The free ample monoid.

*(English)*Zbl 1192.20041A semigroup \(S\) is ‘ample’ if it is both left and right ample (also termed, respectively, left type A and right type A). While variations of these definitions have a long history, the left ample property itself is especially well motivated, being the abstract characterization of the submonoids of symmetric inverse semigroups that are closed under the unary operation \(\alpha\to\alpha^+=\alpha\alpha^{-1}\). They may naturally be regarded as semigroups with an additional unary operation \(a\to a^+\); thus ample semigroups may be regarded as semigroups with two additional unary operations, \(a\to a^+\) and \(a\to a^*\).

The free left ample semigroups were first constructed by the first author [Glasg. Math. J. 33, No. 2, 135-148 (1991; Zbl 0737.20032)] and, perhaps unsurprisingly, are embeddable in the corresponding free inverse semigroups. Since ample semigroups are not, in general, \((^+,^*)\)-embeddable in inverse semigroups, the main theorem of the paper under review is less unsurprising. Let \(FI_X\) and \(FG_X\) denote the free inverse semigroup and free group, respectively, on a set \(X\). Then within \(FI_X\), the free ample semigroup on \(X\) can be recognized as the complete pre-image of the free monoid on \(X\), under the homomorphism \(FI_X\to FG_X\).

The authors provide two proofs of this fact, the second being based on certain semidirect products of monoids and semilattices, induced by double actions. They also observe that the free ample semigroup is in fact weakly ample, moreover even weakly \(E\)-ample. While the distinctions are too subtle to go into here, it should be noted that the class of weakly \(E\)-ample semigroups, which have also gone under the name of ‘restriction’ semigroups (among other names), have a substantial history of their own and the description of their free members is therefore a notable achievement.

The free left ample semigroups were first constructed by the first author [Glasg. Math. J. 33, No. 2, 135-148 (1991; Zbl 0737.20032)] and, perhaps unsurprisingly, are embeddable in the corresponding free inverse semigroups. Since ample semigroups are not, in general, \((^+,^*)\)-embeddable in inverse semigroups, the main theorem of the paper under review is less unsurprising. Let \(FI_X\) and \(FG_X\) denote the free inverse semigroup and free group, respectively, on a set \(X\). Then within \(FI_X\), the free ample semigroup on \(X\) can be recognized as the complete pre-image of the free monoid on \(X\), under the homomorphism \(FI_X\to FG_X\).

The authors provide two proofs of this fact, the second being based on certain semidirect products of monoids and semilattices, induced by double actions. They also observe that the free ample semigroup is in fact weakly ample, moreover even weakly \(E\)-ample. While the distinctions are too subtle to go into here, it should be noted that the class of weakly \(E\)-ample semigroups, which have also gone under the name of ‘restriction’ semigroups (among other names), have a substantial history of their own and the description of their free members is therefore a notable achievement.

Reviewer: Peter R. Jones (Milwaukee)

##### MSC:

20M05 | Free semigroups, generators and relations, word problems |

20M10 | General structure theory for semigroups |

##### Keywords:

ample semigroups; restriction semigroups; type A semigroups; free monoids; semidirect products; \(E\)-ample semigroups; free ample monoids; symmetric inverse semigroups
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\textit{J. Fountain} et al., Int. J. Algebra Comput. 19, No. 4, 527--554 (2009; Zbl 1192.20041)

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