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\$.

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### Bibtex

```@TechReport{catsim,
author =	 {Dorel Lucanu},
title =	 {On the Axiomatization of the Category of Symmetries},
institution =	 {University ``Al.I.Cuza'' of Iac si, Faculty of
Computer Science},
year =	 {1998},
number =   {TR-98-03},
note =	 {URL:http://www.infoiasi.ro/~tr/tr.pl.cgi},
}```