# 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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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 . Thus we may assume pi and qi are linear: say, pi = nj=1 cij xj and qi = nj=1 dij xj , cij , dij rational.

5(iii) to prove Gumm’s result that every congruence permutes with the projection congruences of a direct product. 5. Suppose α ≥ β and [α, β] = 0. Then for u, v ∈ α prove that the function x → d(x, u, v) is an isomorphism from M(β, u) onto M(β, v). 6. 5(iii). 5(i) and (ii) hold for this d. 7. , x · y is an Abelian group operation. 44 5. THE FUNDAMENTAL THEOREM ON ABELIAN ALGEBRAS 8. Let G be the quasigroup with multiplication table · 0 1 2 3 0 3 2 1 0 1 2 3 0 1 2 0 1 2 3 3 1 0 3 2 Show G is Abelian (but x · y is neither commutative nor associative).

We claim first that (β0 ∧ β1 ) ∨ ∆ is the unique smallest congruence of B2 strictly above ∆. To see it, first 0 = β0 ∧ η1 ≤ β0 ∧ β1 , ∆ ∧ η1 = 0, so β0 ∧ β1 ≤ ∆. Suppose that λ > ∆ in Con B2 . Notice that β1 is the unique atom in 1/η1 ∼ = Con B, hence since 1/η1 transposes down onto η0 /0, η0 ∧ β1 is the unique atom below η0 . Likewise for β0 ∧ η1 below η1 . Now ∆ ∨ (β0 ∧ η1 ) ≥ η0 ∧ β1 because choosing x, y ∈ β − 0B , we have x, x ∆ y, y β0 ∧ η1 x, y , and hence (∆ ∨ (β0 ∧ η1 )) ∧ η0 = 0, so it contains η0 ∧ β1 .