**Dorel Lucanu**

In this paper we investigate the possibilities to obtain complete axiomatizations for categories of symmetries. The key point consists in associating a rewrite theory $\calR(\S,E)$ with the equational specification by turning the equations into rewrite rules.

The elegant construction of the free $\calR$-grupoid given in \cite{concrew} provides already an axiomatization of the free $(\S,E)$-system (the non-coherent category of symmetries).

The problem of finding axioms which expresses the commutativity of the diagrams still remains.

We show that if equations $E$, viewed as rewrite rules, form a convergent (confluent and terminating) rewriting system then these axioms are obtained by computing all critical pairs.

Each confluent rewriting generated by a critical pair produce an equation.

The set of all equations obtained in this way forms a specification of the commutative diagrams. The method can be generalized to the case when $E$ is convergent modulo a theory $T$.

### Bibtex

@TechReport{catsim, author = {Dorel Lucanu}, title = {On the Axiomatization of the Category of Symmetries}, institution = {University ``A.I.Cuza'' of Iac{s}i, Faculty of Computer Science}, year = {1998}, number = {TR-98-03}, url = {https://publications.info.uaic.ro/technical-reports/archive/tr98-03-1998-on-the-axiomatization-of-categories-of-symmetries/}, }