REDUCING FUZZY ALGEBRA TO CLASSICAL ALGEBRA

Abstract

This paper presents three main ideas. They are the Metatheorem, the lattice embedding for sets, and the lattice embedding for algebras. The Metatheorem allows you to convert existing theorems about classical subsets into corresponding theorem about fuzzy subsets. The concept of a fuzzyfiable operation on a powerset is defined. The main result states that any implication or identity which can be stated using fuzzyfiable operations is true about fuzzy subsets if and only if it is true about classical subsets. The lattice embedding theorem for sets shows that for any set X, there is a set Y such that the lattice of fuzzy subsets of X is isomorphic to a sublattice of the classical subsets of Y. In fact it is further proved that if X is infinite, then we can choose Y = X and get the surprising result that the lattice of fuzzy subsets of X is isomorphic to a sublattice of the classical subsets of X itself. The idea is illustrated with an example explicitly showing how the lattice of fuzzy subsets of the closed unit interval 𝕀 = [0,1] embeds into the lattice of classical subsets of 𝕀. The lattice embedding theorem for algebras shows that under certain circumstances the lattice of fuzzy subalgebras of an algebra A embeds into the lattice of classical subalgebras of a closely related algebra A′. The following sample use of this embeding theorem is given. It is a well known fact that the lattice of normal subgroups of a group is a modular lattice. The embeding theorem is used here to conclude that lattice of fuzzy normal subgroups of a group is a modular lattice too.