Structure of Decidable Locally Finite Varieties pp 171-192 | Cite as

# The decomposition theorem

Chapter

## 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*V*_{1}, an affine variety*V*_{2}, and a discriminator variety*V*_{3}. (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)
Every subdirect product C ≤ C

_{1}× C_{2}× C_{3}with C_{i}∈*V*_{i}(for 1 ≤*i ≤*3) is direct, i.e., C = C_{1}× C_{2}× C_{3}. - (2)
Every subdirect product C ≤ C

_{1}× C_{2}× C_{3}with C_{i}∈*V*_{i}(for 1 ≤*i ≤*3) and C_{i}finite is direct. - (3)
If C = F

_{V}(3) and C_{i}= F_{Vi}(3), then C ≅ C_{1}× C_{2}× C_{3}. - (4)
There exists a term

*t*(*x*_{1},*x*_{2},*x*_{3}) such that*t*(*x*_{1},*x*_{2},_{3}) ≈*x*_{i}is an identity of V; (for 1 ≤*i ≤*3); i.e., the triple of varieties (*V*_{1},*V*_{2},*V*_{3}) is independent.

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© Birkhäuser Boston, Inc. 1989