Chapter 4 is the heart of this book in that it establishes lattice algebra as a new branch of mathematics. The foundation of lattice algebra rests on two pillars, the first being the idea of lattice semigroups and lattice groups, while the second is the not so well-known minimax algebra. Lattice semigroups and lattice groups are defined in Section 4.1 and minimax algebra is defined in Section 4.2. Section 4.3 merges the two pillars and creates lattice algebra. Some highlights of this algebra are the concepts of lattice vector spaces, lattice independence, bases and dual bases of lattice vector spaces, and the span of subsets of lattice vector spaces. It is important to note that the definition of a lattice vector space mimics that of a vector space in linear algebra, but they are definitely not mathematicly equivalent. To avoid confusion, a lattice-based vector space is denoted by £-vector space. Similarly, lattice independence and lattice-based spans are denoted £-independence and £-span, respectively.

The final section of the chapter provides a concise geometric description of the £-span S (X) for any nonempty subset X of an £-vector space. The geometry of S (X) plays a key role in the subsequent three chapters.