The generalizations of fuzzy monoids and vague monoids

TL;DR

This paper introduces fuzzy and vague monoids using aggregation operators, classifies fuzzy sets, and explores properties of submonoids, providing a unified framework for fuzzy algebraic structures.

Contribution

It extends the concept of monoids to fuzzy and vague contexts using aggregation operators, including classifications, properties, and special cases like uninorms and nullnorms.

Findings

Classification of fuzzy sets based on aggregation operators

Properties of submonoids of t-norm and t-conorm

Redefinition of vague monoids with aggregation operators

Abstract

In this paper, we present the fuzzy monoids and vague monoids by using aggregation operators. The unit interval with a $t$-norm or a $t$-conorm is a special monoid, so we mainly talk about fuzzy subsets of monoids. Firstly, the classification of fuzzy sets based on some special aggregation operators is discussed. At the same time, we give two basic propositions about submonoids of $t$-norm and $t$-conorm. The fuzzification by uninorm and nullnorm are denoted and some properties can be drawn. Next, we briefly present fuzzy subsets on lattice. Finally, the vague monoids on aggregation operators are redefined and further consider the special cases of uninorms and nullnorms.