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Axioms
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Non-classical Logics and Related Algebra Systems
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Non-classical Logics and Related Algebra Systems

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Logic".

Deadline for manuscript submissions: closed (31 August 2023) | Viewed by 21367

Special Issue Editors

Prof. Dr. Xiaohong Zhang

E-Mail Website
Guest Editor
School of Mathematics and Data Science, Shaanxi University of Science and Technology, Xi'an 710021, China
Interests: fuzzy logic; fuzzy sets; non-classical logic algebras; data mining; artifical intelligence
Prof. Dr. Eunsuk Yang

E-Mail Website
Guest Editor
Department of Philosophy & Institute of Critical Thinking and Writing, Colleges of Humanities & Social Science Blvd., Jeonbuk National University, Rm 417, Jeonju 54896, Korea
Interests: non-classical logics; fuzzy logic; substructural logic; relevance logic; algebraic logic

Special Issue Information

Dear Colleagues,

One of important features of the foundation of mathematics is that logic, a central research area, studies basic features of not only reasoning but also many other scientific objects. In particular, logic and algebra, having more general structures, have been investigated in this area, for example, t-norm-based fuzzy logic, and residuated lattices (including MV-algebra, BL-algebra, and MTL-algebra). At the same time, non-classical logic represented by fuzzy logic is widely used in intelligent information processing. Recently, artificial intelligence and big data have become hot spots of science and technology; data intelligence is the integration of the two, which requires a variety of non-classical logic approaches to provide basic theory. In order to promote close communication and cooperation in the research fields of fuzzy logic, various non-classical logic and related algebraic systems, as well as applications in data intelligence, we are planning a Special Issue of Axioms, and we welcome relevant experts and scholars to contribute.

Prof. Dr. Xiaohong Zhang
Prof. Dr. Eunsuk Yang
Guest Editors

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Keywords

  • fuzzy logic
  • general implication logic
  • semilinear logic and substructure logic
  • approximation reasoning
  • various non-classical logic approaches
  • residuated lattice
  • T-norm and micanorm
  • overlap function
  • grouping function
  • BCK/BCI/BZ-algebra
  • L-algebra and quantum B-algebra
  • effect algebra and pseudo effect algebra
  • AG/CA/TA/T2CA-groupoid
  • varoius hyper-algebraic structures

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Published Papers (13 papers)

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Research

22 pages, 410 KiB  
Article
Certain Bounds of Formulas in Free Temporal Algebras
by Francisco Miguel García-Olmedo, Antonio Jesús Rodríguez-Salas and Pedro González-Rodelas
Axioms 2023, 12(12), 1111; https://doi.org/10.3390/axioms12121111 - 11 Dec 2023
Viewed by 1203
Abstract
In this paper, we give a basic structure theorem based on the study of extreme cases for the value of ≺ (the classical precedence relation between ultrafilters), i.e., ≺=∅ and no isolated element in ≺. This gives rise, respectively, to the [...] Read more.
In this paper, we give a basic structure theorem based on the study of extreme cases for the value of ≺ (the classical precedence relation between ultrafilters), i.e., ≺=∅ and no isolated element in ≺. This gives rise, respectively, to the temporal varieties O and W, with the result that O generates a variety of temporal algebras. We also characterize the simple temporal algebras by means of arithmetical properties related to basical temporal operators; we conclude that the simplicity of the temporal algebra lies in being able to make 0 any element less than 1 by repeated application to it of the L operator. We then present an algebraic construction similar to a product but in which the temporal operations are not defined componentwise. This new “product” is shown to be useful in the study of algebra order and finding of bounds by means of something similar to a lifting process. Finally, we give an alternative proof of an already known result on atoms counting in free temporal algebras. Full article
(This article belongs to the Special Issue Non-classical Logics and Related Algebra Systems)
18 pages, 372 KiB  
Article
Residuated Basic Logic
by Zhe Lin and Minghui Ma
Axioms 2023, 12(10), 966; https://doi.org/10.3390/axioms12100966 - 13 Oct 2023
Viewed by 1010
Abstract
Residuated basic logic (RBL) is the logic of residuated basic algebras, which constitutes a conservative extension of basic propositional logic (BPL). The basic implication is a residual of a non-associative binary operator in RBL. The conservativity is shown [...] Read more.
Residuated basic logic (RBL) is the logic of residuated basic algebras, which constitutes a conservative extension of basic propositional logic (BPL). The basic implication is a residual of a non-associative binary operator in RBL. The conservativity is shown by relational semantics. A Gentzen-style sequent calculus GRBL, which is an extension of the distributive full non-associative Lambek calculus, is established for residuated basic logic. The calculus GRBL admits the mix-elimination, subformula, and disjunction properties. Moreover, the class of all residuated basic algebras has the finite embeddability property. The consequence relation of GRBL is decidable. Full article
(This article belongs to the Special Issue Non-classical Logics and Related Algebra Systems)
12 pages, 287 KiB  
Article
Hybrid near Algebra
by Harika Bhurgula, Narasimha Swamy Pasham, Ravikumar Bandaru and Amal S. Alali
Axioms 2023, 12(9), 877; https://doi.org/10.3390/axioms12090877 - 13 Sep 2023
Cited by 1 | Viewed by 989
Abstract
The objective of this paper is to study the hybrid near algebra. It has been summarized with the proper definitions and theorems of hybrid near algebra, hybrid near algebra homomorphism and direct product of hybrid near algebra. It has been proved that a [...] Read more.
The objective of this paper is to study the hybrid near algebra. It has been summarized with the proper definitions and theorems of hybrid near algebra, hybrid near algebra homomorphism and direct product of hybrid near algebra. It has been proved that a homomorphic image of a hybrid near algebra is a hybrid near algebra. It also investigated the intersection of two hybrid near algebras is a hybrid near algebra. Full article
(This article belongs to the Special Issue Non-classical Logics and Related Algebra Systems)
12 pages, 292 KiB  
Article
Some Model Theoretic Properties for Pavelka-Style Gödel Logic, RGL* and Gödel Logic with Δ
by Nazanin Roshandel Tavana
Axioms 2023, 12(9), 858; https://doi.org/10.3390/axioms12090858 - 5 Sep 2023
Viewed by 708
Abstract
Pavelka-style (rational) Gödel logic is an extension of Gödel logic which is denoted by RGL*. In this article, due to the approximate Craig interpolation property for RGL*, the Robinson theorem and approximate Beth theorem are presented and proved. Then, the omitting types theorem [...] Read more.
Pavelka-style (rational) Gödel logic is an extension of Gödel logic which is denoted by RGL*. In this article, due to the approximate Craig interpolation property for RGL*, the Robinson theorem and approximate Beth theorem are presented and proved. Then, the omitting types theorem for this logic is expressed and proved. At the end, as a reduction, the omitting types theorem for standard Gödel logic with Δ is studied. Full article
(This article belongs to the Special Issue Non-classical Logics and Related Algebra Systems)
16 pages, 875 KiB  
Article
First Entire Zagreb Index of Fuzzy Graph and Its Application
by Umapada Jana and Ganesh Ghorai
Axioms 2023, 12(5), 415; https://doi.org/10.3390/axioms12050415 - 24 Apr 2023
Cited by 4 | Viewed by 1784
Abstract
The first entire Zagreb index (FEZI) is a graph parameter that has proven to be essential in various real-life scenarios, such as networking businesses and traffic management on roads. In this research paper, the FEZI was explored for a variety of fuzzy graphs, [...] Read more.
The first entire Zagreb index (FEZI) is a graph parameter that has proven to be essential in various real-life scenarios, such as networking businesses and traffic management on roads. In this research paper, the FEZI was explored for a variety of fuzzy graphs, including star, firefly graph, cycle, path, fuzzy subgraph, vertex elimination, and edge elimination. This study presented several results, including determining the relationship between two isomorphic fuzzy graphs and between a path and cycle (connecting both end vertices of the path). This research also deals with the analysis of α-cut fuzzy graphs and establishes bounds for some fuzzy graphs. To apply these findings to modern life problems, the research team utilized the results to identify areas that require more development in internet systems. These results have practical implications for enhancing the efficiency and effectiveness of internet systems. The conclusion drawn from this research can be used to inform future research and aid in the development of more efficient and effective systems in various fields. Full article
(This article belongs to the Special Issue Non-classical Logics and Related Algebra Systems)
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13 pages, 256 KiB  
Article
On Two Intuitionistic Fuzzy Modal Topological Structures
by Krassimir Atanassov, Nora Angelova and Tania Pencheva
Axioms 2023, 12(5), 408; https://doi.org/10.3390/axioms12050408 - 22 Apr 2023
Cited by 1 | Viewed by 1109
Abstract
The concept of an Intuitionistic Fuzzy Modal Topological Structure (IFMTS) was introduced previously, and some of its properties were studied. So far, there are two different IFMTSs based on the classical intuitionistic fuzzy operations: “union” (∪) and “intersection” (∩). In the present paper, [...] Read more.
The concept of an Intuitionistic Fuzzy Modal Topological Structure (IFMTS) was introduced previously, and some of its properties were studied. So far, there are two different IFMTSs based on the classical intuitionistic fuzzy operations: “union” (∪) and “intersection” (∩). In the present paper, two new IFMTSs are developed. They are based on new intuitionistic fuzzy topological operators from closure and interior types, introduced here for the first time, and on the two standard intuitionistic fuzzy modal operators □ and ◊. Some basic properties of the new IFMTSs are discussed. The newly presented IFMTSs could be considered as a basis for the next research on the IFMTSs. Some ideas for the future development of the IFMTS theory and open problems are formulated, related to the existence of other intuitionistic fuzzy operations that can generate new intuitionistic fuzzy topological operators and, respectively, new IFMTSs. Full article
(This article belongs to the Special Issue Non-classical Logics and Related Algebra Systems)
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16 pages, 326 KiB  
Article
Fuzzy Algebras of Concepts
by Manuel Ojeda-Hernández, Domingo López-Rodríguez and Pablo Cordero
Axioms 2023, 12(4), 324; https://doi.org/10.3390/axioms12040324 - 26 Mar 2023
Viewed by 1315
Abstract
Preconcepts are basic units of knowledge that form the basis of formal concepts in formal concept analysis (FCA). This paper investigates the relations among different kinds of preconcepts, such as protoconcepts, meet and join-semiconcepts and formal concepts. The first contribution of this paper, [...] Read more.
Preconcepts are basic units of knowledge that form the basis of formal concepts in formal concept analysis (FCA). This paper investigates the relations among different kinds of preconcepts, such as protoconcepts, meet and join-semiconcepts and formal concepts. The first contribution of this paper, is to present a fuzzy powerset lattice gradation, that coincides with the preconcept lattice at its 1-cut. The second and more significant contribution, is to introduce a preconcept algebra gradation that yields different algebras for protoconcepts, semiconcepts, and concepts at different cuts. This result reveals new insights into the structure and properties of the different categories of preconcepts. Full article
(This article belongs to the Special Issue Non-classical Logics and Related Algebra Systems)
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19 pages, 378 KiB  
Article
Partial Residuated Implications Induced by Partial Triangular Norms and Partial Residuated Lattices
by Xiaohong Zhang, Nan Sheng and Rajab Ali Borzooei
Axioms 2023, 12(1), 63; https://doi.org/10.3390/axioms12010063 - 6 Jan 2023
Cited by 13 | Viewed by 1745
Abstract
This paper reveals some relations between fuzzy logic and quantum logic on partial residuated implications (PRIs) induced by partial t-norms as well as proposes partial residuated monoids (PRMs) and partial residuated lattices (PRLs) by defining partial adjoint pairs. First of all, we introduce [...] Read more.
This paper reveals some relations between fuzzy logic and quantum logic on partial residuated implications (PRIs) induced by partial t-norms as well as proposes partial residuated monoids (PRMs) and partial residuated lattices (PRLs) by defining partial adjoint pairs. First of all, we introduce the connection between lattice effect algebra and partial t-norms according to the concept of partial t-norms given by Borzooei, together with the proof that partial operation in any commutative quasiresiduated lattice is partial t-norm. Then, we offer the general form of PRI and the definition of partial fuzzy implication (PFI), give the condition that partial residuated implication is a fuzzy implication, and prove that each PRI is a PFI. Next, we propose PRLs, study their basic characteristics, discuss the correspondence between PRLs and lattice effect algebras (LEAs), and point out the relationship between LEAs and residuated partial algebras. In addition, like the definition of partial t-norms, we provide the notions of partial triangular conorms (partial t-conorms) and corresponding partial co-residuated lattices (PcRLs). Lastly, based on partial residuated lattices, we define well partial residuated lattices (wPRLs), study the filter of well partial residuated lattices, and then construct quotient structure of PRMs. Full article
(This article belongs to the Special Issue Non-classical Logics and Related Algebra Systems)
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21 pages, 384 KiB  
Article
The Single Axiomatization on CCRL-Fuzzy Rough Approximation Operators and Related Fuzzy Topology
by Yaoliang Xu, Dandan Zou and Lingqiang Li
Axioms 2023, 12(1), 37; https://doi.org/10.3390/axioms12010037 - 28 Dec 2022
Viewed by 1357
Abstract
It is well known that lattice-valued rough sets are important branches of fuzzy rough sets. The axiomatic characterization and related topology are the main research directions of lattice-valued rough sets. For L=(L,⊛), a complete co-residuated lattice [...] Read more.
It is well known that lattice-valued rough sets are important branches of fuzzy rough sets. The axiomatic characterization and related topology are the main research directions of lattice-valued rough sets. For L=(L,⊛), a complete co-residuated lattice (CCRL), Qiao recently defined an L-fuzzy lower approximation operator (LFLAO) on the basis of the L-fuzzy relation. In this article, we give a further study on Qiao’s LFLAO around the axiomatic characterization and induced L-topology. Firstly, we investigate and discuss three new LFLAO generated by ⊛-transitive, ⊛-Euclidean and ⊛-mediated L-fuzzy relations. Secondly, we utilize a single axiom to characterize the LFLAO generated by serial, symmetric, reflexive, ⊛-transitive and ⊛-mediate L-fuzzy relations and their compositions. Thirdly, we present a method to generate Alexandrov L-topology (ALTPO) from LFLAO and construct a bijection between ALTPO and ⊛-preorder (i.e., reflexive and ⊛-transitive L-fuzzy relation) on the same underlying set. Full article
(This article belongs to the Special Issue Non-classical Logics and Related Algebra Systems)
29 pages, 1877 KiB  
Article
Pseudo Overlap Functions, Fuzzy Implications and Pseudo Grouping Functions with Applications
by Xiaohong Zhang, Rong Liang, Humberto Bustince, Benjamin Bedregal, Javier Fernandez, Mengyuan Li and Qiqi Ou
Axioms 2022, 11(11), 593; https://doi.org/10.3390/axioms11110593 - 26 Oct 2022
Cited by 20 | Viewed by 2186
Abstract
Overlap and grouping functions are important aggregation operators, especially in information fusion, classification and decision-making problems. However, when we do more in-depth application research (for example, non-commutative fuzzy reasoning, complex multi-attribute decision making and image processing), we find overlap functions as well as [...] Read more.
Overlap and grouping functions are important aggregation operators, especially in information fusion, classification and decision-making problems. However, when we do more in-depth application research (for example, non-commutative fuzzy reasoning, complex multi-attribute decision making and image processing), we find overlap functions as well as grouping functions are required to be commutative (or symmetric), which limit their wide applications. For the above reasons, this paper expands the original notions of overlap functions and grouping functions, and the new concepts of pseudo overlap functions and pseudo grouping functions are proposed on the basis of removing the commutativity of the original functions. Some examples and construction methods of pseudo overlap functions and pseudo grouping functions are presented, and the residuated implication (co-implication) operators derived from them are investigated. Not only that, some applications of pseudo overlap (grouping) functions in multi-attribute (group) decision-making, fuzzy mathematical morphology and image processing are discussed. Experimental results show that, in many application fields, pseudo overlap functions and pseudo grouping functions have greater flexibility and practicability. Full article
(This article belongs to the Special Issue Non-classical Logics and Related Algebra Systems)
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10 pages, 756 KiB  
Article
Comparison of Overlap and Grouping Functions
by Songsong Dai
Axioms 2022, 11(8), 420; https://doi.org/10.3390/axioms11080420 - 20 Aug 2022
Cited by 1 | Viewed by 1377
Abstract
This paper investigates the pointwise comparability of overlap and grouping functions which obtained by Bustince et al.’s and Bedregal et al.’s generator pairs, respectively. Some necessary and sufficient conditions for the comparison of these functions are proved. We also introduce some compositions of [...] Read more.
This paper investigates the pointwise comparability of overlap and grouping functions which obtained by Bustince et al.’s and Bedregal et al.’s generator pairs, respectively. Some necessary and sufficient conditions for the comparison of these functions are proved. We also introduce some compositions of these functions and study the order preservation of these compositions. Full article
(This article belongs to the Special Issue Non-classical Logics and Related Algebra Systems)
12 pages, 262 KiB  
Article
On Implicative Derivations of MTL-Algebras
by Jianxin Liu, Yijun Li, Yongwei Yang and Juntao Wang
Axioms 2022, 11(7), 340; https://doi.org/10.3390/axioms11070340 - 15 Jul 2022
Cited by 1 | Viewed by 1509
Abstract
This paper introduces the implicative derivations and gives some of their characterizations on MTL-algebras. Furthermore, we provide some representation of MTL-algebras by implicative derivations and obtain some representation of Boolean algebra via the algebra of all implicative derivations. Finally, we explore the relationship [...] Read more.
This paper introduces the implicative derivations and gives some of their characterizations on MTL-algebras. Furthermore, we provide some representation of MTL-algebras by implicative derivations and obtain some representation of Boolean algebra via the algebra of all implicative derivations. Finally, we explore the relationship between implicative derivation and other operators on MTL-algebras and show that there exists a bijection between the sets of multiplier and implicative derivations on IMTL-algebras. The results of this paper can provide the common properties of implicative derivations in the t-norm-based fuzzy logical algebras. Full article
(This article belongs to the Special Issue Non-classical Logics and Related Algebra Systems)
8 pages, 423 KiB  
Article
On r-Ideals and m-k-Ideals in BN-Algebras
by Sri Gemawati, Musnis Musraini, Abdul Hadi, La Zakaria and Elsi Fitria
Axioms 2022, 11(6), 268; https://doi.org/10.3390/axioms11060268 - 2 Jun 2022
Cited by 1 | Viewed by 2189
Abstract
A BN-algebra is a non-empty set X with a binary operation “∗” and a constant 0 that satisfies the following axioms: (B1) x∗x=0,  [...] Read more.
A BN-algebra is a non-empty set X with a binary operation “∗” and a constant 0 that satisfies the following axioms: (B1) x∗x=0, (B2) x∗0=x, and (BN) (x∗y)∗z=(0∗z)∗(y∗x) for all x, y, z ∈X. A non-empty subset I of X is called an ideal in BN-algebra X if it satisfies 0∈X and if y∈I and x∗y∈I, then x∈I for all x,y∈X. In this paper, we define several new ideal types in BN-algebras, namely, r-ideal, k-ideal, and m-k-ideal. Furthermore, some of their properties are constructed. Then, the relationships between ideals in BN-algebra with r-ideal, k-ideal, and m-k-ideal properties are investigated. Finally, the concept of r-ideal homomorphisms is discussed in BN-algebra. Full article
(This article belongs to the Special Issue Non-classical Logics and Related Algebra Systems)
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