On Prime Hesitant Fuzzy Ideals in Semigroups

Introduction

In 1998, Zhang (Zhang, 1998) defined the fuzzy prime ideals and prime fuzzy ideals of rings and discussed the operations on prime fuzzy ideals. In 1999, Dutta and Biswas (Dutta & Biswas, 1998) characterized fuzzy semiprime ideals in the ring of integers This concept has been studied further in (Martinez, 1999). In 2001, Jun and Xin (Jun & Xin, 2001) investigated relationships between fuzzy prime ideals and invertible fuzzy ideals in commutative BCK-algebras. In 2006, Davvaz (Davvaz, 2006) introduced the notion of the new sort of fuzzy subnear-ring (ideal and prime ideal) of a near-ring. In 2007, Zhan and Dudek (Zhan & Dudek, 2007) investigated the same fuzzy ideal in some class of hemirings and proved that a fuzzy subset f of a hemiring R is a prime fuzzy left (right) h-ideal of R if and only if f is two-valued, f(0)=1, and the set of all x in R such that f(x)=1 is a prime (left) right h-ideal of R. In 2008, Kazanci and Davvaz (Kazanci & Davvaz, 2008) introduced and studied the notations of rough prime (primary) ideals and rough fuzzy prime (primary) ideals in rings. In 2009, Kedukodi et al. (Kedukodi et al., 2009) introduced the ideas of equiprime fuzzy ideals, 3-prime fuzzy ideals and c-prime fuzzy ideals of nearrings. In 2010, Kumbhojkar (Kumbhojkar, 2010) introduced and studied the notation of prime fuzzy h-ideals of hemirings. In 2012, Navarro et al. (Navarro et al., 2012) introduced the notion of prime fuzzy ideals over non commutative rings. In 2014, Abdullah (Abdullah, 2014) defined the n-dimensional prime (a,b)-fuzzy ideals in hemirings. In 2001, Xie (Xie, 2001) introduced the ideas of prime, quasi-prime and weakly quasi-prime fuzzy left ideals of semigroups and characterized strongly semisimple semigroups in terms of quasi-prime fuzzy left ideals. In 2006, Xiao and Zhang (Xiao & Zhang, 2006) for example introduced and examined the notions of rough prime ideals and rough fuzzy prime ideals in semigroups. In 2007, Kehayopulu (Kehayopulu, 2007) investigated the same fuzzy ideal in some class of ordered semigroups and proved the fuzzy ideal of an ordered semigroup is prime if and only if it is both semiprime and weakly prime and that in commutative ordered semigroups the prime and weakly prime fuzzy ideals. In 2008, Xie and Tang (Xie & Tang, 2008) gave the concepts of weakly prime fuzzy ideals, completely prime fuzzy ideals, completely semiprime fuzzy ideals and weakly completely prime fuzzy ideals of ordered semigroups. In 2009, Kazanci and Yamak (Kazanci & Yamak, 2009) investigated relationships between different definitions of semiprime fuzzy ideals of fuzzy semigroups. In 2010, Khan and Shabir (Khan & Shabir, 2010) proved that an ordered semigroup is regular if and only if every intuitionistic fuzzy left (respectively, right) ideal of S is idempotent. In particular, they characterized of different classes of ordered semigroups by using intuitionistic fuzzy ideals. In 2011, Tang (Tang, 2011) defined and examined completely semiprime fuzzy ideals of ordered semigroups. In particular, he characterized an ordered semigroup that is a semilattice of simple ordered semigroups in terms of completely semiprime fuzzy ideals of ordered semigroups. In 2012, Rezvi and Mehmood (Rezvi & Mehmood, 2012) studied prime and semiprime fuzzy bi-ideals and proved that in case of the totally ordered set of fuzzy bi-ideals of semigroups, the concept of irreducible prime and strongly irreducible prime coincides. In 2013, Kar et al. (Kar et al., 2013) proved that if 𝜇 is an (i-v) fuzzy ideal of a semigroup S, then 𝜇 is an (i-v) prime fuzzy ideal of S if and only if 𝜇^c is an (i–v) fuzzy m-system of S. In 2016, Wang and Zhan (Wang & Zhan, 2016) studied rough semigroups, rough ideals, rough prime ideals, rough fuzzy semigroups, rough fuzzy ideals and rough fuzzy prime ideals according to the definitions of rough sets and rough fuzzy sets. In 2017, Sarkar and Kar (Sarkar & Kar, 2017) first studied (i-v) left (right) primary fuzzy ideals on semigroups. The theory of prime fuzzy sets on semigroups has been recently developed (Kim, 2009; Shabir et al., 2010), and Γ-semigroups (Chinram, 2009). In 2020, Yiarayong (Yiarayong, 2020) introduced the concept of hesitant fuzzy bi-ideals and hesitant fuzzy interior ideals in ternary semigroups. and investigated some of their interesting properties.