Definability in substructure orderings, III: Finite distributive lattices

Let ${{\mathcal D}}$ be the ordered set of isomorphism types of finite distributive lattices, where the ordering is by embeddability. We study first-order definability in this ordered set. We prove among other things that for every finite distributive lattice D, the set {d, d ^opp} is definable, where d and d ^opp are the isomorphism types of D and its opposite (D turned upside down). We prove that the only non-identity automorphism of ${{\mathcal D}}$ is the opposite map. Then we apply these results to investigate definability in the closely related lattice of universal classes of distributive lattices. We prove that this lattice has only one non-identity automorphism, the opposite map; that the set of finitely generated and also the set of finitely axiomatizable universal classes are definable subsets of the lattice; and that for each element K of the two subsets, {K, K ^opp} is a definable subset of the lattice.     

Definability in substructure orderings, I: finite semilattices

We investigate definability in the set of isomorphism types of finite semilattices ordered by embeddability; we prove, among other things, that every finite semilattice is a definable element in this ordered set. Then we apply these results to investigate definability in the closely related lattice of universal classes of semilattices; we prove that the lattice has no non-identical automorphisms, the set of finitely generated and also the set of finitely axiomatizable universal classes are definable subsets and each element of the two subsets is a definable element in the lattice.     

Definability in Substructure Orderings,II: Finite Ordered Sets

Let \({{\uppercase {\mathcal{p}}}} \) be the ordered set of isomorphism types of finite ordered sets (posets), where the ordering is by embeddability. We study first-order definability in this ordered set. We prove among other things that for every finite poset P, the set \(\{p,p^{\partial}\}\) is definable, where p and \(p^{\partial}\) are the isomorphism types of P and its dual poset. We prove that the only non-identity automorphism of \({{\uppercase {\mathcal{p}}}}\) is the duality map. Then we apply these results to investigate definability in the closely related lattice of universal classes of posets. We prove that this lattice has only one non-identity automorphism, the duality map; that the set of finitely generated and also the set of finitely axiomatizable universal classes are definable subsets of the lattice; and that for each member K of either of these two definable subsets, \(\{K,K^{\partial}\}\) is a definable subset of the lattice. Next, making fuller use of the techniques developed to establish these results, we go on to show that every isomorphism-invariant relation between finite posets that is definable in a certain strongly enriched second-order language \(\textup{\emph L}_2\) is, after factoring by isomorphism, first-order definable up to duality in the ordered set \({{\uppercase {\mathcal{p}}}}\). The language \(\textup{\emph L}_2\) has different types of quantifiable variables that range, respectively, over finite posets, their elements and order-relation, and over arbitrary subsets of posets, functions between two posets, subsets of products of finitely many posets (heteregenous relations), and can make reference to order relations between elements, the application of a function to an element, and the membership of a tuple of elements in a relation.     

On subtractive varieties IV: Definability of principal ideals

This paper deals with notions of (equational) definability of principal ideals in subtractive varieties. These notions are first characterized in several different ways. The strongest notion (EDPI) is then further investigated. We introduce the variety of MINI algebras (a generalization of Hilbert algebras) and we show that they are a paradigm for subtractive EDPI varieties. Finally we deal with principal ideal operations, and in particular with the cases of meet and join of principal ideals being equationally definable. Received November 7, 1996; accepted in final form December 17, 1997.     

Finite distributive lattices are congruence lattices of almost-geometric lattices

A semimodular lattice L of finite length will be called an almost-geometric lattice if the order J(L) of its nonzero join-irreducible elements is a cardinal sum of at most two-element chains. We prove that each finite distributive lattice is isomorphic to the lattice of congruences of a finite almost-geometric lattice.     

Avoidable structures, I: Finite ordered sets, semilattices and lattices

We find all finite unavoidable ordered sets, finite unavoidable semilattices and finite unavoidable lattices. While working on this paper, the second and third authors were supported by US NSF grant DMS-0604065. The second author was also supported by the Grant Agency of the Czech Republic, grant #201/05/0002 and by the institutional grant MSM0021620839 financed by MSMT.     

Finite modular congruence lattices

We consider the variety of modular lattices generated by all finite lattices obtained by gluing together some M₃’s. We prove that every finite lattice in this variety is the congruence lattice of a suitable finite algebra (in fact, of an operator group). Received February 26, 2004; accepted in final form December 16, 2004.     

Finite lattices as relative congruence lattices of finite algebras

Let A be a finite algebra and a quasivariety. By A is meant the lattice of congruences θ on A with . For any positive integer n, we give conditions on a finite algebra A under which for any n-element lattice L there is a quasivariety such that . The author was supported by INTAS grant 03-51-4110.     

Finite groups with planar subgroup lattices

It is natural to ask when a group has a planar Hasse lattice or more generally when its subgroup graph is planar. In this paper, we completely answer this question for finite groups. We analyze abelian groups, p-groups, solvable groups, and nonsolvable groups in turn. We find seven infinite families (four depending on two parameters, one on three, two on four), and three “sporadic” groups. In particular, we show that no nonabelian group whose order has three distinct prime factors can be planar.     

Avoidable structures, II: Finite distributive lattices and nicely structured ordered sets

Let be the ordered set of isomorphism types of finite distributive lattices, where the ordering is by embeddability. We characterize the order ideals in that are well-quasi-ordered by embeddability, and thus characterize the members of that belong to at least one infinite anti-chain in . While working on this paper, the second and third authors were supported by US NSF grant DMS-0604065. The second author was also supported by the Grant Agency of the Czech Republic, grant #201/05/0002 and by the institutional grant MSM0021620839 financed by MSMT.     

Finite pseudocomplemented lattices: The spectra and the Glivenko congruence

Recently, Gr?tzer, Gunderson and Quackenbush have characterized the spectra of finite pseudocomplemented lattices, solving a problem raised by G. Gr?tzer in his first monograph on lattice theory from 1971. In this note we discuss the tight connection between the spectra and the Glivenko congruence of finite pseudocomplemented lattices.