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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Show that for each c, A, +c is an Abelian group but that A, ·, /, \ is not Abelian. 10. A subloop H of a loop G is called a normal subloop if there is some congruence on G whose class containing the identity element is H. Exactly as for groups there is a one-to-one correspondence between normal subloops and congruences. Thus if H and K are normal subloops we can define [H, K] to be the normal subloop corresponding to the commutator of the congruences associated with H and K. This exercise will show that this correspondence is not as nice as one might hope.

The proof combines ideas from several people. Credits will be given at the end of the chapter. We begin with some useful preliminary results. 5. For each modular variety V there is a ternary term d, called a difference term, satisfying the following. (i) d(x, x, y) ≈ y is an identity of V. (ii) If x, y ∈ θ ∈ Con A, A ∈ V, then d(x, y, y) [θ, θ] x. (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.

2. The proof of the above theorem is based on Lakser, Taylor and Tschantz [59]. The congruence identity used in the proof is due to Steve Tschantz and is actually equivalent to modularity (see the next theorem). Gumm’s original proof produced a similar congruence implication. 5. A variety V is modular if and only if for all A ∈ V and all α, β and γ ∈ Con A, (8) (α ◦ β) ∧ γ ≤ (γ ∧ α + γ ∧ β) ◦ β ◦ α Proof. 4 showed that (8) holds if V is modular. If (8) holds let F = FV(x, y, z), and let α = Cg(x, y), β = Cg(y, z), and γ = Cg(x, z) in Con F.