Implication Algebra Research Articles

Lack of associative filters in lattice implication algebras

Jun et al. (Bull Korean Math Soc 35(1):53–61, 1998) introduced the notion of associative filters in lattice implication algebra and studied some of its properties. In this paper, we show that there is no associative filter in lattice implication algebra.

Fuzzy Prime Filters of Lattice Implication Algebras

In this paper, some properties of fuzzy filters are given. Besides, the structure of fuzzy filters are further studied. And finally, the concept of fuzzy prime filter is proposed with some equivalent conditions of fuzzy prime filters obtained.

Formula omitted] resolution method for a lattice-valued first-order logic

This paper focuses on resolution-based automated reasoning approaches in a lattice-valued first-order logic LF( X) with truth-values defined in a logical algebraic structure—lattice implication algebra (LIA), which aims at providing the logic foundation to represent and handle both imprecision and incomparability. In order to improve the efficiency of α - resolution approach proposed for LF( X), firstly the concepts of α - lock resolution principle and deduction are introduced for lattice-valued propositional logic LP( X) based on LIA, along with its soundness and weak completeness theorems. Then all the results are extended into LF( X) by using Lifting Lemma. Finally an α - lock resolution automated reasoning algorithm in LF( X) is proposed for the implementation purpose. This work provides a theoretical foundation for more efficient resolution-based automated reasoning algorithm in lattice-valued logic LF( X).

Quasivarieties and Congruence Permutability of Łukasiewicz Implication Algebras

In this paper we study some questions concerning ?ukasiewicz implication algebras. In particular, we show that every subquasivariety of ?ukasiewicz implication algebras is, in fact, a variety. We also derive some characterizations of congruence permutable algebras. The starting point for these results is a representation of finite ?ukasiewicz implication algebras as upwardly-closed subsets in direct products of MV-chains.

Congruence Relations on Lattice Implication Algebras

Lattice implication algebra is an important logic algebra, congruence relations is one of important contents in it. The basic properties and the structures of general congruence relations on lattice implication algebras are discussed; The results that a lattice implication algebra is congruence-permutable is obtained.

On -satisfiability and its -lock resolution in a finite lattice-valued propositional logic

Automated reasoning issues are addressed for a finite lattice-valued propositional logic LnP(X) with truth-values in a finite lattice-valued logical algebraic structure—lattice implication algebra. We investigate extended strategies and rules from classical logic to LnP(X) to simplify the procedure in the semantic level for testing the satisfiability of formulas in LnP(X) at a certain truth-value level α (α-satisfiability) while keeping the role of truth constant formula played in LnP(X). We propose a lock resolution method at a certain truth-value level α (α-lock resolution) in LnP(X) and have proved its theorems of soundness and weak completeness, respectively. We provide more efficient resolution based automated reasoning in LnP(X) and key supports for α-resolution-based automated reasoning approaches and algorithms in lattice based linguistic truth-valued logic.

New Types of Fuzzy Filter on Lattice Implication Algebras

Extending the {\it belongs} to ($\in$) relation and {\itquasi-coincidence with}($q$) relation between fuzzy points and a fuzzy subsets, the concept of $(\alpha, \beta)$-fuzzy filters and $(\overline{\alpha}, \overline{\beta})$-fuzzy filters of lattice implication algebras are introduced, where$\alpha,\beta\in\{\in_{h},q_{\delta},\in_{h}\veeq_{\delta},\in_{h}\wedge q_{\delta}\}$,$\overline{\alpha},\overline{\beta}\in\{\overline{\in_{h}},\overline{q_{\delta}},\overline{\in_{h}}\vee\overline{q_{\delta}},\overline{\in_{h}}\wedge\overline{q_{\delta}}\}$ but $\alpha\neq \in_{h}\wedge q_{\delta}$, $\overline{\alpha}\neq\overline{ \in_{h}}\wedge\overline{q_{\delta}}$, respectively, and some related properties are investigated. Some equivalent characterizations of these generalized fuzzy filters are derived. Finally, the relations among these generalized fuzzy filters are discussed. Special attention to $(\in_{h},\in_{h}\vee q_{\delta})$-fuzzy filter and $(\overline{\in_{h}},\overline{\in_{h}}\vee\overline{q_{\delta}})$-fuzzy filter are generalizations of $(\in,\in\vee q)$-fuzzy filter and$(\overline{\in},\overline{\in}\vee \overline{q})$-fuzzy filter,respectively.

A new type of fuzzy filter on lattice implication algebras

A new type of fuzzy filter on lattice implication algebras

ALGEBRAICALLY EXPANDABLE CLASSES OF IMPLICATION ALGEBRAS

An algebraically expandable class is a class of algebras axiomatizable by a set of sentences of the form ∀∃! ∧ p = q. We find all algebraically expandable classes in the variety of implication algebras. A representation result for finite implication algebras is proved, and we also give a characterization of the class of congruence permutable implication algebras.

Rough implication operator based on strong topological rough algebras

The role of topological De Morgan algebra in the theory of rough sets is investigated. The rough implication operator is introduced in strong topological rough algebra that is a generalization of classical rough algebra and a topological De Morgan algebra. Several related issues are discussed. First, the two application directions of topological De Morgan algebras in rough set theory are described, a uniform algebraic depiction of various rough set models are given. Secondly, based on interior and closure operators of a strong topological rough algebra, an implication operator (called rough implication) is introduced, and its important properties are proved. Thirdly, a rough set interpretation of classical logic is analyzed, and a new semantic interpretation of Łukasiewicz continuous-valued logic system Ł uk is constructed based on rough implication. Finally, strong topological rough implication algebra ( STRI-algebra for short) is introduced. The connections among STRI-algebras, regular double Stone algebras and RSL-algebras are established, and the completeness theorem of rough logic system RSL is discussed based on STRI-algebras.

Zariski‐type topology for implication algebras

AbstractIn this work we provide a new topological representation for implication algebras in such a way that its one‐point compactification is the topological space given in [1]. Some applications are given thereof (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A decision method based on uncertainty reasoning of linguistic truth-valued concept lattice

Decision making with linguistic information is a research hotspot now. This paper begins by establishing the theory basis for linguistic information processing and constructs the linguistic truth-valued concept lattice for a decision information system, and further utilises uncertainty reasoning to make the decision. That is, we first utilise the linguistic truth-valued lattice implication algebra to unify the different kinds of linguistic expressions; second, we construct the linguistic truth-valued concept lattice and decision concept lattice according to the concrete decision information system and third, we establish the internal and external uncertainty reasoning methods and talk about the rationality of them. We apply these uncertainty reasoning methods into decision making and present some generation methods of decision rules. In the end, we give an application of this decision method by an example.

Determination of α-resolution in lattice-valued first-order logic LF(X)

Key issues for resolution-based automated reasoning in lattice-valued first-order logic LF(X) are investigated with truth-values in a lattice-valued logical algebraic structure–lattice implication algebra (LIA). The determination of resolution at a certain truth-value level (called α-resolution) in LF(X) is proved to be equivalently transformed into the determination of α-resolution in lattice-valued propositional logic LP(X) based on LIA. The determination of α-resolution of any quasi-regular generalized literals and constants under various cases in LP(X) is further analyzed, specified, and subsequently verified. Hence the determination of α-resolution in LF(X) can be accordingly solved to a very broad extent, which not only lays a foundation for the practical implementation of automated reasoning algorithms in LF(X), but also provides a key support for α-resolution-based automated reasoning approaches and algorithms in LIA based linguistic truth-valued logics.

Some properties of epimorphisms of Hilbert algebras

Abstract This paper represents a start in the study of epimorphisms in some categories of Hilbert algebras. Even if we give a complete characterization for such epimorphisms only for implication algebras, the following results will make possible the construction of some examples of epimorphisms which are not surjective functions. Also, we will show that the study of epimorphisms of Hilbert algebras is equivalent with the study of epimorphisms of Hertz algebras.

Linguistic truth-valued lattice-valued propositional logic system ℓ P( X) based on linguistic truth-valued lattice implication algebra

In the semantics of natural language, quantification may have received more attention than any other subject, and syllogistic reasoning is one of the main topics in many-valued logic studies on inference. Particularly, lattice-valued logic, a kind of important non-classical logic, can be applied to describe and treat incomparability by the incomparable elements in its truth-valued set. In this paper, we first focus on some properties of linguistic truth-valued lattice implication algebra. Secondly, we introduce some concepts of linguistic truth-valued lattice-valued propositional logic system ℓ P( X), whose truth-valued domain is a linguistic truth-valued lattice implication algebra. Then we investigate the semantic problem of ℓ P( X). Finally, we further probe into the syntax of linguistic truth-valued lattice-valued propositional logic system ℓ P( X), and prove the soundness theorem, deduction theorem and consistency theorem.

ANTI FUZZY IDEALS IN WEAK BCC-ALGEBRAS

Abstract. We characterize the main types of anti fuzzy ideals in weakBCC-algebras. 1. IntroductionBCC-algebras (called also BIK + -algebras) are an algebraic model of BIK -logic, i.e., implicational logic based on modus ponens and some axioms schemecontaining the combinators B, I, and K. Weak BCC-algebras (called also BZ-algebras) have the same partial order as BCC-algebras and BCK-algebras butdo not have a minimal element. Many mathematicians studied various typesof algebras such as BCI-algebras, B-algebras, implication algebras, G-algebras,Hilbert algebras and do on. All these algebras have one distinguished element,satisfy some common identities and have a similar partial order. In fact, allthese algebras are a generalization or a special case of weak BCC-algebras. So,results obtained for weak BCC-algebras are, in some sense, fundamental forthese algebras, especially for BCC/BCH/BCI/BCK-algebras.A very important role in the theory of such algebras plays ideals. In BCK-algebras ideals are induced by partial order or by homomorphisms. All idealsdetermine congruences. In BCC-algebras there are congruences which are notdetermined by ideals [3]. Moreover, in BCC-algebras relations determined byideals (in the same way as in BCK-algebras) are not congruences, in general.So, in BCC-algebras the new concept of ideals should br introduced. Similarlyin weak BCC-algebras.2. PreliminariesIn this section, we give basic de nitions and facts on weak BCC-algebras.De nition 1. A weak BCC-algebra Xis an abstract algebra (X;;0) of type(2;0) satisfying the following axioms(i) ((xy) (zy)) (xz) = 0,(ii) xx= 0,

(m, n)-FOLD p-IDEALS IN WEAK BCC-ALGEBRAS

Abstract. Various characterizations of (m;n)-fold p-ideals of weak BCC-algebras are presented. 1. IntroductionBCC-algebras, introduced by Y. Komori (see [10] and [11]), are an algebraicmodel of BIK + -logic, i.e., implicational logic whose axioms scheme are theprincipal type-scheme of the combinators B, I, and K, and whose inferencerules are modus ponens and modus ponens 2, where p !q is inferred fromp !(r !q) and r. Several years later some authors introduced independentlymore extensive algebraic system using di erent names. This new algebraicsystems have the same partial order as BCC-algebras and BCK-algebras buthas no minimal element. Such obtained system is called a BZ-algebra [7, 15] ora weak BCC-algebra [2, 4, 13]. From the mathematical point of view the lastname is more corrected but more popular is the rst.Many mathematicians studied such algebras as BCI-algebras, B-algebras,di erence algebras, implication algebras, G-algebras, Hilbert algebras, d-algebrasand many others. All these algebras have one distinguished element and satisfysome common identities playing a crucial role in these algebras and, in fact, aregeneralization or a special case of weak BCC-algebras. So, results obtained forweak BCC-algebras are in some sense fundamental for these algebras, especiallyfor BCC/BCH/BCI/BCK-algebras.A very important role in the theory of such algebras plays ideals. Many typesof ideals in these algebras have been studied with various relations betweenthem (see for example [5] and [16]). In [14] X.H.Zhang, J.Hao and S.A. Bhattistudied p-ideals of BCI-algebras. In [8] Y.Huang and Z.Chen introduced thefoldness of some ideals in BCK-algebras. In [12] Kordi and Moussavi studied(m;n)-fold p-ideals and fuzzy (m;n)-fold p-ideals in BCI-algebras.This paper is a continuation of our study of p-ideals initiated in [5].

Decision Making with Uncertainty Information Based on Lattice-Valued Fuzzy Concept Lattice

For the processing of decision making with uncertainty information, this paper establishes a decision model based on lattice-valued logic and researches the algorithm for extracting the maximum decision rules. Firstly, we further research the lattice-valued fuzzy concept lattice by combining the lattice implication algebra and classical concept lattice; secondly, we define the lattice-valued decision context as the equivalent form of decision information system and establish the single-target decision model and talk about some properties of the decision rules; finally, we give the calculating methods of decision rules with different decision values and the algorithm for extracting the maximum decision rules.

FUZZIFYING FILTERS IN LATTICE IMPLICATION ALGEBRAS

We introduce the concepts of a fuzzifying fllter and a fuzzifying implicative fllter in lattice implication algebras.

Conditions for Permutability of Congruences in Implication Algebras

Fil: Castano, Diego Nicolas. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Centro Cientifico Tecnologico Conicet - Bahia Blanca. Instituto de Matematica Bahia Blanca. Universidad Nacional del Sur. Departamento de Matematica. Instituto de Matematica Bahia Blanca; Argentina