Chain partitions of ordered sets

For natural numbers s and r and a finite poset P of height h, there exists a finite poset P of height H(s, r, h) such that for an arbitrary r-colouring of the s-chains of P, a monochromatically embedded copy of P can be found in P. A best possible upper bound for H(s, r, h) in terms of the well-known Ramsey numbers is given.     

Group closures of partial transformations

Let \(X\) be an infinite set, \(f\) a partial one-to-one transformation of \(X\), and \(H\) a normal subgroup of G _X , the group of all permutations of \(X\). We investigate when \(H\) is equal to \(G_{<f:H>}\). That is, we are interested when \(H\) is the full group of normalizers of the semigroup of transformations on \(X\) generated by conjugates of \(f\) by elements of \(H\).     

Strongly regular Cayley graphs from partitions of subdifference sets of the Singer difference sets

In this paper, we give a new lifting construction of “hyperbolic” type of strongly regular Cayley graphs. Also we give new constructions of strongly regular Cayley graphs over the additive groups of finite fields based on partitions of subdifference sets of the Singer difference sets. Our results unify some recent constructions of strongly regular Cayley graphs related to m-ovoids and i-tight sets in finite geometry. Furthermore, some of the strongly regular Cayley graphs obtained in this paper are new or nonisomorphic to known strongly regular graphs with the same parameters.     

Difference systems of sets based on cosets partitions

Difference systems of sets （DSS） are combinatorial configurations that arise in connection with code synchronization. This paper proposes a new method to construct DSSs, which uses known DSSs to partition some of the cosets of Zv relative to subgroup of order k, where v = km is a composite number. As applications, we obtain some new optimal DSSs.     

Hyperplane partitions and difference systems of sets

Difference Systems of Sets (DSS) are combinatorial configurations that arise in connection with code synchronization. This paper gives new constructions of DSS obtained from partitions of hyperplanes in a finite projective space, as well as DSS obtained from balanced generalized weighing matrices and partitions of the complement of a hyperplane in a finite projective space.     

A decomposition of partitions and numerical sets

Semigroup Forum - The aim of this work is to exhibit a decomposition of partitions of natural numbers and numerical sets. In particular, we obtain a decomposition of a sparse numerical set into the...     

A class of matroids derived from saturated chain partitions of partially ordered sets

The simultaneously k- and (k − 1)-saturated chain partitions of a finite partially ordered set P determine a matroid G_k(P). This matroid is a gammoid. The identity on P induces a strong map from G_k(P) to G_{k + 1}(P). This strong map has a linear representation.     

Largest and smallest maximal sets of pairwise disjoint partitions

Let $\text{X}$ be a maximal set of pairwise disjoint partitions of n into t distinct parts. Let M_t(n) (resp. m_t(n)) denote the size of the largest (resp. smallest) such maximal set $\text{X}$. Upper and lower bounds for $\text{M}{}_{\mathrm{t}}\text{(n)}\text{n}$ and $\text{m}{}_{\mathrm{t}}\text{(n)}\text{n}$ are established.     

A test for conjugacy of partitions of finite sets

In this note we present a simple test, involving a sequence of integers, which assures the conjugacy of a given partition P of a finite set S when our operations lead only to nonnegative integers. If negative integers appear in our operations, our test is inconclusive. The test, when conclusive, and an elementary property of permutations determine a conjugate for P.     

Comparing partitions of two sets of units based on the same variables

We propose a procedure based on a latent variable model for the comparison of two partitions of different units described by the same set of variables. The null hypothesis here is that the two partitions come from the same underlying mixture model. We define a method of “projecting” partitions using a supervised classification method: once one partition is taken as a reference; the individuals of the second data set are allocated to the clusters of the reference partition; it gives two partitions of the same units of the second data set: the original and the projected one and we evaluate their difference by usual measures of association. The empirical distributions of the association measures are derived by simulation.     

Balanced partitions of two sets of points in the plane

We prove the following theorem. Let m≥2 and q≥1 be integers and let S and T be two disjoint sets of points in the plane such that no three points of ST are on the same line, |S|=2q and |T|=mq. Then ST can be partitioned into q disjoint subsets P₁,P₂,…,P_q satisfying the following two conditions: (i) conv(P_i)∩conv(P_j)=φ for all 1≤i<j≤q, where conv(P_i) denotes the convex hull of P_i; and (ii) |P_i∩S|=2 and |P_i∩T|=m for all 1≤i≤q.     

Odd‐sized partitions of Russell‐sets

In the setting of ZF, i.e., Zermelo–Fraenkel set theory without the Axiom of Choice (AC), we study partitions of Russell‐sets into sets each with exactly n elements (called n ‐ary partitions), for some integer n. We show that if n is odd, then a Russell‐set X has an n ‐ary partition if and only if |X | is divisible by n. Furthermore, we establish that it is relative consistent with ZF that there exists a Russell‐set X such that |X | is not divisible by any finite cardinal n > 1 (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Test closures of sets in Banach spaces generated by linear functionals

Some criteria are proved for an element of a Banach space to belong to the test or Korovkin closure of a given set. The operations of closure in question are determined by linear functionals. The cases of arbitrary linear functionals and of positive functionals are considered. Some sufficient conditions are given for the test closure to coincide with the Korovkin closure. Bibliography: 33 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 247, 1997, pp. 114–145. Translated by S. Yu. Pilyugin.