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Commutator Theory for Congruence Modular Varieties Ralph Freese

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Commutator Theory for Congruence Modular Varieties Ralph Freese Commutator Theory for Congruence Modular Varieties Ralph Freese and Ralph McKenzie Contents Introduction 1 Chapter 1. The Commutator in Groups and Rings 7 Exercise 10 Chapter 2. Universal Algebra 11 Exercises 19 Chapter 3. Several Commutators 21 Exercises 22 Chapter 4. One Commutator in Modular Varieties;Its Basic Properties 25 Exercises 33 Chapter 5. The Fundamental Theorem on Abelian Algebras 35 Exercises 43 Chapter 6. Permutability and a Characterization ofModular Varieties 47 Exercises 49 Chapter 7. The Center and Nilpotent Algebras 53 Exercises 57 Chapter 8. Congruence Identities 59 Exercises 68 Chapter 9. Rings Associated With Modular Varieties: Abelian Varieties 71 Exercises 87 Chapter 10. Structure and Representationin Modular Varieties 89 1. Birkhoff-J´onsson Type Theorems For Modular Varieties 89 2. Subdirectly Irreducible Algebras inFinitely Generated Varieties 92 3. Residually Small Varieties 97 4. Chief Factors and Simple Algebras 102 Exercises 103 Chapter 11. Joins and Products of Modular Varieties 105 Chapter 12. Strictly Simple Algebras 109 iii iv CONTENTS Chapter 13. Mal’cev Conditions for Lattice Equations 115 Exercises 120 Chapter 14. A Finite Basis Result 121 Chapter 15. Pure Lattice Congruence Identities 135 1. The Arguesian Equation 139 Related Literature 141 Solutions To The Exercises 147 Chapter 1 147 Chapter 2 147 Chapter 4 148 Chapter 5 150 Chapter 6 152 Chapter 7 156 Chapter 8 158 Chapter 9 161 Chapter 10 165 Chapter 13 165 Bibliography 169 Index 173 Introduction In the theory of groups, the important concepts of Abelian group, solvable group, nilpotent group, the center of a group and centraliz- ers, are all defined from the binary operation [x, y]= x−1y−1xy. Each of these notions, except centralizers of elements, may also be defined in terms of the commutator of normal subgroups. The commutator [M, N] (where M and N are normal subgroups of a group) is the (nor- mal) subgroup generated by all the commutators [x, y] with x ∈ M, y ∈ N. Thus we have a binary operation in the lattice of normal subgroups. This binary operation, in combination with the lattice op- erations, carries much of the information about how a group is put together. The operation is also interesting in its own right. It is a com- mutative, monotone operation, completely distributive with respect to joins in the lattice. There is an operation naturally defined on the lattice of ideals of a ring, which has these properties. Namely, let [J, K] be the ideal generated by all the products jk and kj, with j ∈ J and k ∈ K. The congruity between these two contexts extends to the following facts: [M, M] is the smallest normal subgroup U of M for which M/U is a commutative group; [J, J] is the smallest ideal K of J for which the ring J/K is a commutative group; that is, a ring with trivial multiplication. Now it develops, amazingly, that a commutator can be defined rather naturally in the congruence lattices of every congruence mod- ular variety. This operation has the same useful properties that the commutator for groups (which is a special case of it) possesses. The resulting theory has many general applications and, we feel, it is quite beautiful. In this book we present the basic theory of commutators in congru- ence modular varieties and some of its strongest applications. The book by H. P. Gumm [41] offers a quite different approach to the subject. Gumm developed a sustained analogy between commutator theory and affine geometry which allowed him to discover many of the basic facts about the commutator. We take a more algebraic approach, using some of the shortcuts that Taylor and others have discovered. 1 2 INTRODUCTION Historical remarks. The lattice of normal subgroups of a group, with the commutator operation, is a lattice ordered monoid. It is a residuated lattice (in G. Birkhoff’s terminology) because (a : b) (equal to the largest lattice element x such that [b, x] ≤ a) always exists. The concept of a lattice ordered monoid, which arose naturally in ideal theory, has been studied by Krull, Birkhoff, Ward and Dilworth, and numerous others. (See Birkhoff’s Lattice Theory, Ch. XIV, especially Section 8 on commutation lattices.) The theory presented in this book does not lie in the tradition of these ongoing axiomatic studies, although it may throw some new light on the subject. The worth of our theory stems, rather, from its very broad applicability combined with depth. We shall be working within a broad class of algebras of diverse kinds, with a general definition of com- mutator which turns out to determine a unique operation over every algebra. The commutator proves to be an excellent tool for arriving at deep algebraic results in this very general setting. The first commutator theory approaching this level of generality was created by the English mathematician J. D. H. Smith, and presented in his 1976 book Mal’cev Varieties. Smith dealt with an important subclass of the congruence modular varieties, namely the varieties in whose algebras all congru- ences permute, which he called Mal’cev Varieties. His theory rapidly evolved into the theory we shall present. The work was done chiefly by the German mathematicians J. Hagemann and C. Herrmann, and pre- sented in their papers Hagemann-Herrmann [44] and Herrmann [45]. Many important details and simplifications were worked out later by other people, notably by H. P. Gumm and W. Taylor. General remarks. The operation we shall study is a binary op- eration defined for pairs of congruences hθ, ψi of an algebra and giving another congruence [θ, ψ]. This operation can be easily defined for congruences of any algebra, but it seems to be especially well behaved only for algebras in congruence modular varieties. Two of its properties contribute most obviously to its strength in applications. Let A be an algebra in a congruence modular variety, L be its lattice of congruences and θ and ψ be two members of L. If the commutator of θ and ψ is as large as the theory allows it to be – that is, if it is identical with the intersection of θ and ψ – then in the neighborhood of this pair of congruences L is very like a distributive lattice. Whenever ψ ≤ ψ1 ∨ ψ2, for example, then θ ∧ ψ ≤ (θ ∧ ψ1) ∨ (θ ∧ ψ2) (where ∨ and ∧ denote join and meet in L). There is a sophisticated way to state these facts. Consider this binary relation on L: θ ∼ ψ if and only if θ ∨ ψ is solvable over θ ∧ ψ. This is an equivalence relation and it respects commutators and lattice joins and meets. The quotient lattice INTRODUCTION 3 L/ ∼ is a distributive commutation lattice in which [α, β]= α ∧ β. We may remark that at one extreme end of the spectrum of congruence modular varieties, the congruence distributive varieties are precisely those in which the commutator is identical with the intersection in all congruence lattices. The second important property concerns Abelian congruences. In a quotient algebra B = A/[θ, ψ], the congruence β = θ ∧ ψ/[θ, ψ] is an Abelian congruence, that is [β, β] = 0. This implies that each equivalence class of β is an Abelian group and every operation of the algebra B, with its variables restricted to range over fixed equivalence classes of β is a group homomorphism (plus a constant). In the extreme case, θ = ψ =1A (= A × A) and [1A, 1A]=0A. The algebra A is then polynomially equivalent to a module over a ring. Our presentation of commutator theory is essentially different from earlier published accounts. Hagemann and Herrmann defined the com- mutator of two congruences on an algebra A, and derived its prop- erties, by applying the modular law in a quite nontrivial fashion to congruences on subalgebras of A, A2 and A3. A congruence of A is itself an algebra (a subalgebra of A2), and in fact they dealt mainly with congruences on congruences of A. Gumm’s approach to commu- tator theory (in [38], [40] and [40]) is heavily flavored with geometric analogies developed in his earlier papers [36] and [39], and is closer in spirit to what Smith did. Gumm made good use of A. Day’s derived operations, whose existence for a variety is equivalent to congruence modularity. Our path to commutator theory is more direct than either of these. We define the commutator in any algebra by considering the set of all term operations of the algebra. Then assuming that algebra belongs to a congruence modular variety, we use Day’s operations to present a set of generators for [θ, ψ]. Following this result, the basic properties of commutators are easily derived. This book is an expanded version of a manuscript entitled The commutator, an overview that we used as the text for a week long workshop on commutator theory held at the Puebla Conference on Universal Algebra and Lattice Theory in January 1982. The first chapter of the present manuscript uses groups and rings to motivate the definition of the commutator. The second chapter gives a brief introduction to the results and definitions of universal algebra which will be required later. It also gives a proof of Day’s characterization of varieties with modular congruence lattices. In the course of the proof we prove two useful lemmas. The third chapter gives several definitions of the commutator and proves some simple results which do not depend on congruence modularity. The fourth chapter 4 INTRODUCTION shows that all of these definitions of the commutator are equivalent in congruence modular varieties and proves some of the fundamental facts about the commutator. The fifth chapter proves Herrmann’s fundamental result on Abelian algebras and introduces Gumm’s 3-ary term and his characterization of Abelian congruences.
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