On the lattice and the algebra of fuzzy subsets of a universal algebra

Abstract

For a given universal algebra, the set of all its fuzzy subsets is endowed with two structures: a structure of algebra called the fuzzy algebra (or the algebra of fuzzy subsets) and a structure of bounded lattice called the lattice of fuzzy subsets. In this paper, first of all, we construct some subuniverses of this fuzzy algebra and give a way to generate such subuniverses. We characterize the subuniverse generated by a fuzzy point of this fuzzy algebra. Some subalgebras containing the subalgebra generated by a finite set of fuzzy points of the fuzzy algebra are specified. We also describe some subuniverses of the initial universal algebra induced by those of its fuzzy algebra, and vice versa. We then give some properties of fuzzy subalgebras of this universal algebra, describe some fuzzy subalgebras generated by others fuzzy subalgebras, and give a partial characterization of universal algebras in which the set of all fuzzy subalgebras is a subuniverse of their fuzzy algebras. After that, we characterize some properties which can be transfered between the codomain lattice of fuzzy subsets (the lattice of truth values) and the lattice of fuzzy subsets. Later, we show that there exists some subsets of the set of all fuzzy subsets which can be both subuniverses of the fuzzy algebra and sublattices of the lattice of fuzzy subsets: that means, these subsets are endowed with a double structure like the set of all fuzzy subsets of the given universal algebra; and finally, we introduce the residuation of some of them by defining the residual operations in a certain way.