By Ralph Freese, Ralph McKenzie

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

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If two congruences have a block in common, they are equal. In Chapter 5 we saw that Abelian algebras were closely connected with Abelian groups. Nilpotent algebras have a close connection with nilpotent loops. For suppose A ∈ U is nilpotent. Let 0 be an arbitrary element of A and define x + y = p(x, 0, y). 4. this defines a loop with null element 0. 3. the left and right division operations (perhaps we should call them subtraction operations) are also polynomials on A. 7 if a ζ 0 and x, y ∈ A then a + x = x + a, a + (x + y) = (a + x) + y, and (x + a) + y = x + (a + y).

However for loops this is not the case; [H, H] is not determined from just H. One needs the whole congruence associated with H to determine [H, H]. To see this let Z4 = {0, 1, 2, 3} denote the group of integers under addition modulo 4. Let G = Z4 ×Z4 and define a binary operation on G by a, b · c, d = a + c, b + d unless b = d = 1 EXERCISES 45 in which case the operation is defined by the following table: · 01 11 21 31 01 12 02 22 32 11 02 22 32 12 21 22 32 12 02 31 32 12 02 22 Show that G = G, · is a loop with identity element 0, 0 and that the second projection is a homomorphism.

Otherwise two of x, y and z lie in one group and the other lies in the other group. In this case p(x, y, z) is the one that is alone. Show V (A) is permutable and A is solvable. 7 cannot be extended to solvable algebras. 8. 3. 9. 7). CHAPTER 8 Congruence Identities If an equation ε of the language of lattices holds in Con A for every A ∈ V (where V is a variety) then we write V |=con ε, and say that ε is a congruence identity of V. The most important congruence identities are the modular law and the distributive law (and of course this whole book is about the consequences of congruence modularity).