Abstract of Paper |

*Factorizing Codes and Schutzenberger Conjectures*

by Clelia De Felice

**Abstract:**

In this paper we mainly deal with factorizing codes $C$ over $A$, i.e., codes verifying the famous still open factorization conjecture formulated by Sch\"utzenberger. Suppose $A =\{a,b\}$ and denote $a^n$ the power of $a$ in $C$. We show how we can construct $C$ starting with factorizing codes $C'$ with $a^{n'}\in C'$ and $n'$ less than $n$, under the hypothesis that all words $a^iwa^j$ in $C$, with $w^n bA^*b\cup\{b\}$, satisfy $i,j$ less than $n$. The operation involved, already introduced by another author, is also used to show that all maximal codes $C= P(A-1)S+1$ with $P,S\in Z\langle A\rangle$ and $P$ or $S$ in $Z\langle a\rangle$ can be constructed by means of this operation starting from prefix and suffix codes. Inspired by another early Sch\"utzenberger conjecture, we propose here an open problem related to the results obtained and to the operation considered in this paper.