On properties of fuzzy left h-ideals in hemirings with t-norms

In [23], Zadeh introduced the notion of fuzzy sets and fuzzy set operations. Since then, fuzzy set theory developed by Zadeh and others has evoked great interest among researchers working in different branches of mathematics. Semirings play an important role in studying matrices and determinants. Many aspects of the theory of matrices and determinants over semirings have been studied by Beasley and Pullman [4, 5], Ghosh [10], and others. Although ideals in semirings are useful for many purposes, they do not in general coincide with the usual ring ideals if S is a ring, and for this reason, their use is somewhat limited in trying to obtain analogues of ring theorems for semirings. Indeed, many results in rings apparently have no analogues in semirings using only ideals. Henriksen [12] defined a more restricted class of ideals in semirings, which is called k-ideals, with the property that if the semiring S is a ring, then a complex in S is a k-ideal if and only if it is a ring ideal. A still more restricted class of ideals in hemirings has been given by Iizuka [13]. However, a definition of ideals in any additively commutative semiring S can be given which coincides with Iizuka’s definition provided that S is a hemiring, and it is called h-ideal. La Torre [19] investigated h-ideals and k-ideals in hemirings in an effort to obtain analogues of familiar ring theorems. Several authors have discussed a fuzzy theory in semirings (see [2, 3, 7, 8, 9, 11, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25]). In this paper, we introduce the concept of T-fuzzy left h-ideals in a hemiring and investigated some basic properties. Using a collection of left h-ideals of a hemiring S, T-fuzzy left h-ideals of S are established. Moreover, we introduce the notion of a finitevalued T-fuzzy left h-ideal and give its characterization. At last, we discuss the T-fuzzy relation on a hemiring S.