Contractive presentations: A family of inverse monoids and semigroups with finite \({\mathcal R}\)-classes (Q1185779)

Some decidability results are obtained for inverse monoids presented by generators and a special class of relations, contractive and strictly contractive relations. These relations are defined using graphical constructions (Munn trees). A fundamental characteristic of the monoids investigated is: if \(F\) is a free inverse monoid and \(M=F/\theta\) then for \(m\in F\), the \(\mathcal R\)-class of \(m\theta\) has no more elements than the \(\mathcal R\)-class of \(m\). The word problem for a finitely based contractive variety of inverse semigroups is decidable and the free objects in such a variety have finite \(\mathcal D\)-classes and are residually finite.