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

• Main Authors: Muhammad Gulistan, Feng Feng, Madad Khan, Aslıhan Sezgin
• Format: Article
• Language: English
• Published: MDPI AG 2018-11-01
• Series: Mathematics
• Subjects: semigroups; cubic soft sets; cubic soft subsemigroups; cubic soft ideals; lattices
• Online Access: https://www.mdpi.com/2227-7390/6/12/293

Cubic sets are the very useful generalization of fuzzy sets where one is allowed to extend the output through a subinterval of <inline-formula> <math display="inline"> <semantics> <mrow> <mo>[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </semantics> </math> </inline-formula> and a number from <inline-formula> <math display="inline"> <semantics> <mrow> <mo>[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </semantics> </math> </inline-formula>. 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.