Journal of Algebraic Hyperstructures and Logical Algebras
Volume:6 Issue: 1, Winter 2025

We define various types of states on implicative involutive BE-algebras (Jauch-Piron state, (P)-state, (B)-state, subadditive  state, valuation), and we investigate the relationships between these states. Moreover, we introduce the unital, full, and  rich sets of states, and we prove certain properties involving these notions. In the case when an implicative involutive BE- algebra possesses a rich or a full set of states, we prove that it is an implicative-orthomodular lattice. If an implicative  involutive BE-algebra possesses a rich set of (P)-states or a full set of valuations, then it is an implicative-Boolean algebra.  Additionally, based on their deductive systems, we give characterizations of implicative-orthomodular lattices and  implicative-Boolean algebras.

The notion of fuzzy points is used to study implicative filters in equality algebras. The concept of implicative (∈, ∈)-fuzzy  filters is introduced, and its properties are investigated. The relationship between (∈, ∈)-fuzzy filters, implicative (∈, ∈)- fuzzy filters and positive implicative (∈, ∈)-fuzzy filters are established. Characterizations of implicative (∈, ∈)-fuzzy filters  are discussed, and the conditions for the (positive implicative) (∈, ∈)-fuzzy filter to be an implicative (∈, ∈)-fuzzy filter are  explored. The extension property for the implicative (∈, ∈)- fuzzy filter is established. The properties associated with ∈tset  and qt-set of a fuzzy set Ψ in E and also look for conditions under which they can become implicative filters are  examined.

Using the fuzzy points, the concept of (∈, ∈)-fuzzy filters of equality algebras is introduced, and related properties are investigated. Characterizations of (∈, ∈)-fuzzy filters are explored, and the conditions under which ∈t-set, qt-set, and positive set can be filters are investigated.

In this paper, we combine the notions of near sets and lattices. We initiate the study of near approximation in a lattice. Then we investigate nearness approximation space in upper rough ideals and lower rough ideals. In particular, we give some properties of them.

This paper introduces Z◦J -ideals in MV -algebras, extending the concept of Z◦ -ideals. Additionally, a broader class of Z◦J -ideals, known as Z◦NJ -ideals, is defined. We then proceed to explore the relationships between the Z◦J - ideals and other  types of ideals, focusing on their interactions within the framework of special MV -algebras, for example, semisimple MV - algebras, MV -chain and hyperarchimedean. In general, the aim of this paper is to provide a comprehensive analysis of  these ideals in MV - algebras.

This paper presents a comprehensive study of internal and external MBJ-neutrosophic sets, detailing their definitions, properties, and potential applications. By Initially introducing foundational concepts, including interval numbers and neutrosophic sets, the paper explores the MBJ-neutrosophic framework, which integrates truth, indeterminate, and false membership functions. We categorize MBJ-neutrosophic sets into internal and external types according to specific membership conditions, presenting a series of propositions and theorems that clarify the relationships among different MBJ-neutrosophic set types. Additionally, we examine criteria under which the intersection and union of these sets maintain their internal or external characteristics. The findings significantly enhance the theoretical structure of neutrosophic set theory and highlight its utility in decision-making processes and fuzzy logic systems.

The aim of this paper is to introduce the concepts of m-systems and n-systems in BCI-algebras. These concepts are related to the concepts of prime ideals and ∧-closed systems play an important role in studying the structure of BCI-algebras, so it seems to be interesting to study them.

In this paper, we construct a topology using the concept of Hoop algebras and investigate its topological properties. These include examining specific types of topological spaces, such as Hausdorff, $T_0$-spaces, and $T_1$-spaces, as well as exploring the concept of connectedness. Additionally, we analyse the relationship between closed and compact sets within this topology. Finally, by incorporating the binary operation $\ri$ and the defined topology on Hoop algebras, we introduce the notion of semi-topological algebra and demonstrate that every Hoop algebra is a right semi-topological algebra.

This paper explores a hierarchy of algebraic structures beginning with ordered semigroups and extending through their generalizations and fuzzifications. An ordered semigroup is a semigroup equipped with a partial order that is compatible with its binary operation. Building on this foundation, we investigate fuzzy ideals within ordered semigroups, where fuzzy set theory introduces gradation and uncertainty into ideal theory. We then extend the framework to ordered semihypergroups, where the binary operation is replaced by a hyperoperation mapping element pairs to subsets, allowing the modeling of systems with multi-valued outputs under an order-preserving structure. From there, we examine methods for constructing ordered semigroups derived from ordered semihypergroups, using representative elements and congruences to recover associativity in a classical sense. Finally, we introduce the fuzzification of ordered semihypergroups, integrating fuzzy logic into hyperoperations to develop fuzzy hyperideals, hyperfilters and explore their properties. This comprehensive study highlights the interplay between order, hyperstructure, and fuzziness, providing a generalized framework for both theoretical advancement and practical applications in areas involving uncertainty and complex relationships.