Intervals in lattices of quasiorders

We investigate the structure of intervals in the lattice of all closed quasiorders on a compact or discrete space. As a first step, we show that if the intervalI has no infinite chains then the underlying space may be assumed to be finite, and in particular,I must be finite, too. We compute several upper bounds for its size in terms of its heighth, which in turn can be computed easily by means of the least and the greatest element ofI. The cover degreec of the interval (i.e. the maximal number of atoms in a subinterval) is less than 4h. Moreover, ifc4(n–1) thenI contains a Boolean subinterval of size 2 ⁿ , and ifI is geometric then it is already a finite Boolean lattice. While every finite distributive lattice is isomorphic to some interval of quasiorders, we show that a nondistributive finite interval of quasiorders is neither a vertical sum nor a horizontal sum of two lattices, with exception of the pentagon. Many further lattices are excluded from the class of intervals of quasiorders by the fact that no join-irreducible element of such an interval can have two incomparable join-irreducible complements. Up to isomorphism, we determine all quasiorder intervals with less than 9 elements and all quasiorder intervals with two complementary atoms or coatoms.     

The order dimension of two levels of the Boolean lattice

LetB _n(s, t) denote the partially ordered set consisting of alls-subsets andt-subsets of ann-element underlying set where these sets are ordered by inclusion. Answering a question of Trotter we prove that dim(B _n(k, n–k))=n–2 for 3k<(1/7)n ^1/3. The proof uses extremal hypergraph theory.     

The dimension of suborders of the Boolean lattice

We consider the order dimension of suborders of the Boolean latticeB _n . In particular we show that the suborder consisting of the middle two levels ofB _n dimension at most of 6 log₃ n. More generally, we show that the suborder consisting of levelss ands+k ofB _n has dimensionO(k ² logn).The research of the second author was supported by Office of Naval Research Grant N00014-90-J-1206.The research of the third author was supported by Grant 93-011-1486 of the Russian Fundamental Research Foundation.     

A study of the order dimension of a poset using matrices

In every finite poset (X, ≤) we assign the so called order-matrix , where α_ij ∈ {?2, 0, 1, 2}. Using this matrix, we characterize the order dimension of an arbitrary finite poset.     

The dimension of interior levels of the Boolean lattice

LetP(k,r;n) denote the containment order generated by thek-element andr-element subsets of ann-element set, and letd(k,r;n) be its dimension. Previous research in this area has focused on the casek=1.P(1,n–1;n) is the standard example of ann-dimensional poset, and Dushnik determined the value ofd(1,r;n) exactly, whenr2 . Spencer used the Erdös-Szekeres theorem to show thatd(1, 2;n) lg lgn, and he used the concept of scrambling families of sets to show thatd(1,r;n)=(lg lgn) for fixedr. Füredi, Hajnal, Rödl and Trotter proved thatd(1, 2;n)=lg lgn+(1/2+o(1))lg lg lgn. In this paper, we concentrate on the casek2. We show thatP(2,n–2;n) is (n–1)-irreducible, and we investigated(2,r;n) whenr2 , obtaining the exact value for almost allr.The research was supported in part by NSF grant DMS 9201467.The research was supported in part by the Universities in Russia program.     

The generalized Füredi conjecture holds for finite linear lattices

We say that a rank-unimodal poset P has rapidly decreasing rank numbers, or the RDR property, if above (resp. below) the largest ranks of P, the size of each level is at most half of the previous (resp. next) one. We show that a finite rank-unimodal, rank-symmetric, normalized matching, RDR poset of width w has a partition into w chains such that the sizes of the chains are one of two consecutive integers. In particular, there exists a partition of the linear lattices L_n(q) (subspaces of an n-dimensional vector space over a finite field, ordered by inclusion) into chains such that the number of chains is the width of L_n(q) and the sizes of the chains are one of two consecutive integers.     

Tensorial decomposition of concept lattices

A tensor product for complete lattices is studied via concept lattices. A characterization as a universal solution and an ideal representation of the tensor products are given. In a large class of concept lattices which contains all finite ones, the subdirect decompositions of a tensor product can be determined by the subdirect decompositions of its factors. As a consequence, one obtains that the tensor product of completely subdirectly irreducible concept lattices of this class is again completely subdirectly irreducible. Finally, applications to conceptual measurement are discussed.Dedicated to Ernst-August Behrens on the occasion of his seventieth birthday.     

Cover-preserving order embeddings into Boolean lattices

It is not known which finite graphs occur as induced subgraphs of a hypercube. This is relevant in the theory of parallel computing. The ordered version of the problem is: Which finite posets P occur as cover-preserving subposets of a Boolean lattice? Our main Theorem gives (for 0,1-posets) a necessary and sufficient condition, which involves the chromatic number of a graph associated to P. It is applied respectively to upper balanced, meet extremal, meet semidistributive, and semidistributive lattices P. More specifically, we consider isometric embeddings of posets into Boolean lattices. In particular, answering a question of Ivan Rival to the positive, a nontrivial invariant for the covering graph of a poset is found.     

On the greedy dimension of a partial order

This paper introduces a new concept of dimension for partially ordered sets. Dushnik and Miller in 1941 introduced the concept of dimension of a partial order P, as the minimum cardinality of a realizer, (i.e., a set of linear extensions of P whose intersection is P). Every poset has a greedy realizer (i.e., a realizer consisting of greedy linear extensions). We begin the study of the notion of greedy dimension of a poset and its relationship with the usual dimension by proving that equality holds for a wide class of posets including N-free posets, two-dimensional posets and distributive lattices.     

The order dimension of relatively free lattices

If V is a nontrivial lattice variety and is an infinite cardinal, then the order dimension of FV() is the smallest cardinal such that 2.This research was supported in part by NSF Grants DMS 87-03540 (Nation) and DMS 86-01576 (Schmerl).     

Incidence posets of trees in posets of large dimension

In this paper, we investigate substructures of partially ordered sets which must be present whenever the dimension is large. We show that for eachn1, ifT is a tree onn vertices and ifP is any poset having dimension at least 4n ⁶, then eitherP or its dual contains the incidence poset ofT as a suborder.     

A partial order for the set of meanders

In this paper, we present a new approach for studying meanders in terms of noncrossing partitions. We show how this approach leads to a natural partial order on the set of meanders. In particular, meanders form a graded poset with regard to this partial order.     

The dimension of orthomodular posets constructed by pasting Boolean algebras

The main aim of this paper is the calculation of the dimension of certain atomic amalgams. These consist of finite Boolean algebras (blocks) pasted together in such a way that a pair of blocks intersects either trivially in the bounds, or the intersection consists of the bounds, an atom, and its complement.     

Posets with large dimension and relatively few critical pairs

The dimension of a poset (partially ordered set)P=(X, P) is the minimum number of linear extensions ofP whose intersection isP. It is also the minimum number of extensions ofP needed to reverse all critical pairs. Since any critical pair is reversed by some extension, the dimensiont never exceeds the number of critical pairsm. This paper analyzes the relationship betweent andm, when 3tmt+2, in terms of induced subposet containment. Ifmt+1 then the poset must containS _t , the standard example of at-dimensional poset. The analysis form=t+2 leads to dimension products and David Kelly's concept of a split. Whent=3 andm=5, the poset must contain eitherS ₃, or the 6-point poset called a chevron, or the chevron's dual. Whent4 andm=t+2, the poset must containS _t , or the dimension product of the Kelly split of a chevron andS _t–3, or the dual of this product.     

On the nonexistence of free complete distributive lattices

We prove that there is no free object over a countable set in the category of complete distributive lattices with homomorphisms preserving binary meets and arbitrary joins.     

Interval dimension and MacNeille completion

Equality between the interval dimensions of a poset and its MacNeille completion, announced in [7], has been obtained by the authors as a byproduct of their study of Galois lattices in [8]. The purpose of this note is to give a direct proof, similar to the classical proof of Baker's result stating that the dimension (in the Dushnik-Miller sense) of a poset and its MacNeille completion are the same.Supported by French PRC Math-Info.     

Partitioning Boolean lattices into chains of subsets

We prove that the Boolean lattice of all subsets of an n-set can be partitioned into chains of size four if and only if n9.Research supported in part by N.S.F. grant DMS-8401281.Research supported in part by N.S.F. grant DMS-8406451.     

Subgraphs of hypercubes and subdiagrams of boolean lattices

In this paper we extensively treat the following problems: When is a given graph a subgraph (resp. induced subgraph) of a hypercube and when is an ordered set a subdiagram (resp. induced subdiagram) of a Boolean lattice? We present characterizations for that in terms of suitable edge-colorings of the graphs and, for ordered sets, of their covering graphs.     

Bigeneration in complete lattices and principal separation in ordered sets

By a recent observation of Monjardet and Wille, a finite distributive lattice is generated by its doubly irreducible elements iff the poset of all join-irreducible elements has a distributive MacNeille completion. This fact is generalized in several directions, by dropping the finiteness condition and considering various types of bigeneration via arbitrary meets and certain distinguished joins. This leads to a deeper investigation of so-called L-generators resp. C-subbases, translating well-known notions of topology to order theory. A strong relationship is established between bigeneration by (minimal) L-generators and so-called principal separation, which is defined in order-theoretical terms but may be regarded as a strong topological separation axiom. For suitable L, the complete lattices with a smallest join-dense L-subbasis consisting of L-primes are the L-completions of principally separated posets.     

Weak orders admitting a perpendicular linear order

Two orders on the same set are perpendicular if the constant maps and the identity map are the only maps preserving both orders. We characterize the finite weak orders admitting a perpendicular linear order.