ABSTRACT
This chapter provides a brief review of pertinent concepts of lattice theory and its relationships with other branches of mathematics. The chapter is divided into three sections, with the first section examining the history and development of lattice theory. Section 3.2 provides an overview of order relations on sets, the notion of partial order, order-preserving functions, maximal and minimal elements of ordered sets, Hasse diagrams, and the notion of lattices in terms of partially ordered sets. The definition of a lattice is followed by theorems that establish several important properties of the binary min-max operations. The remaining part of Section 2 discusses well-known types of lattices that range from distributive and modular lattices to Boolean lattices.
Section 3.3 presents examples of relations between lattice theory and other branches of mathematics that are relevant in the remainder of the book. The specific branches discussed in this section are topology, measure theory, probability, and fuzzy lattices and similarity measures. Since most engineers have not taken a course in mathematical measure theory, the basic elements of this branch are also defined in this section.
