Journal of Algebraic Hyperstructures and Logical Algebras
Volume:4 Issue: 2, Spring 2023

We consider properties of L-fuzzy relations and L-normal operators for a residuated lattice L in detail and show that the class RL(U) of all L-fuzzy relations on U and the class NL(U) of all L-normal operators are residuated lattices and they are isomorphic as lattices. Moreover, we prove that for any L-normal operators F, it is reflexive (or transitive) if and only if the L-fuzzy relation RF induced by F is reflexive (or transitive) respectively.

This article aims to provide a method for defining L-fuzzy algebraic substructures on general algebras. Concretely, the  properties of L-fuzzy sets are first reviewed, and their representations are then provided. Next, algebraic substructures  are generalised as the closure systems on the power set of the algebra, and the properties of the prime and maximal elements in the above closure system are investigated. Based on these facts, L-fuzzy algebraic substructures with respect to the closure system are defined and studied. Two equivalence characterisations of the sup property of the ordered set L are provided using L-fuzzy substructures. Similarly, some properties of L-fuzzy prime and maximal substructures with respect to the closure system are discussed. Finally, to demonstrate the broad applicability of the theory of L-fuzzy  algebraic substructures, the theory is applied to some specific algebraic structures, such as groups and pseudo MV-algebras.

In this paper, modal operators on BCK-algebras, especially BCK-algebras with condition (S) are introduced and several properties and characterizations of them are investigated. Also, it is investigated under what conditions these modal  operators form a lattice. Furthermore, some special modal operators are introduced and their properties and  characterizations of them are obtained, especially in some classes of BCK-algebras such as positive implicative BCK- algebras.

The main goal of this paper is to introduce integral hyper EQ-algebras, integral (pre)filters and obstinate (pre)filters of hyper EQ-algebras. In the following, some characterizations of these (pre)filters in hyper EQ-algebras are investigated and it is proved that the quotient hyper EQ-algebras induced by a filter F is an integral hyper EQ-algebra if and only if F is an integral filter. Moreover, the concept obstinate (pre)filter in hyper EQ-algebras is introduced and some related properties are provided. Finally, the relationship among obstinate (pre)filters and some type of other (pre)filters such integral, maximal, (positive) implicative and fantastic (pre)filters in hyper EQ-algebras are studied.

In this paper, two related quotient structures are investigated utilizing the concept of coset. At first, a new hypervector space F/V = (F/V,\circ,\circledcirc,K) is created, which is composed of all cosets of a bipolar fuzzy soft set (F;A) over a hypervector space V . Then it will be shown that dim F/V = dim V/W, where the quotient hypervector space V/W includes all cosets of an especial subhyperspace W of V. Also, three bipolar fuzzy soft sets over the quotient hypervector space V/W are presented and in this way some new bipolar fuzzy soft hypervector spaces are defined.

Rough is an exceptional mathematical tool for effectively analyzing and addressing the complexities of vague action descriptions in decision problems. This paper explores the concept of an L-algebra, which leads to the introduction of  lower and upper approximations. The properties of these approximations are also discussed and elucidated. Furthermore, it is proven that the lower and upper approximations serve as interior and closure operators, respectively. Additionally, by employing A-lower and A-upper approximations, this paper presents and examines conditions for a nonempty  subset to be definable. Furthermore, we investigated the circumstances under which the A-lower and A-upper approximations can be rough ideals. Finally, we define an operation ‎" -->"‎ on the set of all upper approximations of L  ‎and ‎prove ‎that ‎it ‎is ‎made ‎an ‎‎L-algebra.

The present study aims at introducing a new concept of layer of LB-valued general fuzzy automata (LB-valued GFA)  where B is regarded as a set of propositions about the GFA, in which its underlying structure has been a lattice-ordered  monoid. In general, it demonstrates that the layer plays a key role in the algebraic study of LB-valued GFA by  characterizing the concepts of subautomata and separated subautomata of an LB-valued GFA in terms of its layers. In  other words, it highlights that every LB-valued general fuzzy automaton has at least one strongly connected  subautomaton. In specific, the characterization of some algebraic concepts such as subautomaton, retrievability and  connectivity of an LB-valued GFA in terms of its layers is provided. In addition, it is shown that the maximal layer of a  cyclic LB-valued general fuzzy automaton and minimal layer of a directable LB-valued general fuzzy automaton are  unique. Finally, we investigate the different poset structures associated with an LB-valued general fuzzy automaton,  demonstrating some of these posets as finite upper semilattice, and introducing the isotone Galois connections between  some of the pairs of the posets/finite upper semilattices introduced.

We introduce a new regular relation δ on a given group G and show that δ is a congurence relation on G, with respect to  module the commutator subgroup of G. Then we show that the effect of this relation on the fundamental relation β is equal to the fundamental relation γ, and we conclude that, if ρ is an arbitrary strongly regular relation on the hypergroup  H, then the effect of δ on ρ, results in a strongly regular relation on H such that its quotient is an abelian group.

‎This paper introduces the concepts of reproduced general hyperring and valued-orderable general hyperring and investigates some properties of these classes of general hyperrings‎. ‎It presents the notions of‎ ‎zero divisors and zero  divisor graphs are founded on the absorbing elements of general hyperrings‎. ‎General hyperrings can have more than one zeroing element‎, ‎and therefore‎, ‎based on the zeroing elements‎, ‎multiple zero divisors can be obtained‎. ‎In this study‎, ‎we  discuss the isomorphism of zero divisor graphs based on the diversity of divisors of zero divisors‎. ‎The non-empty  intersection of the set of absorbing elements and the hyperproduct of zero divisors of general hyperrings play a major  role in the production of zero divisor graphs‎. ‎Indeed it investigated a type classification of zero divisor graphs based on  the finite general hyperrings‎. ‎We discuss the finite reproduced general hyperrings‎, ‎investigate their zero divisor graphs‎, ‎and show that an infinite reproduced general hyperring can have a finite zero divisor graph‎.