Table of Contents
ISRN Combinatorics
Volume 2014, Article ID 275260, 5 pages
http://dx.doi.org/10.1155/2014/275260
Research Article

Integer Semigroups Associated with Dumont-Thomas Numeration Systems

Departamento de Matemáticas, Universidad Simón Bolívar, Apartado 89000, Caracas 1086, Venezuela

Received 30 September 2013; Accepted 13 November 2013; Published 23 January 2014

Academic Editors: C. da Fonseca, A. Glen, P. E. Jorgensen, A. V. Kelarev, and C.-K. Lin

Copyright © 2014 Víctor F. Sirvent. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Given a primitive substitution, we define different binary operations on infinite subsets of the nonnegative integers. These binary operations are defined with the help of the Dumont-Thomas numeration system; that is, a numeration system associated with the substitution. We give conditions for these semigroups to have an identity element. We show that they are not finitely generated. These semigroups define actions on the set of positive integers. We describe the orbits of these actions. We also estimate the density of these sets as subsets of the positive integers.