Universal Algebra

Chapter 2

Lattices

Lattices play a curious double role in universal algebra. On the one hand the lattice concept provides a framework for studying many derived objects that apply to any algebraic structure: the collections of all subalgebras, congruences, varieties, and clones all form lattices. On the other hand, lattices are algebras in their own right. As algebras they have some strong properties, and, more important, properties that are rather different from the ones we are familiar with from the study of groups and rings.

Thus we devote a whole chapter to lattices. While we won’t delve deeply into the study of lattices-as-algebras, we develop enough of the theory to allow us to use it later in our examples.

2.1 Ordered sets

Let