# The decomposition theorem

• Ralph McKenzie
• Matthew Valeriote
Chapter
Part of the Progress in Mathematics book series (PM, volume 79)

## Abstract

In this chapter, we assume that V is a structured locally finite variety. It follows from the work of Parts I and II that V is the join of a strongly Abelian variety V1, an affine variety V2, and a discriminator variety V3. (See Definition 1.1 and Theorems 4.1, 5.4 and 9.6.) In this chapter, we shall prove that V is the product of these three varieties. There are several equivalent ways to formulate the result (see Theorem 0.5):
1. (1)

Every subdirect product C ≤ C1 × C2 × C3 with CiVi (for 1 ≤ i ≤ 3) is direct, i.e., C = C1× C2 × C3.

2. (2)

Every subdirect product C ≤ C1 × C2 × C3 with CiVi (for 1 ≤ i ≤ 3) and Ci finite is direct.

3. (3)

If C = FV(3) and Ci= FVi(3), then C ≅ C1 × C2 × C3.

4. (4)

There exists a term t(x1,x2, x3) such that t(x1, x2, 3) ≈ xi is an identity of V; (for 1 ≤ i ≤ 3); i.e., the triple of varieties (V1,V2, V3) is independent.