## Local Groups in Free Burnside Groupoids

### Authors: Alair Pereira do Lago

ABSTRACT

The study of repeated sequences is the basis of the study of \fb{groups} and \fb{semigroups}, and a better knowledge on Burnside algebras can be very useful in the analysis of many properties of sequences containing repetitions. The main theorem announced here establishes an important connection between \fb{groups} and \fb{semigroups}. In order to establish this connection, we use interesting properties on graphs, categories, combinatorics on words to obtain the algebraic main theorem.

Let $\grafo{G}$ be~a (possibly infinite) strongly connected graph and let $\T$ be a set of monoid identities such that any monoid satisfying $\T$ is also a group. Let $\cat{B}$ be~the free groupoid on $\grafo{G}$ satisfying $\T$. Then, the local groups $\local{B}{v}$, for $v\in \vertices{G}$, are all isomorphic to a free group satisfying $\T$. Furthermore, it is free over a generating set which can be effectively characterized and whose cardinality is the cyclomatic number of the graph $\grafo{G}$.

Paper Available at: ftp://dimacs.rutgers.edu/pub/dimacs/TechnicalReports/TechReports/2001/2001-22.ps.gz