Sublattices of complete lattices with continuity conditions

Various embedding problems of lattices into complete lattices are solved. We prove that for any join-semilattice S with the minimal join-cover refinement property, the ideal lattice Id S of S is both algebraic and dually algebraic. Furthermore, if there are no infinite D-sequences in J(S), then Id S can be embedded into a direct product of finite lower bounded lattices. We also find a system of infinitary identities that characterize sublattices of complete, lower continuous, and join-semidistributive lattices. These conditions are satisfied by any (not necessarily finitely generated) lower bounded lattice and by any locally finite, join-semidistributive lattice. Furthermore, they imply M. Erné’s dual staircase distributivity.On the other hand, we prove that the subspace lattice of any infinite-dimensional vector space cannot be embedded into any ℵ₀-complete, ℵ₀-upper continuous, and ℵ₀-lower continuous lattice. A similar result holds for the lattice of all order-convex subsets of any infinite chain.Dedicated to the memory of Ivan RivalReceived April 4, 2003; accepted in final form June 16, 2004.This revised version was published online in August 2005 with a corrected cover date.     

On the number of join-irreducibles in a congruence representation of a finite distributive lattice

For a finite lattice L, let $ \trianglelefteq_L $ denote the reflexive and transitive closure of the join-dependency relation on L, defined on the set J(L) of all join-irreducible elements of L. We characterize the relations of the form $ \trianglelefteq_L $, as follows: Theorem. Let $ \trianglelefteq $ be a quasi-ordering on a finite set P. Then the following conditions are equivalent:(i) There exists a finite lattice L such that $ \langle J(L), \trianglelefteq_L $ is isomorphic to the quasi-ordered set $ \langle P, \trianglelefteq \rangle $.(ii) $ |\{x\in P|p \trianglelefteq x\}| \neq 2 $, for any $ p \in P $.For a finite lattice L, let $ \mathrm{je}(L) = |J(L)|-|J(\mathrm{Con} L)| $ where Con L is the congruence lattice of L. It is well-known that the inequality $ \mathrm{je}(L) \geq 0 $ holds. For a finite distributive lattice D, let us define the join- excess function:$ \mathrm{JE}(D) =\mathrm{min(je} (L) | \mathrm{Con} L \cong D). $We provide a formula for computing the join-excess function of a finite distributive lattice D. This formula implies that $ \mathrm{JE}(D) \leq (2/3)| \mathrm{J}(D)|$ , for any finite distributive lattice D; the constant 2/3 is best possible.A special case of this formula gives a characterization of congruence lattices of finite lower bounded lattices.Dedicated to the memory of Gian-Carlo Rota     

Lattices isomorphic with their free powers

For any N3 there exists a lattice L isomorphic with NL (the free product of its own N copies) but not isomorphic with kL for any k–2,...,N-1.     

Generalizing semidistributivity

A latticeL is called congruence normal if it can be generated by doubling of convex sets starting with the one-element lattice. In the special case of intervals, the lattice is called bounded. It has been proven thatL is bounded if and only ifL is congruence normal and semidistributive.In this paper we study the connection between certain classes of convex sets and generalized semidistributive laws. These so-called doubling classes are pseudovarieties which can be described by implications as well as by forbiden substructures. In the end, we examine the structure of the lattice of all doubling classes.     

Largest extension of a finite convex geometry

We present a new embedding of a finite join-semidistributive lattice into a finite atomistic join-semidistributive lattice. This embedding turns out to be the largest extension, when applied to a finite convex geometry.In Celebration of the Sixtieth Birthday of Ralph N. McKenzieReceived September 18, 2002; accepted in final form September 29, 2003.     

Reflections on lower bounded lattices

Lattices in the variety of lower bounded lattices of rank k are characterized. A sufficient condition for a lattice to be lower bounded is given, and used to produce a new example of a non-finitely-generated lower bounded lattice. Lattices that are subdirect products of finite lower bounded lattices are characterized.In memory of Ivan RivalReceived September 18, 2003; accepted in final form October 5, 2004.This revised version was published online in August 2005 with a corrected cover date.     

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.     

Finite lattices and congruences. A survey

In the early forties, R.P. Dilworth proved his famous result: Every finite distributive lattice D can be represented as the congruence lattice of a finite lattice L. In one of our early papers, we presented the first published proof of this result; in fact we proved: Every finite distributive lattice D can be represented as the congruence lattice of a finite sectionally complemented lattice L.We have been publishing papers on this topic for 45 years. In this survey paper, we are going to review some of our results and a host of related results by others: Making L nice.If being nice is an algebraic property such as being semimodular or sectionally complemented, then we have tried in many instances to prove a stronger form of these results by verifying that every finite lattice has a congruence-preserving extension that is nice. We shall discuss some of the techniques we use to construct nice lattices and congruence-preserving extensions.We shall describe some results on the spectrum of a congruence of a finite sectionally complemented lattice, measuring the sizes of the congruence classes. It turns out that with very few restrictions, these can be as bad as we wish.We shall also review some results on simultaneous representation of two distributive lattices. We conclude with the magic wand construction, which holds out the promise of obtaining results that go beyond what can be achieved with the older techniques.In Celebration of the Sixtieth Birthday of Ralph N. McKenzieReceived November 26, 2002; accepted in final form June 18, 2004.     

Many varieties of lattices with nonsurjective epimorphisms

We use dominions to show that many varieties of lattices have nonsurjective epimorphisms. The variety D of distributive lattices is treated in detail. We show that the dominion in D of a sublattice is the closure of M under relative complementation in L. This dominion is also the largest sublattice of L in which M is epimorphically embedded. In any variety of lattices larger than D, the dominion of M in L is just M. Received May 1, 2001; accepted in final form October 4, 2005.     

When do abstract bases generate continuous lattices and L-domains

In this note, the new concepts of C-bases (resp., BC-bases, L-bases) which are special kinds of abstract bases are introduced. It is proved that the round ideal completion of a C-basis (resp., BC-basis, L-basis) is a continuous lattice (resp., bc-domain, L-domain). Furthermore, representation theorems of continuous lattices (resp., bc-domains, L-domains) by means of the round ideal completions of C-bases (resp., BC-bases, L-bases) are obtained. Supported by the NSF of China (10371106, 60774073) and by the Fund (S0667-082) from Nanjing University of Aeronautics and Astronautics.     

Small sublattices with large dominions

Two examples of finite sublattices with infinite dominions are given. It is proven that this cannot occur in a finitely generated lattice variety. Received May 10, 2001; accepted in final form October 6, 2005.     

Cube-like structures generated by filters

The join semi-lattice of faces of an n-cube has a rich structure. In considering generalizations of these structures we are led to looking at interval algebras constructed using Boolean filters. We look at the structure of these algebras and their automorphism groups.Dedicated to the memory of Gian-Carlo Rota     

III-Homomorphisms and III-congruences on posets

A new homomorphism between two partially ordered sets (the III-homomorphism) and a new congruence on a poset (the III-congruence) are introduced. Some properties of these homomorphisms and congruences and their relationship to the other known homomorphisms and congruences on posets are investigated. In contrast to total algebras, there are many different ways to introduce these notions. It is usually required that the respective notions should coincide with the usual definitions whenever lattices or semilattices are treated. The present paper presents an approach which in some sense completes the hierarchy of definitions so far used.     

A note on essentially convex orders

In this note, we determine precisely which partially ordered sets (posets) have the property that, whenever they occur as subposets of a larger poset, they occur there convexly, i.e., as convex subposets. As a corollary, we also determine which lattices have the property that, if they occur as sublattices of a finite distributive lattice L, then they also occur as closed intervals in L. Throughout, all sets will be finite.Dedicated to the memory of Ivan RivalReceived May 5, 2003; accepted in final form October 3, 2004.This revised version was published online in August 2005 with a corrected cover date.     

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.     

Nearlattices

By a nearlattice is meant a join-semilattice where every principal filter is a lattice with respect to the induced order. Alternatively, a nearlattice can be described as an algebra with one ternary operation satisfying eight simple identities. Hence, the class of nearlattices is a variety. We characterize nearlattices every sublattice of which is distributive. Then we introduce the so-called section pseudocomplementation on nearlattices which can also be characterized by identities.     

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.     

Meet decompositions in complete lattices

In the present paper we shall study infinite meet decompositions of an element of a complete lattice. We give here a generalization of some results of papers [2] and [3].     

Combinatorial representation and convex dimension of convex geometries

We develop a representation theory for convex geometries and meet distributive lattices in the spirit of Birkhoff's theorem characterizing distributive lattices. The results imply that every convex geometry on a set X has a canonical representation as a poset labelled by elements of X. These results are related to recent work of Korte and Lovász on antimatroids. We also compute the convex dimension of a convex geometry.Supported in part by NSF grant no. DMS-8501948.