# Commutator theory for congruence modular varieties by Ralph Freese, Ralph McKenzie

By Ralph Freese, Ralph McKenzie

Freese R., McKenzie R. Commutator conception for congruence modular types (CUP, 1987)(ISBN 0521348323)(O)(174s)

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Iii) If α, β, γ ∈ Con A, A ∈ V and α ∧ β ≤ γ, then the implication of Figure 1 holds. x β α y implies γ u x z u′ z γ d(u, u′, y) y u u′ Figure 1. Proof. Since V is modular there are terms m0 (x, y, z, u), . . 2. Define qi (x, y, z), i = 0, 1, . . , n inductively by q0 (x, y, z) = z qi+1 (x, y, z) = mi+1 (qi (x, y, z), y, x, qi(x, y, z)) i odd mi+1 (qi (x, yz), x, qi (x, y, z)) i even and set d(x, y, z) = qn (x, y, z). 2(ii) that qi (x, x, y) ≈ y. Hence d(x, x, y) ≈ y holds in V. 5. THE FUNDAMENTAL THEOREM ON ABELIAN ALGEBRAS 37 To show (ii) assume x, y ∈ θ.

X2i ∈ [Ai , Ai ]. Thus, if a ∈ A = ΠAi has ith component x21 +· · ·+x2i , then a ∈ Π[Ai , Ai ]. However, a ∈ [A, A]. bk in A with a = a1 b1 + · · · + ak bk . Choose n > k and let the nth component of ai and bi be pi and qi respectively. Then x21 + · · · + x2n = p1 q1 + · · · + pk qk . (∗) Let pi be the polynomial obtained from pi by deleting all nonlinear terms and define q i similarly. Since the constant terms of each of these polynomials are zero, the above equation is still valid if we replace pi and qi with pi and qi .

To see this let ∆ be the relation on A2 defined by the second part, that is, ∆′ = { x, y , u, v : x β y α u, v = d(y, x, u)}. 7. For example, to see that this relation 5. THE FUNDAMENTAL THEOREM ON ABELIAN ALGEBRAS 41 is symmetric suppose x β y α u and v = d(y, x, u). Clearly u β v α x and since d commutes with itself. d(v, u, x) = d(d(y, x, u), d(x, x, u), d(x, x, x)) = d(d(y, x, x), d(x, x, x), d(u, u, x) = d(y, x, x) = y. , ∆′ is symmetric. Now ∆β,α is the congruence on A(β) generated by { x, x , u, u : x α u}.