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

Symmetry ◽  
2018 ◽  
Vol 10 (11) ◽  
pp. 573 ◽  
Author(s):  
Xiaohong Zhang ◽  
Rajab Borzooei ◽  
Young Jun
Keyword(s):  
Algebraic Semantics ◽  
Fuzzy Logics ◽  
Quantum Logics ◽  
Filter Theory ◽  
Implication Algebras ◽  
Implication Algebra

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.

scholarly journals Generalized continuous and left-continuous t-norms arising from algebraic semantics for fuzzy logics

2010 ◽  
Vol 180 (8) ◽  
pp. 1354-1372 ◽  
Author(s):  
Carles Noguera ◽  
Francesc Esteva ◽  
Lluís Godo
Keyword(s):  
Algebraic Semantics ◽  
Fuzzy Logics

An Algebraic Aspect of Correspondences Between Implicational Fragment Logics and Fuzzy Logics

2015 ◽  
Vol 19 (6) ◽  
pp. 861-866
Author(s):  
Mayuka F. Kawaguchi ◽  
◽  
Michiro Kondo ◽  
Keyword(s):  
Partially Ordered Set ◽  
Ordered Set ◽  
Unit Interval ◽  
Axiomatic System ◽  
Research Report ◽  
Algebraic Semantics ◽  
Fuzzy Logics ◽  
The Real ◽  
Partially Ordered ◽  
Algebraic Aspect

This research report treats a correspondence between implicational fragment logics and fuzzy logics from the viewpoint of their algebraic semantics. The authors introduce monotone BI-algebras by loosening the axiomatic system of BCK-algebras. We also extend the algebras of fuzzy logics with weakly-associative conjunction from the case of the real unit interval to the case of a partially ordered set. As the main result of this report, it is proved that the class of monotone BI-algebras with condition (S) coincides with the class of weakly-associative conjunctive algebras.

scholarly journals Distinguished algebraic semantics for t-norm based fuzzy logics: Methods and algebraic equivalencies

2009 ◽  
Vol 160 (1) ◽  
pp. 53-81 ◽  
Author(s):  
Petr Cintula ◽  
Francesc Esteva ◽  
Joan Gispert ◽  
Lluís Godo ◽  
Franco Montagna ◽  
...  
Keyword(s):  
Algebraic Semantics ◽  
Fuzzy Logics

scholarly journals Basic Core Fuzzy Logics and Algebraic Routley–Meyer-Style Semantics

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 273
Author(s):  
Eunsuk Yang
Keyword(s):  
Fuzzy Logic ◽  
Substructural Logics ◽  
Algebraic Semantics ◽  
Relational Semantics ◽  
Fuzzy Logics

Recently, algebraic Routley–Meyer-style semantics was introduced for basic substructural logics. This paper extends it to fuzzy logics. First, we recall the basic substructural core fuzzy logic MIAL (Mianorm logic) and its axiomatic extensions, together with their algebraic semantics. Next, we introduce two kinds of ternary relational semantics, called here linear Urquhart-style and Fine-style Routley–Meyer semantics, for them as algebraic Routley–Meyer-style semantics.

scholarly journals Vague Congruences and Quotient Lattice Implication Algebras

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Xiaoyan Qin ◽  
Yi Liu ◽  
Yang Xu
Keyword(s):  
Lattice Implication Algebra ◽  
Implication Algebras ◽  
Homomorphism Theorem ◽  
Similarity Relations ◽  
Implication Algebra ◽  
Congruence Theory ◽  
Congruence Relations

The aim of this paper is to further develop the congruence theory on lattice implication algebras. Firstly, we introduce the notions of vague similarity relations based on vague relations and vague congruence relations. Secondly, the equivalent characterizations of vague congruence relations are investigated. Thirdly, the relation between the set of vague filters and the set of vague congruences is studied. Finally, we construct a new lattice implication algebra induced by a vague congruence, and the homomorphism theorem is given.

Algebraic functions in Łukasiewicz implication algebras

2016 ◽  
Vol 26 (02) ◽  
pp. 223-247 ◽  
Author(s):  
Miguel Campercholi ◽  
Diego Castaño ◽  
José Patricio Díaz Varela
Keyword(s):  
Main Tool ◽  
Algebraic Functions ◽  
Implication Algebras ◽  
Implication Algebra ◽  
Tarski Algebras ◽  
Łukasiewicz Implication Algebras ◽  
Finitely Generated Variety ◽  
Element Chain ◽  
Global Representation ◽  
Partial Description

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.

scholarly journals The logic induced by effect algebras

2020 ◽  
Vol 24 (19) ◽  
pp. 14275-14286 ◽  
Author(s):  
Ivan Chajda ◽  
Radomír Halaš ◽  
Helmut Länger
Keyword(s):  
Quantum Mechanics ◽  
Chain Condition ◽  
Axiom System ◽  
Single Element ◽  
Effect Algebras ◽  
Algebraic Semantics ◽  
Implication Algebras ◽  
Lattice Effect ◽  
Ascending Chain Condition

Abstract Effect algebras form an algebraic formalization of the logic of quantum mechanics. For lattice effect algebras $${\mathbf {E}}$$ E , we investigate a natural implication and prove that the implication reduct of $${\mathbf {E}}$$ E is term equivalent to $${\mathbf {E}}$$ E . Then, we present a simple axiom system in Gentzen style in order to axiomatize the logic induced by lattice effect algebras. For effect algebras which need not be lattice-ordered, we introduce a certain kind of implication which is everywhere defined but whose result need not be a single element. Then, we study effect implication algebras and prove the correspondence between these algebras and effect algebras satisfying the ascending chain condition. We present an axiom system in Gentzen style also for not necessarily lattice-ordered effect algebras and prove that it is an algebraic semantics for the logic induced by finite effect algebras.

scholarly journals Filters in Strong BI-Algebras and Residuated Pseudo-SBI-Algebras

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1513
Author(s):  
Xiaohong Zhang ◽  
Xiangyu Ma ◽  
Xuejiao Wang
Keyword(s):  
Fuzzy Logic ◽  
Special Kind ◽  
Algebra Structure ◽  
Algebraic Structures ◽  
Commutative Case ◽  
Filter Theory ◽  
Fuzzy Implication ◽  
Implication Algebra ◽  
Common Extension ◽  
The Common

The concept of basic implication algebra (BI-algebra) has been proposed to describe general non-classical implicative logics (such as associative or non-associative fuzzy logic, commutative or non-commutative fuzzy logic, quantum logic). However, this algebra structure does not have enough characteristics to describe residual implications in depth, so we propose a new concept of strong BI-algebra, which is exactly the algebraic abstraction of fuzzy implication with pseudo-exchange principle (PEP). Furthermore, in order to describe the characteristics of the algebraic structure corresponding to the non-commutative fuzzy logics, we extend strong BI-algebra to the non-commutative case, and propose the concept of pseudo-strong BI (SBI)-algebra, which is the common extension of quantum B-algebras, pseudo-BCK/BCI-algebras and other algebraic structures. We establish the filter theory and quotient structure of pseudo-SBI- algebras. Moreover, based on prequantales, semi-uninorms, t-norms and their residual implications, we introduce the concept of residual pseudo-SBI-algebra, which is a common extension of (non-commutative) residual lattices, non-associative residual lattices, and also a special kind of residual partially-ordered groupoids. Finally, we investigate the filters and quotient algebraic structures of residuated pseudo-SBI-algebras, and obtain a unity frame of filter theory for various algebraic systems.

scholarly journals Fantastic filters of lattice implication algebras

2000 ◽  
Vol 24 (4) ◽  
pp. 277-281 ◽  
Author(s):  
Young Bae Jun
Keyword(s):  
Extension Property ◽  
Equivalent Condition ◽  
Lattice Implication Algebra ◽  
Implication Algebras ◽  
Implication Algebra ◽  
Fantastic Filter ◽  
Positive Implicative Filter

The notion of a fantastic filter in a lattice implication algebra is introduced, and the relations among filter, positive implicative filter, and fantastic filter are given. We investigate an equivalent condition for a filter to be fantastic, and state an extension property for fantastic filter.

Substructural fuzzy logics

2007 ◽  
Vol 72 (3) ◽  
pp. 834-864 ◽  
Author(s):  
George Metcalfe ◽  
Franco Montagna
Keyword(s):  
Linear Logic ◽  
Substructural Logics ◽  
Unit Interval ◽  
Residuated Lattices ◽  
Cut Elimination ◽  
Algebraic Semantics ◽  
Fuzzy Logics ◽  
The Real ◽  
Gentzen Systems ◽  
Intuitionistic Linear Logic

AbstractSubstructural fuzzy logics are substructural logics that are complete with respect to algebras whose lattice reduct is the real unit interval [0, 1]. In this paper, we introduce Uninorm logic UL as Multiplicative additive intuitionistic linear logic MAILL extended with the prelinearity axiom ((A → B) ∧ t) V ((B → A)∧ t). Axiomatic extensions of UL include known fuzzy logics such as Monoidal t-norm logic MIX and Gödel logic G, and new weakening-free logics. Algebraic semantics for these logics are provided by subvarieties of (representable) pointed bounded commutative residuated lattices. Gentzen systems admitting cut-elimination are given in the framework of hypersequents. Completeness with respect to algebras with lattice reduct [0, 1] is established for UL and several extensions using a two-part strategy. First, completeness is proved for the logic extended with Takeuti and Titani's density rule. A syntactic elimination of the rule is then given using a hypersequent calculus. As an algebraic corollary, it follows that certain varieties of residuated lattices are generated by their members with lattice reduct [0, 1].

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