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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.

     

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Copyright information

© Birkhäuser Boston, Inc. 1989

Authors and Affiliations

  • Ralph McKenzie
    • 1
  • Matthew Valeriote
    • 2
  1. 1.Department of MathematicsUniversity of CaliforniaBerkeleyUSA
  2. 2.Department of Mathematics and StatisticsMcMaster UniversityHamiltonCanada

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