TR98-03 (1998) – On the Axiomatization of Categories of Symmetries

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 ``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},
}