Transactions on Fuzzy Sets and Systems
Volume:1 Issue: 1, Spring - Summer 2022

Orthomodular lattices generalize the Boolean algebras; they have arisen in the study of quantum logic. Quantum-MV  algebras were introduced as non-lattice theoretic generalizations of MV algebras and as non-idempotent generalizations of orthomodular lattices.In this paper, we continue the research in the “world” of involutive algebras of the form (A,⊙,−,1), with 1−= 0, 1 being the last element. We clarify now some aspects concerning the quantum-MV (QMV) algebras as non-idempotent generalizations of orthomodular lattices. We study in some detail the orthomodular lattices (OMLs) and we introduce and study two generalizations of them, the orthomodular softlattices (OMSLs) and the ortho-modular widelattices (OMWLs). We establish systematically connections between OMLs and OMSLs/OMWLs and QMV, pre-MV, metha-MV, orthomodular algebras and ortholattices, orthosoftlattices/orthowidelattices - connections illustrated in 22 Figures. We prove, among others, that the transitive OMLs coincide with the Boolean algebras, that the OMSLs coincide with the OMLs, that the OMLs are included in OMWLs and that the OMWLs are a proper subclass of QMV algebras. The transitive and/or the antisymmetric case is also studied.

Ideals in MV algebras are, by definition, kernels of homomorphism. An ideal is the dual of a filter in some special logical algebras but not in non-regular residuated lattices. Ideals in residuated lattices are defined as natural generalizations of ideals in MV algebras. Spec(L), the spectrum of a residuated lattice L, is the set of all prime ideals of L and it can be endowed with the spectral topology. The main scope of this paper is to characterize Spec(L), called the stable topology. In this paper, we introduce and investigate the notion of pure ideal in residuated lattices and using these ideals we study the related spectral topologies. Also, using the model of MV algebras, for a De Morgan residuated lattice L, we construct the Belluce lattice associated with L. This will provide informations about the pure ideals and the prime ideals space of L. So, in this paper we generalize some results relative to MV algebras to the case of residuated lattices.

In this paper we develop a theory of ⊤-nets and study their relation to ⊤-filters. We show that convergence in strong L-topological spaces can be described by both ⊤-nets and ⊤-filters and that both concepts are equivalent in the sense that definitions and proofs that are given using ⊤-filters can also be given using ⊤-nets and vice versa.

NeutroAlgebra & AntiAlgebra vs. Classical Algebra is alike Realism vs. Idealism. The Classical Algebra does not leave room for partially true axioms nor for partially well-defined operations. Our world is full of indeterminate (unclear, conflicting, unknown etc.) data.This paper is a review of the emerging, development and applications of the NeutroAlgebra and AntiAlgebra [2019-2022] as generalizations and alternatives of the classical algebras.

The aim of this paper is to introduce and study a new concept of convergence almost surely (a.s.), convergence in probability, convergence in mean, and convergence in distribution are four important convergence concepts of random sequence and also discusses some convergence concepts of the fuzzy sequence: convergence almost surely, convergence in credibility, convergence in mean, and convergence in distribution.

The hyperstructures have applications in mathematics and other sciences such as biology, physics, linguistics, sociology, to mention but a few. For this, mainly, the largest class of the hyperstructures, the Hv-structures, is used, which satisfy the weak axioms where the non-empty intersection replaces the equality and they are straightly related to fuzzy set theory. The fundamental relations connect the Hv-structures with the classical ones, moreover, they reveal new concepts as the Hv-fields. Hv-numbers are called the elements of an Hv-field and they are used in representation theory. We introduce the raised finite Hv-fields, and present some results and examples on 2 × 2 representations on them.

Learning, a universal process that all individuals experience, is a fundamental component of human cognition. It combines cognitive, emotional and environmental in uences for acquiring or enhancing ones knowledge and skills. Volumes of research have been written about learning and many theories have been developed for the description of its mechanisms. The goal was to understand objectively how people learn and then develop teaching approaches accordingly. In this paper soft sets, a generalization of fuzzy sets introduced in 1999 by D. Molodstov as a new mathematical tool for dealing with the uncertainty in a parametric manner, are used for assessing student learning skills with the help of the Blooms taxonomy. Blooms taxonomy has been applied and is still applied by generations of teachers as a teaching tool to help balance assessment by ensuring that all orders of thinking are exercised in student learning. The innovative assessment method introduced in this paper is very useful when the assessment has qualitative rather than quantitative characteristics. A classroom application is also presented illustrating its applicability under real conditions.

Given a fuzzy subgroup $\mu$ of a group $G$‎, ‎$x\rhd_uy$ if and only if $\mu(xy) < \mu(yx)$ defines a directed relation with an associated digraph $(G‎, ‎\rhd_u)$‎. ‎We consider $(\mu‎, ‎\nu)$-homomorphisms $\varphi‎: ‎(G‎, ‎\mu)\to (H‎, ‎\nu)$ where $\mu$ and $\nu$ are fuzzy subgroups of $G$ and $H$ respectively and the preservation of properties of the digraphs $(G‎, ‎\rhd_u)$ several of which are also noted here‎, ‎e.g.‎, ‎$(G‎, ‎\rhd_u)$ is an anti-chain if and only if $\mu$ is a fuzzy normal subgroup of the group $G$‎.

An ideal I of a ring R is a lifting ideal if the idempotents of R can be lifted modulo I. A rich literature has been dedicated to lifting ideals. Recently, new algebraic and topological results on lifting ideals have been discovered. This paper aims to generalize some of these results to coherent quantales. We introduce the notion of lifting elements in a quantale and a lot of results about them are proven. Some properties and characterizations of a coherent quantale in which any element is a lifting element are obtained. The formulations and the proofs of our results use the transfer properties of reticulation, a construction that assigns to each coherent quantale a bounded distributive lattice. The abstract results on lifting elements can be applied to study some Boolean lifting properties in concrete algebraic structures: commutative rings, bounded distributive lattices, residuated lattices, MV-algebras, BL-algebras, abelian l-groups, some classes of universal algebras, etc.

In this paper, we extend some results of nonstandard analysis to include concepts from fuzzy mathematics. Then we apply our results to issues from special and general relativity and the theory of light-clocks. The extension includes concepts of fuzzy numbers, continuity, and differentiability. Our goal is to provide a new research area in fuzzy mathematics.

Generally, linear programming (LP) problem is the most extensively utilized technique for solving and optimizing real-world problems due to its simplicity and efficiency. However, to deal with the inaccurate data, the neutrosophic set theory comes into play, which creates a simulation of the human decision-making process by considering all parts of the choice (i.e., agree, not sure, and disagree). Keeping the bene ts in mind, we proposed the neutrosophic LP models based on triangular neutrosophic numbers (TNN) and provided a method for solving them. Fuzzy LP problem can be converted into crips LP problem based on the de ned ranking function. The provided technique has been demonstrated with numerical examples given by Abdelfattah. Finally, we found that, when compared to previous approaches, the suggested method is simpler, more efficient, and capable of solving all types of fuzzy LP models.

Experts usually express their degrees of belief in‎ ‎their statements by the words of a natural language (like ``maybe''‎, ‎``perhaps''‎, ‎etc.) If an expert system contains the degrees of‎ ‎beliefs $t(A)$ and $t(B)$ that correspond to the statements $A$‎ ‎and $B$‎, ‎and a user asks this expert system whether ``$A\,\&\,B$'' is‎ ‎true‎, ‎then it‎ ‎is necessary to come up with a reasonable estimate for the‎ ‎degree of belief of $A\,\&\,B$‎. ‎The operation that processes $t(A)$‎ ‎and $t(B)$ into such an estimate $t(A\,\&\,B)$ is called an $\&$-operation‎. ‎Many‎ ‎different $\&$-operations have been proposed‎. ‎Which of them to‎ ‎choose? This can be (in principle) done by interviewing experts and‎ ‎eliciting a $\&$-operation from them‎, ‎but such a process is very‎ ‎time-consuming and therefore‎, ‎not always possible‎. ‎So‎, ‎usually‎, ‎to choose a $\&$-operation‎, ‎we extend the finite‎ ‎set of actually possible degrees of belief to an infinite set‎ ‎(e.g.‎, ‎to an interval [0,1])‎, ‎define an operation there‎, ‎and‎ ‎then restrict this operation to the finite set‎. ‎In this paper‎, ‎we consider only this original finite set‎. ‎We show that a‎ ‎reasonable assumption that an $\&$-operation is continuous (i.e.‎, ‎that gradual change in $t(A)$ and $t(B)$ must lead to a gradual‎ ‎change in $t(A\,\&\,B)$)‎, ‎uniquely determines $\min$ as an‎ ‎$\&$-operation‎. ‎Likewise‎, ‎$\max$ is the only continuous‎ ‎$\vee$-operation‎. ‎These results are in good accordance with the‎ ‎experimental analysis of ``and'' and ``or'' in human beliefs‎.