New results from an algorithm for counting posets

In this paper we report on the success of a new technique for computing the number of unlabeled partial orders on n elements based on the partial order of partial orders ordered by containment. In addition to the number of partial orders, we obtain complete enumerations of the number of partial orders on n elements with r relations for n<-11, where r takes on all possible values. We point out some interesting sequences that arise in these tables.Supported by Natural Sciences and Engineering Research Council Grant No. OGP8053.     

Tensor products for bounded posets revisited

The category BPC of bounded posets and so-called cut continuous maps has concrete products, and the Dedekind-MacNeille completion gives rise to a reflector from BPC to the full subcategory CLJ of complete lattices and join-preserving maps. Like CLJ, the category BPC has a functional internal hom-functor in the sense of Banaschewski and Nelson. But, in contrast to CLJ, arbitrary universal bimorphisms do not exist in BPC. However, a natural tensor product is defined in terms of so-called G-ideals, such that the desired universal property holds at least for BPC-morphisms into complete lattices. Moreover, this tensor product is associative and distributes over (cartesian) products. The tensor product of an arbitrary family of bounded posets is isomorphic to that of their normal completions; hence, restricted to the subcategory CLJ, it agrees with the usual one.     

Imbedding posets in the integers

An imbedding of a poset P in the integers is a one-to-one order presevring map from P into the integers. Such a map always exists when P is finite and, moreover, certain imbeddings of subsets of finite P can be extended to imbeddings of the whole of P. D. E. Daykin has asked when an imbedding in the integers of a finite subset of a countable poset can be extended to the whole poset. This paper answers Daykin's question and some related questions.     

Asymptotic enumeration of two-dimensional posets

In this paper we show that the number of pairwise nonisomorphic two-dimensional posets with n elements is asymptotically equivalent to 1/2n!. This estimate is based on a characterization, in terms of structural decomposition, of two-dimensional posets having a unique representation as the intersection of two linear extensions.     

Universal correlations in finite posets

Posets A, BX×X, with X finite, are said to be universally correlated (AB) if, for all posets R over X, (i.e., all posets RY×Y with XY), we have P(RA) P(RB)P(RAB) P(R). Here P(RA), for instance, is the probability that a randomly chosen bijection from Y to the totally ordered set with |Y| elements is a linear extension of RA. We show that AB iff, for all posets R over X, P(RA) P(RB)P(RAB) P(R(AB)).Winkler proved a theorem giving a necessary and sufficient condition for AB. We suggest an alteration to his proof, and give another condition equivalent to AB.Daykin defined the pair (A, B) to be universally negatively correlated (A B) if, for all posets R over X, P(RA) P(RB)P(RAB) P(R(AB)). He suggested a condition for AB. We give a counterexample to that conjecture, and establish the correct condition. We write AB if, for all posets R over X, P(RA) P(RB)P(RAB) P(R). We give a necessary and sufficient condition for AB.We also give constructive techniques for listing all pairs (A, B) satisfying each of the relations AB, AB, and AB.     

Topological representation of posets

There is a canonical imbedding of a poset into a complete Boolean lattice and hence into a Boolean lattice. This gives it a representation as a collection of clopen sets of a Boolean space. There are reflective functions from a category of distributive posets to the subcategories of distributive and Boolean lattices and consequently a topological dual equivalence that extends the Stone duality of Boolean lattices.Presented by B. Jonsson.     

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.     

Some correlation inequalities in finite posets

Posets are said to be correlated with respect to another poset R on X (we write A _R B) if P(RA) P(RB)P(RAB) P(R). Here P(S) is the probability that a randomly chosen bijection from X to the totally ordered set with |X| elements is a linear extension of S. We study triples (A, B, R) such that A _R B holds for all extensions S of R (we write A _R B). Two well-known correlation inequalities, the xyz inequality and an inequality of Graham, Yao, and Yao, can be considered as giving cases when this relation holds. We show when the Graham, Yao, and Yao inequality holds strictly. Our main result is a classification of all R such that (a, b) _R (c, d) holds, where a, b, c, d are elements of X.     

Quotients of peck posets

An elementary, self-contained proof of a result of Pouzet and Rosenberg and of Harper is given. This result states that the quotient of certain posets (called unitary Peck) by a finite group of automorphisms retains some nice properties, including the Sperner property. Examples of unitary Peck posets are given, and the techniques developed here are used to prove a result of Lovász on the edge-reconstruction conjecture.Supported in part by a National Science Foundation research grant.     

Algorithms for the computation of functionals defined on the solution of a discrete ill-posed problem

We study a linear, discrete ill-posed problem, by which we mean a very ill-conditioned linear least squares problem. In particular we consider the case when one is primarily interested in computing a functional defined on the solution rather than the solution itself. In order to alleviate the ill-conditioning we require the norm of the solution to be smaller than a given constant. Thus we are lead to minimizing a linear functional subject to two quadratic constraints. We study existence and uniqueness for this problem and show that it is essentially equivalent to a least squares problem with a linear and a quadratic constraint, which is easier to handle computationally. Efficient algorithms are suggested for this problem.     

On forbidden configurations for strong posets

The concept of strong elements in posets is introduced. Several properties of strong elements in different types of posets are studied. Strong posets are characterized in terms of forbidden structures. It is shown that many of the classical results of lattice theory can be extended to posets. In particular, we give several characterizations of strongness for upper semimodular (USM) posets of finite length. We characterize modular pairs in USM posets of finite length and we investigate the interrelationships between consistence, strongness, and the property of being balanced in USM posets of finite length. In contrast to the situation in upper semimodular lattices, we show that these three concepts do not coincide in USM posets.     

Wide posets with fixed height and cutset number

Given an element x of a partial order P, a set S P is said to be a cutset for x if S x meets every maximal chain of P and x is incomparable to every element of S. The cutset number of P is the minimum m such that every element of P has a cutset of size at most m. Let w(m, h) be the maximum width of a poset with height h and cutset number m. We determine the order of growth of w(m, h) for fixed m or fixed h: w(m, h)(h ^m/2) for fixed m and w(m, h)(m ^h) for fixed h.Research supported in part by ONR Grant N00014-85K0570 and by NSA/MSP Grant MDA904-90-H-4011.     

Substitution and atomic extension on greedy posets

In this paper, we study the operations of substitution and atomic extension on greedy posets. For the substitution operation, if P=(P ₁ , x, P ₂ )is a greedy poset, then P ₁ and P ₂ are greedy posets, the converse being false. However, for the atomic extension, P=P ₁ (x, P ₂ )is a greedy poset if and only if P ₁ and P ₂ are greedy posets. We prove also that the class of greedy semi-partitive lattices is the smallest one containing M _n (n2), B ₃ and closed by atomic extension. The class C _n of greedy posets with jump number n is infinite. However, we show that C _n can be obtained, in a very simple way, from a subclass D _n of finite cardinal ity. We construct D _n for n=1, 2.     

On derived equivalences of categories of sheaves over finite posets

A finite poset X carries a natural structure of a topological space. Fix a field k, and denote by D^b(X) the bounded derived category of sheaves of finite dimensional k-vector spaces over X. Two posets X and Y are said to be derived equivalent if D^b(X) and D^b(Y) are equivalent as triangulated categories.We give explicit combinatorial properties of X which are invariant under derived equivalence; among them are the number of points, the Z-congruency class of the incidence matrix, and the Betti numbers. We also show that taking opposites and products preserves derived equivalence.For any closed subset Y⊆X, we construct a strongly exceptional collection in D^b(X) and use it to show an equivalence D^b(X)?D^b(A) for a finite dimensional algebra A (depending on Y). We give conditions on X and Y under which A becomes an incidence algebra of a poset.We deduce that a lexicographic sum of a collection of posets along a bipartite graph S is derived equivalent to the lexicographic sum of the same collection along the opposite .This construction produces many new derived equivalences of posets and generalizes other well-known ones.As a corollary we show that the derived equivalence class of an ordinal sum of two posets does not depend on the order of summands. We give an example that this is not true for three summands.     

Representing posets with k-tournaments

In this paper, we introduce a binary relation on the vertex set of a k-tournament, and using this relation show that every finite poset with cardinality n4 can be represented by a k-tournament for every k satisfying 3kn–1.     

Counting finite posets and topologies

A refinement of an algorithm developed by Culberson and Rawlins yields the numbers of all partially ordered sets (posets) with n points and k antichains for n11 and all relevant integers k. Using these numbers in connection with certain formulae derived earlier by the first author, one can now compute the numbers of all quasiordered sets, posets, connected posets etc. with n points for n14. Using the well-known one-to-one correspondence between finite quasiordered sets and finite topological spaces, one obtains the numbers of finite topological spaces with n points and k open sets for n11 and all k, and then the numbers of all topologies on n14 points satisfying various degrees of separation and connectedness properties, respectively. The number of (connected) topologies on 14 points exceeds 10²³.     

Multiplicative posets

A functionf from the posetP to the posetQ is a strict morphism if for allx, y P withx we havef(x). If there is such a strict morphism fromP toQ we writeP Q, otherwise we writeP Q. We say a posetM is multiplicative if for any posetsP, Q withP M andQ M we haveP ×Q M. (Here (p ₁,q ₁)<(p ₂,q ₂) if and only ifp ₁<p ₂ andq ₁<q ₂.) This paper proves that well-founded trees with height are multiplicative posets.This research was supported in part by NSERC Grant #69-1325.     

Order series of labelled posets

Order series of labelled posets are multi-analogues of the more familiar order polynomials; the corresponding multi-analogues of the related representation polynomials are calledE-series. These series can be used to describe the effect of composition of labelled posets on their order polynomials, whence their interest for us. We give a reciprocity theorem forE-series, and show that for an unlabelled poset the form of theE-series depends only upon the comparability graph of the poset. We also prove that theE-series of any labelled poset is a rational power series (in many indeterminates) and give an algorithm for computing it which runs in polynomial time when the poset is strictly labelled and of bounded width. Finally, we give an explicit product formula for theE-series of strictly labelled interval posets.This research was supported by the Natural Sciences and Engineering Research Council of Canada under operating grant #0105392.     

Series parallel posets with nonfinitely generated clones

In 1986 Tardos proved that for the poset 1+2+2+2+1, the clone of monotone operations is nonfinitely generated. We generalize his result in the class of series parallel posets. We characterize the posets with nonfinitely generated clones in this class by the property that they have a retract of the form either 1+2+2+2+1, 2+2+1, or 1+2+2.Research partially supported by Hungarian National Foundation for Research under grant no. 1903.