Two-sided expansions of monoids
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arXiv
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| Formato: | Preprint |
| Publicado: |
2018
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| _version_ | 1866913564388753408 |
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| author | Kudryavtseva, Ganna |
| author_facet | Kudryavtseva, Ganna |
| contents | We initiate the study of expansions of monoids in the class of two-sided restriction monoids and show that generalizations of the Birget-Rhodes prefix group expansion, despite the absence of involution, have rich structure close to that of respective relatively free inverse monoids.
For a monoid $M$, we define ${\mathcal{FR}}_R(M)$ to be the freest two-sided restriction monoid generated by a bijective copy, $M'$, of the underlying set of $M$, such that the inclusion map $ι\colon M\to {\mathcal{FR}}_R(M)$ is determined by a set of relations, $R$, so that $ι$ is a premorphism which is weaker than a homomorphism. Our main result states that ${\mathcal{FR}}_R(M)$ can be constructed, by means of a partial action product construction, from $M$ and the idempotent semilattice of ${\mathcal{FI}}_R(M)$, the free $M'$-generated inverse monoid subject to relations $R$. In particular, the semilattice of projections of ${\mathcal{FR}}_R(M)$ is isomorphic to the idempotent semilattice of ${\mathcal{FI}}_R(M)$. The result by Fountain, Gomes and Gould on the structure of the free two-sided restriction monoid is recovered as a special case of our result.
We show that important properties of ${\mathcal{FR}}_R(M)$ are well agreed with suitable properties of $M$, such as being cancellative or embeddable into a group. We observe that if $M$ is an inverse monoid, then ${\mathcal{FI}}_s(M)$, the free inverse monoid with respect to strong premorphisms, is isomorphic to the Lawson-Margolis-Steinberg generalized prefix expansion $M^{pr}$. This gives a presentation of $M^{pr}$ and leads to a model for ${\mathcal{FR}}_s(M)$ in terms of the known model for $M^{pr}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1805_05272 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Two-sided expansions of monoids Kudryavtseva, Ganna Rings and Algebras Group Theory We initiate the study of expansions of monoids in the class of two-sided restriction monoids and show that generalizations of the Birget-Rhodes prefix group expansion, despite the absence of involution, have rich structure close to that of respective relatively free inverse monoids. For a monoid $M$, we define ${\mathcal{FR}}_R(M)$ to be the freest two-sided restriction monoid generated by a bijective copy, $M'$, of the underlying set of $M$, such that the inclusion map $ι\colon M\to {\mathcal{FR}}_R(M)$ is determined by a set of relations, $R$, so that $ι$ is a premorphism which is weaker than a homomorphism. Our main result states that ${\mathcal{FR}}_R(M)$ can be constructed, by means of a partial action product construction, from $M$ and the idempotent semilattice of ${\mathcal{FI}}_R(M)$, the free $M'$-generated inverse monoid subject to relations $R$. In particular, the semilattice of projections of ${\mathcal{FR}}_R(M)$ is isomorphic to the idempotent semilattice of ${\mathcal{FI}}_R(M)$. The result by Fountain, Gomes and Gould on the structure of the free two-sided restriction monoid is recovered as a special case of our result. We show that important properties of ${\mathcal{FR}}_R(M)$ are well agreed with suitable properties of $M$, such as being cancellative or embeddable into a group. We observe that if $M$ is an inverse monoid, then ${\mathcal{FI}}_s(M)$, the free inverse monoid with respect to strong premorphisms, is isomorphic to the Lawson-Margolis-Steinberg generalized prefix expansion $M^{pr}$. This gives a presentation of $M^{pr}$ and leads to a model for ${\mathcal{FR}}_s(M)$ in terms of the known model for $M^{pr}$. |
| title | Two-sided expansions of monoids |
| topic | Rings and Algebras Group Theory |
| url | https://arxiv.org/abs/1805.05272 |