Rough fuzzy ternary subsemigroups based on fuzzy ideals with three-dimensional congruence relation

The main objective of the proposed work in this paper is to introduce a generalized form of rough fuzzy subsemigroups, which is rough fuzzy ternary subsemigroups (RFTSs) combining the notions of fuzziness and roughness in ternary semigroups. In RFTSs, we deal with vague and incomplete information in decision-making problems. RFTSs are characterized by lower and upper approximations using fuzzy ideals. In this research, we propose the three-dimensional k-level relation and proved that this relation is a congruence relation on a ternary semigroup. Furthermore, comparing it with the previous literature, we conclude that our proposed technique is better and effective because it deals with vague problems and there are many structures which are not handled using binary multiplication such as all the sets of negative numbers. In addition, we have proved by counterexamples that converses of many parts of many results do not hold which have negated the results proved in Q. Wang’s paper.