Algebraic Aspects of Fuzzy Sets and Fuzzy Logic

This paper is expository. It is mainly a survey of some of our work on the algebraic systems that arise in fuzzy set theory and logic. We include some of the proofs here and refer to the papers [18, 19, 20, 21, 22] for further details. Our point of view is the algebraic one: when are the various systems that arise isomorphic, and what are their symmetries (automorphisms)? The bulk of the material centers around t-norms, and a typical concern is with the unit interval endowed with its natural order structure, a t-norm, and a negation. A fundamental problem is to determine when two such algebraic systems are isomorphic. Section 2.1 introduces the notions involved in the structure of a deMorgan system on the unit interval. Section 2.1 gives some basic facts about the unit interval I = ([0; 1] ; ) with its order structure. This includes some information about various groups that will play a role throughout. Section 2.2 introduces t-norms, gives representation theorems for them, and determines their isomorphy, and their automorphism groups. Section 2.3 does for negations what Section 2.2 does for t-norms. In Section 2.4, deMorgan systems are introduced. These concern the unit interval with its usual order, a t-norm, a negation, and the corresponding t-conorm. In Section 3, isomorphisms of deMorgan systems with strict t-norms are discussed. A typical result is that any deMorgan system with strict t-norm and strong negation is isomorphic to one whose t-norm is multiplication. The non-uniqueness of the negation in a strict deMorgan system is discussed in Section 3.1. The generators of strict t-norms are determined up to elements of the multiplicative group of positive real numbers. This group can be viewed as a subgroup of the automorphism group of the unit interval, and its normalizer in that group yields a special set of t-norms. In Section 3.2, we determine that normalizer and give explicit formulas for the resulting t-norms, t-conorms and negations.