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This paper presents a series of important results from the theory of n-hypergroups. Connections with binary relations and with lattices are presented. Special attention is paid to the fundamental relation and to the commutative fundamental relation. In particular, join n-spaces are analyzed. [ABSTRACT FROM AUTHOR]

An multi-polar fuzzy set is a robust mathematical model to examine multipolar, multiattribute, and multi-index data. The multi-polar fuzzy sets was created as a useful mechanism to portray uncertainty in multiattribute decision making. In this article, we consider the theoretical applications of multi-polar fuzzy sets. We present the notion of multi-polar fuzzy sets in ordered semihypergroups and define multi-polar fuzzy hyperideals (bi-hyperideals, quasi hyperideals) in an ordered semihypergroup. Relations between multi-polar fuzzy hyperideals, multi-polar fuzzy bi-hyperideals and multi-polar fuzzy quasi hyperideals are discussed. [ABSTRACT FROM AUTHOR]

The combination of two elements in a group structure is an element, while, in a hypergroup, the combination of two elements is a non-empty set. The use of hypergroups appears mainly in certain subclasses. For instance, polygroups, which are a special subcategory of hypergroups, are used in many branches of mathematics and basic sciences. On the other hand, in a multi-fuzzy set, an element of a universal set may occur more than once with possibly the same or different membership values. A soft set over a universal set is a mapping from parameters to the family of subsets of the universal set. If we substitute the set of all fuzzy subsets of the universal set instead of crisp subsets, then we obtain fuzzy soft sets. Similarly, multi-fuzzy soft sets can be obtained. In this paper, we combine the multi-fuzzy soft set and polygroup structure, from which we obtain a new soft structure called the multi-fuzzy soft polygroup. We analyze the relation between multi-fuzzy soft sets and polygroups. Some algebraic properties of fuzzy soft polygroups and soft polygroups are extended to multi-fuzzy soft polygroups. Some new operations on a multi-fuzzy soft set are defined. In addition to this, we investigate normal multi-fuzzy soft polygroups and present some of their algebraic properties. [ABSTRACT FROM AUTHOR]

The goal of this paper is to introduce a novel soft hyperstructure called multi-fuzzy soft hyperstructure. We investigate the notion of multi-fuzzy soft hypermodules and some of their structural properties are discussed. We discuss the behavior image and inverse image of a multi-fuzzy soft set under the multi-fuzzy soft function. According to Zadeh's extension principle, we prove that the image and inverse image of a multi-fuzzy soft hypermodules are further multi-fuzzy soft hypermodule. [ABSTRACT FROM AUTHOR]

The concept of j-hyperideals, for all positive integers 1 ≤ j ≤ n and n ≥ 2 , in n-ary semihypergroups, is a generalization of the concept of left, lateral and right hyperideals in ternary semihypergroups. In this paper, we first introduce the concept of j-(0-)simple n-ary semihypergroups and discuss their related properties through terms of j-hyperideals. Furthermore, we characterize the minimality and maximality of j-hyperideals in n-ary semihypergroups and establish the relationships between the (0-)minimal, maximal j-hyperideals and the j-(0-)simple n-ary semihypergroups. Our main results are to extend and generalize the results on semihypergroups and ternary semihypergroups. Moreover, a related question raised by Petchkaew and Chinram is solved. [ABSTRACT FROM AUTHOR]

A fuzzy multiset is a generalization of a fuzzy set. This paper aims to combine the innovative notion of fuzzy multisets and hypergroups. In particular, we use fuzzy multisets to introduce the concept of fuzzy multi-hypergroups as a generalization of fuzzy hypergroups. Different operations on fuzzy multi-hypergroups are defined and discussed and some results known for fuzzy hypergroups are generalized to fuzzy multi-hypergroups. [ABSTRACT FROM AUTHOR]

Fuzzy graphs (FGs) and their generalizations have played an essential role in dealing with real-life problems involving uncertainties. The goal of this article is to show some serious flaws in the existing definitions of several root-level generalized FG structures with the help of some counterexamples. To achieve this, first, we aim to improve the existing definition for interval-valued FG, interval-valued intuitionistic FG and their complements, as these existing definitions are not well-defined; i.e., one can obtain some senseless intervals using the existing definitions. The limitations of the existing definitions and the validity of the new definitions are supported with some examples. It is also observed that the notion of a single-valued neutrosophic graph (SVNG) is not well-defined either. The consequences of the existing definition of SVNG are discussed with the help of examples. A new definition of SVNG is developed, and its improvement is demonstrated with some examples. The definition of an interval-valued neutrosophic graph is also modified due to the shortcomings in the current definition, and the validity of the new definition is proved. An application of proposed work is illustrated through a decision-making problem under the framework of SVNG, and its performance is compared with existing work. [ABSTRACT FROM AUTHOR]