## DIMACS TR: 2001-22

## 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

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