Abstract
Prime fuzzy ideals, prime fuzzyk-ideals, and prime fuzzyh-ideals are roped in one condition. It is shown that this way better fuzzification is achieved. Other major results of the paper are: every fuzzy ideal (resp.,k-ideal,h-ideal) is contained in a prime fuzzy ideal (resp.,k-ideal,h-ideal). Prime radicals and nil radicals of a fuzzy ideal are defined; their relationship is established. The nil radical of a fuzzyk-ideal (resp., anh-ideal) is proved to be a fuzzyk-ideal (resp.,h-ideal). The correspondence theorems for different types of fuzzy ideals of hemirings are established. The concept of primary fuzzy ideal is introduced. Minimum imperative for proper fuzzification is suggested and it is shown that the fuzzifications introduced in this paper are proper fuzzifications.
Highlights
Several attempts have been made to fuzzify the concepts of prime ideals/k-ideals/h-ideals of a semiring [1,2,3,4,5,6,7], prime ideals of a ring [8,9,10,11,12,13,14,15], and prime ideals of a semigroup [16,17,18]
When the zero element of the valuation lattice is not a prime element, even the characteristic function of a prime ideal fails to be a prime fuzzy ideal
While proving major results of the paper, only 2-valued prime fuzzy ideals are used.) In this paper, we show that the problem of fuzzification of left ideal, left k-ideal, and left h-ideal need not be tackled separately
Summary
This paper is, in some sense, an extended version of the article “On Fuzzification of Prime Ideals with Special Reference to Semirings” in SciTopics and something more. We have discussed elsewhere [6], in detail, the deficiencies in the definition of a prime fuzzy h-ideal proposed in [7]. The definition suffers from three major drawbacks It is very restrictive in the sense that the fuzzy h-ideals, which are prime according to the definition, are 2-valued function. The technique adopted for the fuzzification by Zhan and Dudek in [7] and by others in [1,2,3, 5] is identical Their prime fuzzy ideals inherit the same drawbacks. While proving major results of the paper, only 2-valued prime fuzzy ideals are used.) In this paper, we show that the problem of fuzzification of left ideal, left k-ideal, and left h-ideal need not be tackled separately. Fuzzifications introduced in this paper can be labeled as “proper fuzzifications”
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