Multiplication by a Constant and Recognizability in an Abstract Numeration System

One of the main interesting concerns in work about numeration systems is studying the relationship that exist between arithmetic properties of numbers and syntactical properties of their representation. More precisely, we prefer to manipulate set of numbers whose representations are characterized by very simple syntactical rules, that is, forming a regular language. Such sets are called S-recognizable sets, where S is the numeration system we are working with. In particular, we would like to obtain numeration systems in which the whole set N is recognizable, partly due to the fact that, if the set of representations of all integers is regular, there are very simple algorithms making it possible to decide if a word stands or does not stand for a number. Some general questions in this area are the following.
• Characterization of the recognizable parts in a fixed numeration system.
• Determination of the numeration systems for which a given set of numbers is recognizable.
• To examine the stability of recognizability under arithmetic operations.
In this poster we present our research about that last problem. First, we will define the abstract numeration systems based on a regular language. Then we will study the relationship between multiplication by a constant and the S-recognizability for languages of the type a_1^∗ . . . a_k^∗.