Characterizations of Right Weakly Regular Semigroups in Terms of Generalized Cubic Soft Sets

Aslıhan Sezgin,Muhammad Gulistan,Feng Feng,Madad Khan

doi:10.3390/math6120293

Aslıhan Sezgin, Muhammad Gulistan + Show 2 more

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https://doi.org/10.3390/math6120293

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Abstract

Cubic sets are the very useful generalization of fuzzy sets where one is allowed to extend the output through a subinterval of [ 0 , 1 ] and a number from [ 0 , 1 ] . Generalized cubic sets generalized the cubic sets with the help of cubic point. On the other hand Soft sets were proved to be very effective tool for handling imprecision. Semigroups are the associative structures have many applications in the theory of Automata. In this paper we blend the idea of cubic sets, generalized cubic sets and semigroups with the soft sets in order to develop a generalized approach namely generalized cubic soft sets in semigroups. As the ideal theory play a fundamental role in algebraic structures through this we can make a quotient structures. So we apply the idea of neutrosophic cubic soft sets in a very particular class of semigroups namely weakly regular semigroups and characterize it through different types of ideals. By using generalized cubic soft sets we define different types of generalized cubic soft ideals in semigroups through three different ways. We discuss a relationship between the generalized cubic soft ideals and characteristic functions and cubic level sets after providing some basic operations. We discuss two different lattice structures in semigroups and show that in the case when a semigroup is regular both structures coincides with each other. We characterize right weakly regular semigroups using different types of generalized cubic soft ideals. In this characterization we use some classical results as without them we cannot prove the inter relationship between a weakly regular semigroups and generalized cubic soft ideals. This generalization leads us to a new research direction in algebraic structures and in decision making theory.

Highlights

To handle uncertainty in many real world problems the existing methods are not sufficient.To reduce these uncertainties, a few sorts of speculations were presented like hypothesis of fuzzy sets [1], intuitionistic fuzzy sets [2] and rough sets [3]
In the following we provide some applications of generalized cubic soft sets
In this paper we introduced the concept of generalized cubic soft sets which is the most general approach and characterize the right weakly regular semigroups in terms of generalized cubic soft ideals

Summary

Introduction

To handle uncertainty in many real world problems the existing methods are not sufficient. Initiated the new approach namely soft sets which is a new theory and has the ability to capture the uncertainty in a better way After this many researchers used the idea of soft sets in many directions, such as Maji et al [5], Maji et al [6], Aktas and Cagman [7], and Jun et al [8,9]. Another general version of fuzzy sets and intutionistic fuzzy sets was presented by Jun et al [15]. Soft sets and apply on the right weakly regular semigroups studied by Feng et al [35] and Khan et al [36]. We provide an application of the current proposal and conclusion is given at the end

Preliminaries

Cubic Soft Sets

Generalized Cubic Soft Ideals of Semigroups

Application

Conclusions