Research

22 pages, 410 KiB

Open AccessArticle

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 862

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.,

≺ = \emptyset

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.,

≺ = \emptyset

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

Open AccessArticle

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 707

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

Open AccessArticle

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

Viewed by 645

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

Open AccessArticle

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 - 05 Sep 2023

Viewed by 475

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

Open AccessArticle

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 2 | Viewed by 1148

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

Open AccessArticle

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

Viewed by 756

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

Open AccessArticle

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 878

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

Open AccessArticle

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 - 06 Jan 2023

Cited by 12 | Viewed by 1330

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

Open AccessArticle

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 981

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 (

L F L A O

) on the basis of the L-fuzzy relation. In this article, we give a further study on Qiao’s

L F L A O

around the axiomatic characterization and induced L-topology. Firstly, we investigate and discuss three new

L F L A O

generated by ⊛-transitive, ⊛-Euclidean and ⊛-mediated L-fuzzy relations. Secondly, we utilize a single axiom to characterize the

L F L A O

generated by serial, symmetric, reflexive, ⊛-transitive and ⊛-mediate L-fuzzy relations and their compositions. Thirdly, we present a method to generate Alexandrov L-topology (

A L T P O

) from

L F L A O

and construct a bijection between

A L T P O

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

Open AccessArticle

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 15 | Viewed by 1471

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

Open AccessArticle

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 1028

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

Open AccessArticle

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 1135

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

Open AccessArticle

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 - 02 Jun 2022

Cited by 1 | Viewed by 1742

Abstract

A BN-algebra is a non-empty set

X

with a binary operation “

*

” and a constant 0 that satisfies the following axioms:

(B 1) 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:

(B 1) x * x = 0,

(B 2) x * 0 = x,

and

(B N) (x * y) * z = (0 * z) * (y * x)

for all

x, y, z \in X

. A non-empty subset

I

of

X

is called an ideal in BN-algebra X if it satisfies

0 \in X

and if

y \in I

and

x * y \in I

, then

x \in I

for all

x, y \in 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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