ABSTRACT
As we saw in Chapter 1, any algebra A has associated with it a lattice, the lattice Con(A) of all congruence relations on A. We can often use properties of the algebra A itself to deduce properties of the associated congruence lattice, and we can also sometimes use properties of Con(A) to tell us about the algebra A as well. Thus we want to relate properties of lattices, such as permutability, distributivity, modularity, and so on, to properties of algebras and varieties. The first result in this direction was given by A. I. Mal'cev in 1954 ([74], [75]): he showed that all the congruence relations of any algebra in a variety are permutable with respect to the relational product if the variety satisfies a certain identity (equality of terms). The special term used in this identity is called a Mal'cev term, and theorems like this one which relate properties of the congruence lattices of all the algebras in a variety to identities of the variety are usually called Mal'cev-type conditions. In this chapter we investigate a number of properties of congruence lattices, and the corresponding Mal'cev-type conditions. We also investigate these properties in detail for a particular example, the case of varieties generated by algebras of size two; our analysis here will be used in Chapter 10 to describe the lattice of all clones on a two-element set.
