A combinatorial property and Cayley graphs of semigroups

We describe semigroups satisfying a combinatorial property defined in terms of Cayley graphs.     

A combinatorial property of cardinals

(GCH) For every cardinal there exists such that for every , there are such that .

     

A combinatorial property of rho-functions

Acta Mathematica Hungarica - We show that if $$\mathcal{T} $$ is any Hausdorff topology on $$\omega_{1} $$ , then any subset of $$\omega_{1} $$ which is homeomorphic to the rationals under...     

A combinatorial property of convex sets

A known result in combinatorial geometry states that any collection P _n of points on the plane contains two such that any circle containing them contains n/c elements of P _n , c a constant. We prove: Let Φ be a family of n noncrossing compact convex sets on the plane, and let S be a strictly convex compact set. Then there are two elements S _i , S _j of Φ such that any set S' homothetic to S that contains them contains n/c elements of Φ, c a constant (S' is homothetic to S if , where λ is a real number greater than 0 and is a vector of ). Our proof method is based on a new type of Voronoi diagram, called the ``closest covered set diagram' based on a convex distance function. We also prove that our result does not generalize to higher dimensions; we construct a set Φ of n disjoint convex sets in such that for any nonempty subset Φ _H of Φ there is a sphere S _H containing all the elements of Φ _H , and no other element of Φ. Received December 7, 1994, and in revised form March 25, 1996.     

A structural property of Adian inverse semigroups

We prove that an inverse semigroup over an Adian presentation is E-unitary.     

A combinatorial problem involving graphs and matrices

In this paper we discuss a combinatorial problem involving graphs and matrices. Our problem is a matrix analogue of the classical problem of finding a system of distinct representatives (transversal) of a family of sets and relates closely to an extremal problem involving 1-factors and a long standing conjecture in the dimension theory of partially ordered sets. For an integer n ?1, let n denote the n element set {1,2,3,…, n}. Then let A be a k×t matrix. We say that A satisfies property P(n, k) when the following condition is satisfied: For every k-taple (x₁,x₂,…,x_k?n^k there exist k distinct integers j₁,j₂,…,j_k so that x_i= a_ii for i= 1,2,…,k. The minimum value of t for which there exists a k × t matrix A satisfying property P(n,k) is denoted by f(n,k). For each k?1 and n sufficiently large, we give an explicit formula for f(n, k): for each n?1 and k sufficiently large, we use probabilistic methods to provide inequalities for f(n,k).     

A combinatorial approach to the structure of locally inverse semigroups

Semigroup Forum - We provide a simple and short proof of a multidimensional Borg-Levinson type theorem. Precisely, we prove that the knowledge of spectral boundary data determine uniquely the...     

A sufficient condition for the non-finite basis property of semigroups

Recently, Zhang and Luo proved that a certain semigroup L of order six is non-finitely based. The main aim of the present article is to generalize this result to a sufficient condition for the non-finite basis property of semigroups. It follows that the semigroup L is inherently non-finitely based relative to a certain class of semigroups. It is also shown that the variety var?L generated by L contains a unique maximal subvariety that is non-finitely based. Consequently, the variety var ?L is not a limit variety.     

A combinatorial property of points and ellipsoids

For eachd1 there is a constantc _d>0 such that any finite setXR ^d contains a subsetYX, |Y|[1/4d(d+3)]+1 having the following property: ifEY is an ellipsoid, then |E X|c _d |X|.On leave from the Mathematical Institute of the Hungarian Academy of Sciences, 1364 Budapest, P.O. Box 127, Hungary. Supported by a research fellowship from the Science and Engineering Research Council, U.K., and by Hungarian National Foundation for Scientific Research Grant No. 1812.     

Incidence matrices and inequalities for combinatorial designs

In this paper we use incidence matrices of block designs and row–column designs to obtain combinatorial inequalities. We introduce the concept of nearly orthogonal Latin squares by modifying the usual definition of orthogonal Latin squares. This concept opens up interesting combinatorial problems and is expected to be useful in planning experiments by statisticians. © 2002 John Wiley & Sons, Inc. J Combin Designs 10: 17–26, 2002     

On varieties of combinatorial inverse semigroups II

Communicated by L. N. Shevrin     

Invariant factors of combinatorial matrices

The Smith normal forms of an Hadamard matrix of order 4m (m square-free), and of the incidence matrix of a (ν, k, λ) configuration (n=k−λ square-free (n, λ)=1), are determined.     

A degree property of rational matrices

In this paper, the existence of a relationship between the degrees of polynomials in any given rational matrix and the degrees of the polynomials in the inverse of that matrix is explained and proved.