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Sets are fundamental in mathematics. In this chapter we briefly introduce the concepts and notations from set theory we will use throughout the book. We assume that the reader is familiar with the basic concepts of set theory. He may use some kind of naive set theory or a formal theory as ZF or ZFC [18], i.e., the Zermelo-Fraenkel axioms of set theory. As usual, we denote the fact that “ is an element of a set ” by . The set with no elements is called the , and is denoted by . If every element of a set is also an element of the set , we say is a of denoted by .

In this chapter we want to introduce basic concepts from lattice theory we will need throughout this book. For a comprehensive introduction to this theory we refer to [4, 16, 29].

As mentioned in the introduction, for a complete Brouwerian lattice an fuzzy relation between two nonempty sets and is a function from to . Notice, if = B, we get the set of regular binary relations between and . Therefore, we also use the denotation : to indicate that an -fuzzy relation R has source and target .

Usually a binary relation acts between two different sets. Therefore, an algebraic theory for relations should reflect this kind of typing, i.e., the theory should have a suitable notion of source and target of its elements. A convenient framework for that is given by category theory.

The notion of crispness is a basic property of -fuzzy relations and sets such that a suitable algebraic theory should be able to express this property. We have shown that there are some notions of crispness within Dedekind categories, which grasp the notion of 0–1 crispness under an assumption on the underlying lattice. Unfortunately, a general notion, which coincides with 0–1 crispness has not yet been given.

Following our mathematical investigation on the theory of Goguen categories, we now want to focus on an applications in computer science. Throughout this chapter, we will use the notations of Goguen categories even if the relations are concrete -fuzzy relations. Furthermore, is considered to be an operation from a closg (Sc[], I) unless otherwise stated.