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| Natura: | Preprint |
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2014
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| Accesso online: | https://arxiv.org/abs/1401.4224 |
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| _version_ | 1866915732989673472 |
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| author | Egri-Nagy, Attila Nehaniv, Chrystopher L. |
| author_facet | Egri-Nagy, Attila Nehaniv, Chrystopher L. |
| contents | We establish key connections between Green's $\cal J$- and $\cal L$-relations on a finite semigroup and the subduction relation defined on the image sets of an action of the same semigroup when it acts faithfully on a finite set. The construction of the skeleton order, the partial order on equivalence classes of the subduction relation, is shown to depend in a functorial way on transformation semigroups and surjective morphisms, and to factor through the Green's $\leq_{\cal L}$-order and $\leq_{\cal J}$-order on the semigroup and through the inclusion order on image sets. For right regular representations, the correspondence between the $\cal J$-class order and the skeleton order is one of isomorphism. Finally, we characterize the relationship between natural subsystems of a transformation semigroup, permutator groups and the $\cal H$-relation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1401_4224 |
| institution | arXiv |
| publishDate | 2014 |
| record_format | arxiv |
| spellingShingle | Skeleton Key: Subduction Classes in Finite Transformation Semigroups and Green's Relations Egri-Nagy, Attila Nehaniv, Chrystopher L. Group Theory 20M20 We establish key connections between Green's $\cal J$- and $\cal L$-relations on a finite semigroup and the subduction relation defined on the image sets of an action of the same semigroup when it acts faithfully on a finite set. The construction of the skeleton order, the partial order on equivalence classes of the subduction relation, is shown to depend in a functorial way on transformation semigroups and surjective morphisms, and to factor through the Green's $\leq_{\cal L}$-order and $\leq_{\cal J}$-order on the semigroup and through the inclusion order on image sets. For right regular representations, the correspondence between the $\cal J$-class order and the skeleton order is one of isomorphism. Finally, we characterize the relationship between natural subsystems of a transformation semigroup, permutator groups and the $\cal H$-relation. |
| title | Skeleton Key: Subduction Classes in Finite Transformation Semigroups and Green's Relations |
| topic | Group Theory 20M20 |
| url | https://arxiv.org/abs/1401.4224 |