Membership function (MF), also called vertical membership function (VMF), is an essential object of study in fuzzy concepts which models fuzzy sets using different sorts of data or fuzzy hypotheses. Fuzzy membership functions are classified based on the number of fuzzy parameters. According to the definition of VMF, the degree of membership <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mu(x)$</tex> can be ambiguous to the variable <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$x$</tex>. It is necessary to construct horizontal membership functions (HMF) to ensure that each function is unambiguous. HMF concept needs to be developed further to solve the problem containing uncertain variables. HMF are constructed using the definition of VMF. The paper presents the construction of HMF with two parameters like Gaussian, Sigmoid, Rectangular, S-Shaped, and Z-Shaped membership functions. Visualization of construction results are described to observe how similar the visualizations between the two-parameter HMF types. Some examples of fuzzy optimal control problems are given to be solved implementing two-parameter HMF types.

Due to its intrinsic complexity, medical diagnosis presents a challenge to typical computational methods due to its wide range of variables, uncertainties, and imprecise information. In order to arrive at an accurate diagnosis in this dynamic sector, medical practitioners must synthesise a multitude of data, from lab results and imaging reports to patient history and symptoms. Nonetheless, clinical decision-making frequently entails the use of vague or insufficient facts, which adds a subjective aspect and increases the possibility of error. Fuzzy logic-based decision support systems (DSS) have become an effective tool for addressing these issues, as they improve diagnostic accuracy while also facilitating the provision of personalised treatment.

The paper investigates the existence of a mild solution for a fuzzy fractional differential equation involving the Caputo–Katugampola fractional derivative. The main results are established using Mönch's fixed point theorem, along with essential tools like semigroup theory, fractional calculus, and the measure of noncompactness. Finally, our theoretical results are applied by providing an interesting example.

In this paper, F-augmented closure spaces are generalized to F-closure spaces, and the concept of F-closed sets are introduced. Properties of their ordered structures are investigated. Representations of various algebraic domains such as algebraic lattices, algebraic L-domains, BF-domains via F-closure spaces are considered. As applications of these methods, more direct approaches to representing various algebraic domains via classical closure space are given, respectively. F-relations between F-closure spaces are defined and properties of them are examined. It is also proved that the category of algebraic domains with Scott continuous maps is equivalent to that of F-closure spaces with F-relations.

This paper explores the synergy between fuzzy set theory and genetic algorithms in the domain of epidemic modeling, shedding light on the broader challenges inherent in infectious diseases. In response to the evolving landscape of epidemiology, the study advocates for sophisticated modeling techniques to better understand and predict dynamic scenarios. The research thoroughly investigates the global stability of equilibrium solutions within a fuzzy epidemic model, accompanied by an inventive parameter estimation methodology. Utilizing Lyapunov functions, the paper establishes global stability outcomes for both disease-free and endemic equilibria. The novel approach, integrating continued fraction theory and a genetic algorithm, is applied to fit realworld epidemic data, offering insights crucial for public health planning and control measures. By addressing unexplored facets and bridging gaps in previous research, the paper contributes to a more comprehensive understanding of epidemic dynamics, providing a foundation for effective public health strategies in the face of infectious diseases.

In this article, a novel idea of domination degree and index are defined in a fuzzy graph (FG) using weight of strong edges. The strong domination degree (SDD) of a vertex u is defined using the weight of minimal strong dominating set (MSDS) containing u. Methods to obtain an MSDS containing a particular vertex are discussed in the article. Idea of upper strong domination number, strong irredundance number, strong upper irredundance number, strong independent domination number, and strong independence number are explained and illustrated subsequently. Strong domination index (SDI) of an FG is defined using the SDD of each vertex. The concept is applied on various FGs like complete FG, complete bipartite and r-partite FG, fuzzy tree, fuzzy cycle, and fuzzy stars. Bounds involving the SDD and SDI are also obtained. Applications for SDD of a vertex is also provided.

In this paper, a systematic theory for the ideals of an L-ring has been developed. Earlier the authors have introduced the concepts of prime ideals, semiprime ideals, primary ideals, and radical of an ideal in an Lring. They have also introduced and discussed the notion of a maximal ideal in different papers wherein several results pertaining to these notions have been proved. In this paper, the concepts of associated prime ideal, minimal prime ideal, and that of irreducibility of an ideal in an L-ring have been introduced, which in fact, are the continuation of authors' previous works in the theory of ideals in L-rings. prime ideal; semiprime ideal; primary ideal; radical of an ideal; L-subring S o far in the studies of fuzzy algebraic substructures, the definition of a subalgebra has been formulated in the framework of classical algebra. In fact, from Rosenfeld [1] onwards, almost all the researchers defined and studied the notion of a fuzzy subgroup of an ordinary group. Similar is the situation in the theory of fuzzy rings or lattice valued fuzzy rings [2−6] . Here we mention that such studies were initiated by Liu [7] . In Refs. [8, 9] and in the theory of L-subgroups [10−17] , there is a deviation from this approach. Wu [18] gave a hint of pursuing studies of a fuzzy subgroup in a fuzzy group by providing the definition of a fuzzy normal subgroup in a fuzzy group. This idea was further taken up by Martinez [19, 20] . In fact, in Ref. [10], we studied the notion of a normal fuzzy subgroup of a fuzzy group and defined a characteristic fuzzy subgroup of a fuzzy group. This laid the foundation of the theory of fuzzy subalgebras where the parent structures are also fuzzy algebras. The development of fuzzy algebraic structures was hampered mainly because of two reasons. Firstly, it was due to some inherent problem that the analogs of some of the concepts and results of classical algebra were not even formulated in this setting such as transitivity of properties of subsets of an algebra. For example: "a normal subgroup of a characteristic subgroup of a group is a normal subgroup of the given group"; and "a prime ideal of an ideal of a ring is an ideal of the given ring". Moreover, the concept of maximality of a fuzzy substructure of an algebraic structure is not properly formulated. Secondly, since most of the concepts studied in fuzzy algebraic