On multiplicative bases in commutative semigroups

AbstractWe generalize some older results on multiplicative bases of integers to a certain class of commutative semigroups. In particular, we examine the structure of union bases of integers

On the K-theory of crossed products by automorphic semigroup actions

Let P be a semigroup that admits an embedding into a group G. Assume that the embedding satisfies a certain Toeplitz condition and that the Baum-Connes conjecture holds for G. We prove a formula describing the K- theory of the reduced crossed product A \rtimes{\alpha},r P by any automorphic action of P. This formula is obtained as a consequence of a result on the K-theory of crossed products for special actions of G on totally disconnected spaces. We apply our result to various examples including left Ore semigroups and quasi-lattice ordered semigroups. We also use the results to show that for certain semigroups P, including the ax + b-semigroup for a Dedekind domain R, the K-theory of the left and right regular semigroup C*-algebras of P coincide, although the structure of these algebras can be very different

Hypergroups and Hypergroup Algebras

The survey contains a brief description of the ideas, constructions, results, and prospects of the theory of hypergroups and generalized translation operators. Representations of hypergroups are considered, being treated as continuous representations of topological hypergroup algebras.Comment: 52 page

Semigroups with the Erdös-Turán Property

A set X in a semigroup G has the Erdös-Turán property ET if, for any basis A of X, the representation function rA is ubounded, where rA(x) counts the number of representations of x as a product two elements in A. We show that, under some conditions, operations on binary vectors whose value at each coordinate depends only on neighbouring coordinates of the factors give rise to semigroups with the ET{property. In particular countable powers of semigroups with no mutually inverse elements have the ET{property. As a consequence, for each k there is N(k) such that, for every ¯nite subset X of a group G with X \ X¡1 = f1g, the representation function of every basis of XN ½ GN, N ¸ N(k), is not bounded by k. This is in contrast with the known fact that each p{elementary group admits a basis of the whole group whose representation function is bounded by an absolute constan