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

By Ralph Freese, Ralph McKenzie

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

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3 [θ, ψ] ∨ π = [θ ∨ π, ψ ∨ π] ∨ π. Thus we may assume without loss of generality that θ, ψ ≥ π. Then f carries a set of generators of [θ, ψ] ∨ π (namely X(θ, ψ) ∪ π) onto a set of generators for [f (θ), f (ψ)]. Hence f ([θ, ψ] ∨ π) = [f (θ), f (ψ)], which is equivalent to the desired conclusion. For (2), θ|B centralizes ψ|B modulo [θ, ψ]|B , or argue by generators. 5. Let A = Πi∈I Ai and θi ∈ Con Ai , i ∈ I. Then the map (3) (θi )i∈I → { a, b ∈ A2 ; ai, bi ∈ θi , for all i ∈ I, and ai = bi for all but finitely many i ∈ I} is a lattice isomorphism from Πi∈I Con Ai into Con A.

Hence d(x, x, y) ≈ y holds in V. 5. THE FUNDAMENTAL THEOREM ON ABELIAN ALGEBRAS 37 To show (ii) assume x, y ∈ θ. We prove inductively that (1) qi (x, y, y) [θ, θ] mi (y, y, y, x), i odd qi (x, y, y) [θ, θ] mi (y, y, x, x), i even Since mn (x, y, z, u) = u, (ii) will then follow. The case i = 0 is immediate from Day’s identities. Suppose i is odd and that (1) holds for i. Then qi+1 (x, y, y) = mi+1 (qi (x, y, y), y, x, qi(x, y, y)) [θ, θ] mi+1 (mi (y, y, y, x), y, x, mi(y, y, y, x)). 2 mi+1 (mi (y, y, y, x),y, y, mi(y, y, y, x)) = mi (y, y, y, x) = mi+1 (y, y, y, x) = mi+1 (mi (y, y, y, y), y, y, mi(x, x, x, x).

In a modular variety every Abelian algebra is affine, and conversely. As the definition of affine and the remarks following it indicate, Abelian algebras are closely related to modules (in fact, each Abelian algebra is polynomially equivalent to a module). This connection is studied more thoroughly in Chapter 9. 5 was first constructed by Herrmann in [45]. The short proof that it satisfies (ii) is Taylor’s. Gumm [36], [38], using his geometrical methods, constructed a term satisfying (iii) and showed (iii) implies (i) and (ii).

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