##
**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 \(M(X_ 2;R_ 2)\).

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 |

### Keywords:

presentations of inverse monoids; Cayley graph; E-unitary inverse monoid; group; free group; free inverse monoid; McAlister P-semigroup; Green’s relations; category of X-generated E-unitary inverse monoids; category of X-generated groups; free product of E-unitary inverse monoids
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\textit{S. W. Margolis} and \textit{J. C. Meakin}, J. Pure Appl. Algebra 58, No. 1, 45--76 (1989; Zbl 0676.20037)

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