The shape of congruence lattices

183 Pages·2013·1.309 MB·English

by  Keith A. Kearnes, Emil W. Kiss

MEMOIRS
of the
American Mathematical Society
Volume 222 • Number 1046 (fifth of 5 numbers) • March 2013
The Shape
of Congruence Lattices
Keith A. Kearnes
Emil W. Kiss
ISSN 0065-9266 (print) ISSN 1947-6221 (online)
American Mathematical Society
Number 1046
The Shape
of Congruence Lattices
Keith A. Kearnes
Emil W. Kiss
March2013 • Volume222 • Number1046(fifthof5numbers) • ISSN0065-9266
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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 5
1.4. The Results of this Monograph 8
1.5. Thanks! 10
Chapter 2. Preliminary Notions 11
2.1. Algebras, Varieties, and Clones 11
2.2. Lattice Theory 13
2.3. Meet Continuous Lattice Theory 16
2.4. Maltsev Conditions 17
2.5. The Term Condition 20
2.6. Congruence Identities 23
Chapter 3. Strong Term Conditions 29
3.1. Varieties Omitting Strongly Abelian Congruences 29
3.2. Join Terms 39
3.3. Abelian Tolerances and Congruences 45
Chapter 4. Meet Continuous Congruence Identities 49
4.1. Maltsev Conditions from Congruence Identities 50
4.2. Congruence Identities from Maltsev Conditions 57
4.3. Omitted Sublattices 65
4.4. Admitted Sublattices 67
Chapter 5. Rectangulation 71
5.1. Rectangular Tolerances 71
5.2. Rectangular Tolerances and Join Terms 77
5.3. Varieties Omitting Rectangular Tolerances 83
Chapter 6. A Theory of Solvability 95
6.1. Varieties with a Weak Difference Term 96
6.2. ∞-Solvability 98
6.3. An Alternative Development 115
Chapter 7. Ordinary Congruence Identities 117
7.1. A Rank for Solvability Obstructions 117
7.2. Congruence Identities 127
Chapter 8. Congruence Meet and Join Semidistributivity 131
v
vi CONTENTS
8.1. Congruence Meet Semidistributivity 131
8.2. More on Congruence Identities 134
8.3. Congruence Join Semidistributivity 145
Chapter 9. Residually Small Varieties 149
9.1. Residual Smallness and Congruence Modularity 149
9.2. Almost Congruence Distributive Varieties 153
Problems 155
Appendix A. Varieties with Special Terms 159
A.1. Varieties with a Taylor Term 159
A.2. Varieties with a Hobby-McKenzie Term 160
Bibliography 163
Index 167
Abstract
We develop the theories of the strong commutator, the rectangular commu-
tator, the strong rectangular commutator, as well as a solvability 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 identity.
(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 throughout 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) TherectangularandTCcommutatorsarebothtrivialthroughoutV.
We prove that a residually small variety that satisfies a congruence identity is
congruence modular.
ReceivedbytheeditorJune27,2007,and,inrevisedform,November28,2011.
ArticleelectronicallypublishedonSeptember18,2012;S0065-9266(2012)00667-8.
2010 MathematicsSubjectClassification. Primary08B05;Secondary08B10.
Keywordsandphrases. Abelian,almostcongruencedistributivity,commutatortheory,com-
patible semilattice operation, congruence identity, congruence modularity, 8congruence semidis-
tributivity, Maltsev condition, meet continuous lattice, rectangulation, residual smallness, solv-
able,tamecongruencetheory,termcondition,variety,weakdifferenceterm.
ThefirstauthorwassupportedinpartbyNSFGrant#9802922.
ThesecondauthorwassupportedinpartbyOTKAGrants#T043671and#T043034.
Affiliationsattimeofpublication: KeithA.Kearnes,DepartmentofMathematics,University
of Colorado, Boulder, Colorado 80309-0395, email: [email protected]; and Emil W.
Kiss, Lor´and E¨otv¨os University, Department of Algebra and Number Theory, 1117 Budapest,
P´azma´nyP´eters´et´any1/c,Hungary,email: [email protected].
(cid:2)c2012 American Mathematical Society
vii

CHAPTER 1
Introduction
This monograph is concerned with the relationships between Maltsev condi-
tions, 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
thata−b∈mZ. Asthenotationsuggests, congruencemodulo misanequivalence
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∈Z.
RichardDedekindconceivedofamoregeneralnotionof“integer”,whichnowa-
days we call an ideal in a number ring. Dedekind extended the notation (1.1) to
(1.2) a≡b (mod μ)
where a,b ∈ C and μ ⊆ C; (1.2) is defined to hold if a−b ∈ μ. Dedekind called
a subset μ ⊆C a module if it could serve as the modulus of a congruence, i.e., if
thisrelationofcongruencemodulo μisanequivalencerelationonC. Thishappens
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 intersection
and sum. These two operations make the set of modules into a lattice. Dedekind
proposed and investigated the problem of determining 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
(cid:2) (cid:3)
(1.3) α∩ β+(α∩γ) =(α∩β)+(α∩γ).
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 congruence 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 identity, but
rather as a quasi-identity: for all modules α,β,γ ⊆C
(1.4) α⊇γ −→α∩(β+γ)=(α∩β)+γ.1
1Infact,Dedekindusedthesymbols+and−insteadof+and∩.
1