## E-unitary inverse monoids and the Cayley graph of a group presentation.(English)Zbl 0676.20037

Let (X;R) be a presentation for the group G and let $$\Gamma$$ (X;R) be the Cayley graph of this presentation. For each finite subgraph $$\Gamma$$ of $$\Gamma$$ (X;R) and $$g\in G$$, let $$g\cdot \Gamma$$ be the subgraph of $$\Gamma$$ (X;R) with vertex set $$\{$$ gh: h is a vertex of $$\Gamma$$ $$\}$$ ; there is an edge labelled x from gh to ghx in $$g\cdot \Gamma$$ if the edge labelled x from h to hx occurs in $$\Gamma$$. Let M(X;R) consist of the pairs ($$\Gamma$$,g) where $$\Gamma$$ is a finite connected subgraph of $$\Gamma$$ (X;R) containing 1 and g as vertices. On M(X;R) define a multiplication by $$(\Gamma,g)(\Gamma ',g')=(\Gamma \cup g\cdot \Gamma ',gg')$$. It turns out that M(X;R) is an E-unitary inverse monoid generated by the elements $$(\Gamma_ x,x)$$, $$x\in X$$, where $$\Gamma_ x$$ has vertices 1 and x and one edge labelled x.
Given the presentation (X;R) for the group G, a category is constructed whose objects are diagrams. A diagram involving M(X;R) is proven to be an initial object in this category. From this it is proved that every X- generated E-unitary monoid having G as its maximum group image is an idempotent pure homomorphic image of M(X;R). For any variety $${\mathcal V}$$ of groups, let $$\hat {\mathcal V}$$ be the variety consisting of those inverse monoids S for which there exists an E-unitary inverse monoid N with greatest group homomorphic image in $${\mathcal V}$$ and an idempotent separating homomorphism of N onto S. If G is free in $${\mathcal V}$$ on X and has a presentation (X;R), then M(X;R) turns out to be free in $$\hat {\mathcal V}$$ on X. In particular, if G is trivial, then M(X;R) is the free (monoid-) semilattice, and if G is the free group, then M(X;R) is the free inverse monoid.
The authors give a representation in the form of a McAlister P-semigroup for such monoids, investigate Green’s relations, explain the connection with Munn’s construction of the free inverse monoid in terms of birooted word trees, prove that the M(X;R) are completely semisimple and residually finite. A group is isomorphic to a maximal subgroup of M(X;R) if and only if it is finite and embeddable into the group G with presentation (X;R). There exists an obvious functor of the category of X- generated E-unitary inverse monoids into the category of X-generated groups, which associates with every E-unitary inverse monoid its greatest group image; the above construction yields a left adjoint of this functor.
The paper concludes with several concrete examples and a detailed description of the free product of E-unitary inverse monoids. In particular, if $$(X_ 1;R_ 1)$$ and $$(X_ 2;R_ 2)$$ are group presentations such that $$X_ 1$$ and $$X_ 2$$ are disjoint, then $$M(X_ 1\cup X_ 2;R_ 1,R_ 2)$$ is isomorphic to the free product of $$M(X_ 1;R_ 1)$$ and $$M(X_ 2;R_ 2)$$.
Reviewer: F.Pastijn

### MSC:

 20M05 Free semigroups, generators and relations, word problems 20F05 Generators, relations, and presentations of groups 20E10 Quasivarieties and varieties of groups 05C25 Graphs and abstract algebra (groups, rings, fields, etc.) 20M07 Varieties and pseudovarieties of semigroups
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