New Types of Fuzzy Interior Ideals of Ordered Semigroups Based on Fuzzy Points
Journal: Matrix Science Mathematic (MSMK)
Author: Faiz Muhammad Khan, Nor Haniza Sarmin, Asghar Khan and Hidayat Ullah Khan
This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited
Subscribing to the Zadeh’s idea on fuzzy sets, many researchers strive to identify the key attributes of these sets for new finding in mathematics. In this perspective, new types of fuzzy interior ideals called (∈, ∈ ∨qk)-fuzzy interior ideals of ordered semigroups are reported. Several classes of ordered semigroups such as regular ordered semigroups, intra-regular, simple and semi-simple ordered semigroups are characterized by (∈, ∈ ∨qk)-fuzzy interior ideals and (∈, ∈ ∨qk)-fuzzy ideals. We also prove that in regular (resp. intra-regular and semisimple) ordered semigroups the concept of (∈, ∈ ∨qk)-fuzzy ideals and (∈, ∈ ∨qk)-fuzzy interior ideals coincide. Further, we show that an ordered semigroup S is simple if and only if it is (∈, ∈ ∨qk)-fuzzy simple. The characterization of intra-regular and semi-simple ordered semigroups in terms of (∈, ∈ ∨qk)-fuzzy ideals and (∈, ∈ ∨qk)-fuzzy interior ideals are provided. We define semiprime(∈, ∈ ∨qk)-fuzzy ideals and prove that S is left regular if and only if every(∈, ∈ ∨qk)-fuzzy left ideal is semiprime and S is intra-regular if and only if every (∈, ∈ ∨qk )-fuzzy ideal is semiprime. The concept of upper/lower parts of an (∈, ∈ ∨qk)-fuzzy interior ideal and some interesting results are discussed.