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The Shape of Congruence Lattices Keith A. Kearnes Emil W. Kiss

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The Shape of Congruence Lattices Keith A. Kearnes Emil W. Kiss The Shape of Congruence Lattices Keith A. Kearnes Emil W. Kiss Author address: Department of Mathematics, University of Colorado, Boulder, CO 80309-0395, USA E-mail address: [email protected] URL: http://spot.colorado.edu/~kearnes (Emil W. Kiss) Lorand´ Eotv¨ os¨ University, Department of Algebra and Number Theory, 1117 Budapest, Pazm´ any´ Peter´ set´ any´ 1/c, Hungary E-mail address: [email protected] URL: http://www.cs.elte.hu/~ewkiss Dedicated to Bjarni J´onsson, For his work on congruence varieties, To Ralph McKenzie, For his work on commutator theories, And to Walter Taylor, For his work on Maltsev conditions. Contents Chapter 1. Introduction 1 1.1. Shapes of Congruence Lattices 1 1.2. Maltsev Conditions 3 1.3. Commutator Theories 6 1.4. The Results of This Monograph 11 Chapter 2. Preliminary Notions 15 2.1. Algebras, Varieties, and Clones 15 2.2. Lattice Theory 17 2.3. Meet Continuous Lattice Theory 22 2.4. Maltsev Conditions 23 2.5. The Term Condition 27 2.6. Congruence Identities 31 Chapter 3. Strong Term Conditions 37 3.1. Varieties Omitting Strongly Abelian Congruences 37 3.2. Join Terms 50 3.3. Abelian Tolerances and Congruences 58 Chapter 4. Meet Continuous Congruence Identities 63 4.1. Maltsev Conditions From Congruence Identities 65 4.2. Congruence Identities From Maltsev Conditions 73 4.3. Omitted Sublattices 83 4.4. Admitted Sublattices 85 Chapter 5. Rectangulation 91 5.1. Rectangular Tolerances 91 5.2. Rectangular Tolerances and Join Terms 99 5.3. Varieties Omitting Rectangular Tolerances 107 Chapter 6. A Theory of Solvability 121 6.1. Varieties With A Weak Difference Term 122 6.2. 1-Solvability 126 6.3. An Alternative Development 146 Chapter 7. Ordinary Congruence Identities 151 vii viii CONTENTS 7.1. A Rank for Solvability Obstructions 151 7.2. Congruence Identities 163 Chapter 8. Congruence Meet and Join Semidistributivity 167 8.1. Congruence Meet Semidistributivity 167 8.2. More on Congruence Identities 171 8.3. Congruence Join Semidistributivity 185 Chapter 9. Residually Small Varieties 191 9.1. Residual Smallness and Congruence Modularity 191 9.2. Almost Congruence Distributive Varieties 195 Problems 199 Bibliography 205 Index 209 Abstract We develop the theories of the strong commutator, the rectangular commutator, the strong rectangular commutator, as well as a solvabil- ity theory for the nonmodular TC commutator. These theories are used to show that each of the following sets of statements are equivalent for a variety V of algebras. (I) (a) V satisfies a nontrivial congruence identity. (b) V satisfies an idempotent Maltsev condition that fails in the variety of semilattices. (c) The rectangular commutator is trivial throughout V. (II) (a) V satisfies a nontrivial meet continuous congruence iden- tity. (b) V satisfies an idempotent Maltsev condition that fails in the variety of sets. (c) The strong commutator is trivial throughout V. (d) The strong rectangular commutator is trivial through- out V. (III) (a) V is congruence semidistributive. (b) V satisfies an idempotent Maltsev condition that fails in the variety of semilattices and in any nontrivial variety of modules. (c) The rectangular and TC commutators are both trivial throughout V. We prove that a residually small variety that satisfies a congruence identity is congruence modular. 1991 Mathematics Subject Classification. Primary 08B05; Secondary 08B10. Key words and phrases. Abelian, almost congruence distributivity, commutator theory, compatible semilattice operation, congruence identity, congruence modular- ity, congruence semidistributivity, Maltsev condition, meet continuous lattice, rect- angulation, residual smallness, solvable, tame congruence theory, term condition, variety, weak difference term. The first author was supported in part by NSF Grant #9802922. The second author was supported in part by OTKA Grants #T043671 and #T043034. ix CHAPTER 1 Introduction This monograph is concerned with the relationships between Mal- tsev conditions, commutator theories and the shapes of congruence lattices in varieties of algebras. 1.1. Shapes of Congruence Lattices Carl F. Gauss, in [23], introduced the notation (1.1) a ≡ b (mod m); which is read as \a is congruent to b modulo m", to mean that the integers a and b have the same remainder upon division by the integer modulus m, equivalently that a − b 2 mZ. As the notation suggests, congruence modulo m is an equivalence relation on Z. It develops that congruence modulo m is compatible with the ring operations of Z, and that the only equivalence relations on Z that are compatible with the ring operations are congruences modulo m for m 2 Z. Richard Dedekind conceived of a more general notion of \integer", which nowadays we call an ideal in a number ring. Dedekind extended the notation (1.1) to (1.2) a ≡ b (mod µ) where a; b 2 C and µ ⊆ C; (1.2) is defined to hold if a − b 2 µ. Dedekind called a subset µ ⊆ C a module if it could serve as the modulus of a congruence, i.e., if this relation of congruence modulo µ is an equivalence relation on C. This happens precisely when µ is closed under subtraction. For Dedekind, therefore, a \module" was an additive subgroup of C. The set of Dedekind's modules is closed under the operations of in- tersection and sum. These two operations make the set of modules into a lattice. Dedekind proposed and investigated the problem of determin- ing the identities of this lattice (the \laws of congruence arithmetic"). In 1900, in [13], he published the discovery that if α; β; γ ⊆ C are modules, then (1.3) α \ β + (α \ γ) = (α \ β) + (α \ γ): 1 2 1. INTRODUCTION This 3-variable law of the lattice of modules is now called the modular law. Dedekind went on to prove that any equational law of congru- ence arithmetic that can be expressed with at most 3 variables is a consequence of the modular law and the laws valid in all lattices. Dedekind did not write the law in the form (1.3), which is an iden- tity, but rather as a quasi-identity: for all modules α; β; γ ⊆ C (1.4) α ⊇ γ −! α \ (β + γ) = (α \ β) + γ:1 Dedekind also discovered a useful \omitting sublattices" version of the modular law. It is the assertion that there do not exist modules α; β; γ ⊆ C which generate a sublattice isomorphic to N5 (Theorem I.7.12 of [1]). β + α = β + γ @ r @ @ α β r J γ r J J r J β \ α =r β \ γ Figure 1.1. The lattice N5 More generally, a congruence on an arbitrary algebra A is an equivalence relation on the universe of A that is compatible with the operations of A. Equivalently, it is the kernel of a homomorphism with domain A. The set of all congruences is a sublattice of the lattice of equivalence relations on A.A congruence identity of A is an identity that holds in the lattice Con(A) of all congruences of A. Dedekind's observation is that the modular law is a congruence identity satisfied by C as an abelian group. In fact, it is a congruence identity of any group, ring, vector space, Boolean algebra or lattice. Dedekind's result initiated many lines of research in the 20th cen- tury, but to avoid losing focus we mention only a few. Garrett Birkhoff observed that the congruences of any group permute, meaning that α ◦β = β ◦α, and that any lattice of permuting equivalence relations is modular. This \explains" Dedekind's result to some degree, but in [22] N. Funayama and T. Nakayama showed that the lattice of congruences of a lattice satisfies the distributive law: for all α, β, γ (1.5) α \ (β + γ) = (α \ β) + (α \ γ); 1In fact, Dedekind used the symbols + and − instead of + and \. 1.2. MALTSEV CONDITIONS 3 which is stronger than the modular law, yet congruences of lattices need not permute. Thus permutability implies modularity, but not vice versa. Bjarni J´onssonthen refined the results of Birkhoff and Dedekind by showing in [37] that any lattice of permuting equivalence relations satisfies the arguesian law, which is stronger than the modular law. Conversely, he showed that any complemented lattice satisfying the arguesian law is embeddable into the lattice of congruences of an abelian group. Here the arguesian law is the 6-variable law asserting that for all αi, βi, i = 0; 1; 2, (1.6) (α0 +β0)\(α1 +β1)\(α2 +β2) ≤ α0 \(γ +α1) + β0 \(γ +β1) where γ = (α0 +α1)\(β0 +β1)\(δ0 +δ1), δ0 = (α0 +α2)\(β0 +β2) and δ1 = (α1 + α2) \ (β1 + β2). Thus J´onssonsolved the special instance of Dedekind's problem (of determining the laws of modules) which concerns only complemented lattices of modules. The full problem is still open, although in the same paper J´onssongave an example of a finite noncomplemented arguesian lattice that is not embeddable in the congruence lattice of a group, and in [30] Mark Haiman proved that lattices of permuting equivalence relations satisfy laws stronger than the arguesian law. 1.2. Maltsev Conditions Let R be the variety of rings and A be the variety of abelian groups. The fact that rings have a term-definable underlying abelian group structure may be denoted A ≤ R, i.e., rings have the structure of abelian groups and more. A. I. Maltsev proved in [63] that a variety V of algebras has the property that the congruence lattices of its members consist of permuting equivalence relations if and only if P ≤ V where P is the variety with one ternary basic operation symbol p that is axiomatized by the Maltsev identities: (1.7) p(x; y; y) ≈ x and p(y; y; x) ≈ x: In other words, V consists of congruence permutable algebras if and only if there is a ternary V-term p, called a Maltsev term, such that the identities (1.7) hold in V.
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