebra is proposed, and quotient structures are constructed by q-filters (in contrast, although the notion of filter in quantum B-algebra has been defined before this paper, but corresponding quotient structures cannot be constructed according to the usual methods). Moreover, a new, more general, implication algebra is proposed, which is called basic implication algebra and can be regarded as a unified frame of general fuzzy logics, including nonassociative fuzzy logics (in contrast, quantum B-algebra is not applied to nonassociative fuzzy logics). The filter theory of basic implication algebras is also established.

On weak implication algebra

In this paper, we introduce the notion of implication filter as a generalization of Boolean filter of first kind in Hilbert algebra and study it in detail. Also, we prove that F is an implication filter of a Hilbert algebra H if and only if every implicative filter of quotient algebra H / F is an implication filter. Finally, we generalized Boolean algebra and introduced a weak implication algebra, we prove that F is an implication filter of a Hilbert algebra H if and only if the quotient algebra H / F is a weak implication algebra. By suitable diagrams, we summarize the results of this paper and the previous results in these fields.

On A-subsets in lattice implication algebras

On A-subsets in lattice implication algebras

Maximal filter and its topology in lattice implication algebras

Maximal filter and its topology in lattice implication algebras

On the injective hulls of quantum B-algebras

Quantum B-algebras as implicational subreducts of quantales were introduced by Rump and Yang, which cover the majority of implicational algebras and provide a unified semantics for non-commutative algebraic logic. Due to the adjunction between multiplication and its residuals, they can be regarded as the implicational counterpart of partially ordered semigroups. By means of the injective hulls (envelopes) of quantum B-algebras, Rump gave the injective hulls of partially ordered semigroups. In this paper, one part of our work is to give new examples of quantum B-algebras where the lower sets do not form a quantale. Another part of our work is to consider the inverse problem of Rump's way, that is, we can also give the injective hulls of quantum B-algebras in terms of the injective hulls of partially ordered semigroups.

On extended implication groupoids

In this paper we generalize the notion of an implication groupoid and introduce ei-groupoid, and investigate related properties. We study the notion of filters in this structure and we give the construction of implication algebra from ei-groupoid. Finally, we prove that for distributive ei-groupoids filters coincide with ideals.

The Unitality of Quantum B-algebras

Quantum B-algebras as a generalization of quantales were introduced by Rump and Yang, which cover the majority of implicational algebras and provide a unified semantic for a wide class of substructural logics. Unital quantum B-algebras play an important role in the classification of implicational algebras. The main purpose of this paper is to construct unital quantum B-algebras from non-unital quantum B-algebras.

Implicative Almost Distributive Lattice

In this paper, we introduce the concept of Implicative Almost Distributive Lattices (IADLs) as a generalization of implicative algebra in the class of Almost Distributive Lattices. We discuss some properties of IADL and derive some equivalent conditions in IADLs. We also discuss some characterizations of IADL to become an implicative algebra.

Implicative subreducts of MV-algebras: free and weakly projective objects

In this article, we explore in some detail the free and weakly projective objects of the variety of Łukasiewicz implication algebras (the implicative subreducts of MV-algebras). We review the two already known descriptions of finitely generated free algebras, giving new insights into their structure and their connection, as well as providing new proofs of the characterizations. We give a representation theorem for weakly projective algebras as algebras of certain McNaughton functions restricted to rational polyhedra and prove that finitely generated weakly projective algebras coincide with finitely presented ones. We also prove that finite chains are the only totally ordered weakly projective examples in this variety.

Implicative algebras and Heyting algebras can be residuated lattices

The commutative residuated lattices were first introduced by M. Ward and R.P. Dilworth as generalization of ideal lattices of rings. Complete studies on residuated lattices were developed by H. Ono, T. Kowalski, P. Jipsen and C. Tsinakis. Also, the concept of lattice implication algebra is due to Y. Xu. And Luitzen Brouwer founded the mathematical philosophy of intuitionism, which believed that a statement could only be demonstrated by direct proof. Arend Heyting, a student of Brouwer’s, formalized this thinking into his namesake algebras. In this paper, we investigate the relationship between implicative algebras, Heyting algebras and residuated lattices. In fact, we show that implicative algebras and Heyting algebras can be described as residuated lattices.

Distributive residuated frames and generalized bunched implication algebras

We show that all extensions of the (non-associative) Gentzen system for distributive full Lambek calculus by simple structural rules have the cut elimination property. Also, extensions by such rules that do not increase complexity have the finite model property, hence many subvarieties of the variety of distributive residuated lattices have decidable equational theories. For some other extensions, we prove the finite embeddability property, which implies the decidability of the universal theory, and we show that our results also apply to generalized bunched implication algebras. Our analysis is conducted in the general setting of residuated frames.

On BI-Algebras

Abstract In this paper, we introduce a new algebra, called a BI-algebra, which is a generalization of a (dual) implication algebra and we discuss the basic properties of BI-algebras, and investigate ideals and congruence relations.

On derivations of linguistic truth-valued lattice implication algebras

Algebraic characters of linguistic truth-valued lattice implication algebras (L-LIAs) are focused in this paper. Concretely, the concept of derivation coming from analytic theory is introduced in L-LIAs, and then some properties of derivations are given in L-LIAs. Subsequently, some special derivations (such as isotone deri-vation, embedding derivation, isomorphism derivation and regular derivation) are discussed in L-LIAs along with the relations among them. The relations between derivations and lattice implication homomorphisms are also investigated in L-LIAs. At the end, the derivations of some algebraic substructures are described in L-LIAs.

2-dimension linguistic computational model with 2-tuples for multi-attribute group decision making

The 2-dimension linguistic information includes two common linguistic labels. One dimension is used for describing the evaluation result of alternatives provided by the decision maker, and the other is used for describing the self-assessment of the decision maker on the reliability of the given evaluation result. In order to deal with comparability and incomparability of 2-dimension linguistic labels (2DLLs), a 2-dimension linguistic lattice implication algebra (2DL-LIA) is constructed as a linguistic evaluation set with lattice structure. In this paper, a 2DLL is firstly represented by two 2-tuples in a 2DL-LIA for more precise computing and aggregating 2-dimension linguistic information. This model allows a continuous representation of 2DLL on its domain, therefore, it can represent any continuous 2-dimension linguistic information obtained in the aggregation process. Next, two new 2-dimension linguistic aggregation operators, including 2-dimension linguistic weighted arithmetic aggregation (2DLWAA) operator and 2-dimension linguistic ordered weighted arithmetic aggregation (2DLOWAA) operator, are developed, and then some desirable properties of the operators are studied. Subsequently, based on 2DLWAA and 2DLOWAA operators, a decision making approach is provided to solve multi-attribute group decision making (MAGDM) problem with 2DLL assessment. Finally, an illustrative example is provided to show the concrete steps of the developed decision making approach and to demonstrate the practicality and the flexibility of this proposal by comparing with existing decision making approaches.

VAGUE PRIME LI-IDEALS OF LATTICE IMPLICATION ALGEBRAS

In order to investigate a many valued logical system whose proportional value is given in a lattice, in 1993 Y. Xu [12] first established the lattice implication algebra by combining lattice and implication algebra, and explored many useful structures. The ideal theory serves a vital function for the development of lattice implication algebras. Y. Xu, Y.B. Jun and E.H. Roh [6] introduced the notion of LI ideals of a lattice implication algebras. In particular, Y.B. Jun

Algebraic functions in Łukasiewicz implication algebras

In this article we study algebraic functions in [Formula: see text]-subreducts of MV-algebras, also known as Łukasiewicz implication algebras. A function is algebraic on an algebra [Formula: see text] if it is definable by a conjunction of equations on [Formula: see text]. We fully characterize algebraic functions on every Łukasiewicz implication algebra belonging to a finitely generated variety. The main tool to accomplish this is a factorization result describing algebraic functions in a subproduct in terms of the algebraic functions of the factors. We prove a global representation theorem for finite Łukasiewicz implication algebras which extends a similar one already known for Tarski algebras. This result together with the knowledge of algebraic functions allowed us to give a partial description of the lattice of classes axiomatized by sentences of the form [Formula: see text] within the variety generated by the 3-element chain.

Rough soft lattice implication algebras and corresponding decision making methods

Soft set theory and rough set theory are two mathematical tools for dealing with uncertainties. By combining rough sets and soft sets, Feng put forth rough soft set theory. In this paper, we apply rough soft set theory to lattice implication algebras. Rough soft (implicative, associative) filters with respect to a filter over lattice implication algebras are investigated. Finally, we put forward two kinds of decision making methods for rough soft lattice implication algebras. In particular, two applied examples are also given. Maybe it would be served as a foundation of more complicated soft set models in decision making methods.

VAGUE POSITIVE IMPLICATIVE AND VAGUE ASSOCIATIVE LI - IDEALS OF LATTICE IMPLICATION ALGEBRAS

We vaguefy the concepts of positive implicative LI - ideals and associative LI - ideals of lattice implication algebras. We investigate the con- nections to related classes. Also we provide the equivalent conditions for both vague positive implicative LI - ideals and vague associative LI - ideals. Ex- tension property of a vague positive implicative LI - ideal is built. Also We studied some equivalent conditions for vague implicative LI - ideals.And finally, we prove that in a lattice implication algebra, the concepts of vague implicative and vague positive implicative LI - ideals coincide.