Retractions onto series-parallel posets

The poset retraction problem for a poset P is whether a given poset Q containing P as a subposet admits a retraction onto P, that is, whether there is a homomorphism from Q onto P which fixes every element of P. We study this problem for finite series-parallel posets P. We present equivalent combinatorial, algebraic, and topological charaterisations of posets for which the problem is tractable, and, for such a poset P, we describe posets admitting a retraction onto P.     

B-free numbers in short arithmetic progressions

Given a sequence B of relatively prime positive integers with the sum of inverses finite, we investigate the problem of finding B-free numbers in short arithmetic progressions.     

Some new results on k-free numbers

In this paper we obtain an improved asymptotic formula on the frequency of k-free numbers with a given difference. We also give a new upper bound of Barban-Davenport-Halberstam type for the k-free numbers in arithmetic progressions.     

N-Capacity, N-harmonic radius and N-harmonic transplantation

For the N-Laplacian ΔN on a regular domain the N-Green's function exists. This allows us to define the N-Robin's function and the N-harmonic radius. We show their basic properties and extend the method of the harmonic transplantation to that of the N-harmonic transplantation. The method is used to estimate the first eigenvalue of the N-Laplacian ΔN. We also give another proof of the isoperimetric inequality for the N-capacity and give the isoperimetric inequality for the N-harmonic radius.     

Monopole solutions as three-dimensional generalizations of Kronecker series

We investigate the Dirac monopole on a three-dimensional torus as a solution of the Bogomolny equations with nontrivial boundary conditions. We show that a suitable analytic continuation of the obtained solution is a three-dimensional generalization of the Kronecker series, satisfies the corresponding functional equation, and is invariant under modular transformations.     

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.     

Difference posets as commutative directoids with sectional antitone involutions

It is shown that every difference poset can be converted into a total algebra in a manner similar to that which is used for difference lattices. As a tool applied here we have commutative directoids and posets with sectional antitone involutions. This work is supported by the Research and Development Council of the Czech Government via the project MSM6198959214.     

Boxicity of series-parallel graphs

We show that there exist series-parallel graphs with boxicity 3.     

Inverse NO-matrices

Results similar to those of Thomas L. Markham concerning inverse M-matrices are shown to hold for inverse N_Omatrices. A more general result and an alternate method of proof using Schur complements are given for one of Markham's theorems. Conditions are given to transform a nonpositive matrix to an N_O-amtrix by multiplication by Householder transformations. Also, an LU factorization of an inverse N_O-matrix is given.     

Canonical extensions of posets

We study a construction that produces a variety of completions of a given poset. The denseness and compactness properties of the completions obtained in this way are investigated. Next we focus our attention on three specific completions of a given poset that can be obtained through this construction—two of which have been called ‘the canonical extension’ of the poset in the literature. We investigate extensions of maps to these three completions. Although the extensions of unary operators need not be operators on the completions, we show that the extensions of unary residuated maps are residuated. We also investigate extensions of n-ary maps. In particular, we have a closer look at order-preserving n-ary maps and binary residuated maps.     

Parity representations of posets

Parity representations, introduced in this paper, comprise a new method of representation of posets that yields insight into the combinatorics of the poset of all intervals of a poset. Results here generalize some results previously obtained for the face lattices of binary partition polytopes.     

Ultraproducts of continuous posets

We consider local partial clones defined on an uncountable set E having the form Polp(\({\mathfrak{R}}\)), where \({\mathfrak{R}}\) is a set of relations on E. We investigate the notion of weak extendability of partial clones of the type Polp(\({\mathfrak{R}}\)) (in the case of E countable, this coincides with the notion of extendability previously introduced by the author in 1987) which allows us to expand to uncountable sets results on the characterization of Galois-closed sets of relations as well as model-theoretical properties of a relational structure \({\mathfrak{R}}\). We establish criteria for positive primitive elimination sets (sets of positive primitive formulas over \({\mathfrak{R}}\) through which any positive primitive definable relation over \({\mathfrak{R}}\) can be expressed without existential quantifiers) for finite \({\mathfrak{R}}\) as well as for \({\mathfrak{R}}\) having only finite number of positive primitive definable relations of any arity. Emphasizing the difference between countable and uncountable sets, we show that, unlike in the countable case, the characterization of Galois-closed sets InvPol(\({\mathfrak{R}}\)) (that is, all relations which are invariant under all operations from the clone Pol(\({\mathfrak{R}}\)) defined on an uncountable set) cannot be obtained via the application of finite positive primitive formulas together with infinite intersections and unions of updirected sets of relations from \({\mathfrak{R}}\).     

Stable representations of posets

The purpose of this paper is to study stable representations of partially ordered sets (posets) and compare it to the well known theory for quivers. In particular, we prove that every indecomposable representation of a poset of finite type is stable with respect to some weight and construct that weight explicitly in terms of the dimension vector. We show that if a poset is primitive then Coxeter transformations preserve stable representations. When the base field is the field of complex numbers we establish the connection between the polystable representations and the unitary χ-representations of posets. This connection explains the similarity of the results obtained in the series of papers.     

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.     

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.     

Geometric applications of posets

We show the power of posets in computational geometry by solving several problems posed on a set S of n points in the plane: (1) find the n − k − 1 rectilinear farthest neighbors (or, equivalently, k nearest neighbors) to every point of S (extendable to higher dimensions), (2) enumerate the k largest (smallest) rectilinear distances in decreasing (increasing) order among the points of S, (3) given a distance δ > 0, report all the pairs of points that belong to S and are of rectilinear distance δ or more (less), covering k ≥ n/2 points of S by rectilinear (4) and circular (5) concentric rings, and (6) given a number k ≥ n/2 decide whether a query rectangle contains k points or less.     

Representations of primitive posets

We give a new proof of the Kleiner theorem for the case of primitive posets.     

N-reconstructibility of non-reconstructible digraphs

In this paper we prove that all the non-reconstructible digraphs constructed by Stockmeyer in [3] are reconstructible from their point-deleted subdigraphs with the additional knowledge of the degree pair of the deleted point for each point-deleted subdigraph.