Results and problems on congruence lattice representations

We survey some results and problems on congruence lattice representations. We will consider congruence representations and similarity type, and we will consider simultaneous congruence representations. A seemingly innocuous problem unites the sections. This paper is dedicated to Walter Taylor. Received October 9, 2005; accepted in final form January 3, 2006.     

Covering in the lattice of equational theories and some properties of term finite theories

For countable languages, we completely describe those cardinals κ such that there is an equational theory which covers exactly κ other equational theories. For this task understanding term finite theories is helpful. A theoryT isterm finite provided {ψ:Tτϕ≈ψ} is finite for all terms ϕ. We develop here some fundamental properties of term finite theories and use them, together with Ramsey's Theorem, to prove that any finitely based term finite theory covers only finitely many others. We also show that every term finite theory possesses an independent base and that there are such theories whose pairwise joins are not term finite. The paper was written with the support of NSF Grant MCS-80-01778. Presented by B. Jónsson. Received July 22, 1980. Accepted for publication in final form March 19, 1981.     

A second note on the equational theory of modular ortholattices

We show how to alter the material in [4] to prove that every variety of modular ortholattices is generated by its simple members. Supported by NSERC Operating Grant 0041702.     

Simultaneous congruence representations: a special case

We study the problem of representing a pair of algebraic lattices, L₁ and L₀, as Con(A₁) and Con(A₀), respectively, with A₁ an algebra and A₀ a subalgebra of A₁, and we provide such a representation in a special case. Received September 11, 2004; accepted in final form January 7, 2005.     

Varieties of lattice ordered groups that contain no non-abelian o-groups are solvable

It is shown that the variety _n of lattice ordered groups defined by the identity x ⁿ y ⁿ =y ⁿ x ⁿ , where n is the product of k (not necessarily distinct primes) is contained in the (k+1)st power A ^k+1 of the variety A of all Abelian lattice ordered groups. This implies, in particular, that _n is solvable class k + 1. It is further established that any variety V of lattice ordered groups which contains no non-Abelian totally ordered groups is necessarily contained in _n , for some positive integer n.This work was supported in part, by NSERC Grant A4044.     

Small congruence distributive varieties: Retracts, injectives, equational compactness and amalgamation

The relationship between absolute retracts, injectives and equationally compact algebras in finitely generated congruence distributive varieties with 1- element subalgebras is considered and several characterization theorems are proven. Amongst others, we prove that the absolute retracts in such a variety are precisely the injectives in the amalgamation class and that every equationally compact reduced power of a finite absolute retract is an absolute retract. We also show that any elementary amalgamation class is Horn if and only if it is closed under finite direct products. The second author's work was supported by grants from the South African Council for Scientific and Industrial Research and the University of Cape Town Research Committee.     

A solution to Dilworth's congruence lattice problem

We construct an algebraic distributive lattice D that is not isomorphic to the congruence lattice of any lattice. This solves a long-standing open problem, traditionally attributed to R.P. Dilworth, from the forties. The lattice D has a compact top element and ℵω+1 compact elements. Our results extend to any algebra possessing a congruence-compatible structure of a join-semilattice with a largest element.     

The automorphism group of a function lattice: A problem of Jónsson and McKenzie

It is shown that Aut(L ^Q ) is naturally isomorphic to Aut(L) × Aut(Q) whenL is a directly and exponentially indecomposable lattice,Q a non-empty connected poset, and one of the following holds:Q is arbitrary butL is ajm-lattice,Q is finitely factorable and L is complete with a join-dense subset of completely join-irreducible elements, orL is arbitrary butQ is finite. A problem of Jónsson and McKenzie is thereby solved. Sharp conditions are found guaranteeing the injectivity of the natural mapv _P,Q from Aut(P) × Aut(Q) to Aut(P ^Q )P andQ posets), correcting misstatements made by previous authors. It is proven that, for a bounded posetP and arbitraryQ, the Dedekind-MacNeille completion ofP ^Q ,DM(P ^Q ), is isomorphic toDM(P)^Q. This isomorphism is used to prove that the natural mapv _P,Q is an isomorphism ifv _DM(P),Q is, reducing a poset problem to a more tractable lattice problem.Presented by B. Jonsson.The author would like to thank his supervisor, Dr. H. A. Priestley, for her direction and advice as well as his undergraduate supervisor, Prof. Garrett Birkhoff, and Dr. P. M. Neumann for comments regarding the paper. This material is based upon work supported under a (U.S.) National Science Foundation Graduate Research Fellowship and a Marshall Aid Commemoration Commission Scholarship.     

Every finite lattice in $$ \mathcal{V} (M_3) $$

We prove that every finite lattice in the variety generated by M₃ is isomorphic to the congruence lattice of a finite algebra.     

Star-linear equational theories of groupoids

We prove that there are precisely six equational theories E of groupoids with the property that every term is E-equivalent to a unique linear term. Presented by J. Berman. Received November 11, 2004; accepted in final form March 12, 2006. The first and third authors were supported by the Ministry of Science and Environment of Serbia, grant no. 144011; the second and fifth authors were supported by MŠMT, research project MSM 0021620839, and by the Grant Agency of the Czech Republic, grant #201/02/0594; the fourth author was supported by the NSF grant #DMS-0245622.     

Almost distributive sublattices and congruence heredity

A congruence lattice L of an algebra A is hereditary if every 0-1 sublattice of L is the congruence lattice of an algebra on A. Suppose that L is a finite lattice obtained from a distributive lattice by doubling a convex subset. We prove that every congruence lattice of a finite algebra isomorphic to L is hereditary. Presented by E. W. Kiss. Received July 18, 2005; accepted in final form April 2, 2006.     

The Shifting Lemma and shifting lattice identities

Gumm [6] used the Shifting Lemma with high success in congruence modular varieties. Later, some analogous diagrammatic statements, including the Triangular Scheme from [1] were also investigated. The present paper deals with the purely lattice theoretic underlying reason for the validity of these lemmas. The shift of a lattice identity, a special Horn sentence, is introduced. To any lattice identity and to any variable y occurring in we introduce a Horn sentence S(, y). When S(, y) happens to be equivalent to , we call it a shift of . When has a shift then it gives rise to diagrammatic statements resembling the Shifting Lemma and the Triangular Scheme. Some known lattice identities will be shown to have a shift while some others have no shift.     

The equational theory of a nontrivial discriminator variety is co-NP-hard

Discriminator varieties play a central role in the classification of decidable varieties; and they arise naturally in the study of algebraic logics. There are also important connections with the reduction of theorem proving to equational logic. In this paper we show, for any nontrivial discriminator variety, that the problem of determining if an equation holds in the variety is co-NP-hard.Received August 24, 2002; accepted in final form August 5, 2004.     

Join-semidistributive lattices and convex geometries

We introduce the notion of a convex geometry extending the notion of a finite closure system with the anti-exchange property known in combinatorics. This notion becomes essential for the different embedding results in the class of join-semidistributive lattices. In particular, we prove that every finite join-semidistributive lattice can be embedded into a lattice S_P(A) of algebraic subsets of a suitable algebraic lattice A. This latter construction, S_P(A), is a key example of a convex geometry that plays an analogous role in hierarchy of join-semidistributive lattices as a lattice of equivalence relations does in the class of modular lattices. We give numerous examples of convex geometries that emerge in different branches of mathematics from geometry to graph theory. We also discuss the introduced notion of a strong convex geometry that might promise the development of rich structural theory of convex geometries.     

Freely adjoining a relative complement to a lattice

Let K be a lattice, and let a < b < c be elements of K. We adjoin freely a relative complement u of b in [a, c] to K to form the lattice L. For two polynomials A and B over K ∪ {u}, we find a very simple set of conditions under which A and B represent the same element in L, so that in L all pairs of relative complements in [a, c] can be described. Our major result easily follows: Let [a, c] be an interval of a lattice K; let us assume that every element in [a, c] has at most one relative complement. Then K has an extension L such that [a, c] in L, as a lattice, is uniquely complemented.As an immediate consequence, we get the classical result of R. P. Dilworth: Every lattice can be embedded into a uniquely complemented lattice. We also get the stronger form due to C. C. Chen and G. Grätzer: Every at most uniquely complemented bounded lattice has a {0, 1}-embedding into a uniquely complemented lattice. Some stronger forms of these results are also presented.A polynomial A over K ∪ {u} naturally represents an element 〈A 〉 of L. Let us call a polynomial A minimal, if it is of minimal length representing x. We characterize minimal polynomials.Dedicated to the memory of Ivan RivalReceived February 12, 2003; accepted in final form June 18, 2004.This revised version was published online in August 2005 with a corrected cover date.     

A variety of lattice-ordered groups containing all representable covers of the abelian variety

A small variety of representable lattice-ordered groups is constructed, which contains all of the representable covers of the abelian variety.     

Lattices of algebraic subsets

This survey article tackles different aspects of lattices of algebraic subsets, with the emphasis on the following: the theory of quasivarieties, general lattice theory and the theory of closure spaces with the anti-exchange axiom.In Celebration of the Sixtieth Birthday of Ralph N. McKenzieReceived August 24, 2002; accepted in final form October 2, 2003.     

The word problem in free normal valued lattice-ordered groups: a solution and practical shortcuts

In any lattice-ordered group (l-group) generated by a setX, every element can be written (not uniquely) in the form w(x)=⋁ _i ⋀ _j w _ij (x), where eachw _ij (x) is a group word in the elements ofX. An algorithm will be given for deciding whetherw(x) is the identitye in the free normal valuedl-group onX, or equivalently, whether the statement “∀x,w(x)=e” holds in all normal valuedl-groups. The algorithm is quite different from the one given recently by Holland and McCleary for the freel-group, and indeed the solvability of the word problem was established first for the normal valued case. The present algorithm makes crucial use of the fact (due to Glass, Holland, and McCleary) that the variety of normal valuedl-groups is generated by the finite wreath powersZ Wr Z Wr...Wr Z of the integersZ. In general, use of the algorithm requires a fairly large amount of work, but in several important special cases shortcuts are obtained which make the algorithm very quick. This is an expanded version of material developed while the author was on leave at Bowling Green State University in Bowling Green, Ohio, and presented in 1978 at the Conference on Ordered Groups at Boise State University in Boise, Idaho [9]. Presented by L. Fuchs.     

A counterexample to the finite height conjecture

There is an infinite subdirectly irreducible lattice which generates a variety that contains only finitely many subvarietes.The author was supported in part by NSF Grant DMS 94-00511