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This research note constructs S-H fuzzy collections and S-H fuzzy partitions of a finite set. The main objective of the paper lies in defining two different operations on a class of S-H fuzzy partitions of a set and, in turn, proving that these give rise to a monoid.

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Overtime, mathematics had been used as a tool in modeling real life phenomenon. In some cases, these problems cannot fit-into the classical deterministic or stochastic modeling techniques, perhaps due system complexity arising from lack of complete knowledge about the phenomenon or some uncertainty. The uncertainty could either be due to lack of clear boundaries in the description of the object or perhaps due to randomness. In this article, we study a mathematical tool discovered in 1965 by Zadeh suitable for modeling real life phenomenon and examined operations on such a tool. Motivated by the work of Zadeh, we studied operators on Type-1 Fuzzy Sets (T1FSs) and Type-2 Fuzzy sets (T2FSs) and provided examples, one of which is a variant of the Yager complement function for which the complement operator was graphically illustrated. The joint and the meet operators were also studied and examples provided. Non-standard operators were defined on T1FSs and T2FSs and also classified into two groups; the triangular-norm (t-norm) and triangular-conorm (t-conorm). Using tnorm and t-conorm, an example was adopted from Castillo and Aguilar to illustrate the computation of the standard operation on T2FSs. Finally, future research direction was provided based on what is yet to be achieved in fuzzy set theory.

The paper argues that the classical set-theoretic foundation for mathematics is too restrictive to be able to model a large class of real-life problems which intrinsically involve ambiguities. Further, it describes how by relaxing the restrictions of definiteness and distinctness imposed on the nature of objects to form a cantorian set, the notions of fuzzy sets and multisets respectively get introduced. Finally, it explicates the relevance of generalizing fuzzy sets to fuzzy multisets.