Additively Inversive Semirings/ Hemirings Characterized By Their Fuzzy Ideals

The main objective of this study is to construct fuzzy analogous using Zadeh’s notion of the following results of Yusuf (Yusuf, 1965) regarding inversive semirings and hemirings. In case of a collection of left/right/two-sided ideals of an additively inversive semiring their non-empty intersection is also left/right/two-sided ideal. For any additively inversive hemiring, the complex sum of two left/right/two sided ideals is also a left/right/two-sided ideal. For an additively inversive hemiring with zero, the complex sum of two left/right/two-sided ideals is the intersection of all left/right/two-sided ideals that contain the two ideals. For an additively inversive hemiring and its two left/right/two-sided ideals with identical intersections with the set of idempotent elements, the complex sum of the two ideals is the intersection of all left/right/two-sided ideals that contain the two ideals. The following theorems manifest the fuzzy analogous of the above results. (i’) For any collection of left/right/two-sided fuzzy ideals of an additively inversive semiring, their non-empty intersection is also a left/right/two-sided fuzzy ideal. (ii’) For an additively inversive hemiring, the complex sum of two left/right/two-sided fuzzy ideals is also a left/right/two-sided fuzzy ideal. (iii’) For an additively inversive hemiring containing absorbing zero ‘0’ and two fuzzy ideals satisfying 0-1 condition, the complex sum of two left/right/two sided fuzzy ideals is the intersection of all left/right/two sided fuzzy ideals that contain the two ideals. (iv’) For an additively inversive hemiring containing absorbing zero ‘0’ and two fuzzy ideals with identical intersections with the set of idempotent elements, satisfying 0-1 condition, the complex sum of two left/right/two sided fuzzy ideals is the intersection of all left/right/two-sided fuzzy ideals that contain the two ideals. It is worth mentioning that the fundamental concept of a fuzzy set, introduced by Zadeh in 1965, has been applied by many authors to generalize some of the basic notions of algebra.