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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2404.11923 |
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| _version_ | 1866911866106675200 |
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| author | Egri-Nagy, Attila Nehaniv, Chrystopher L. |
| author_facet | Egri-Nagy, Attila Nehaniv, Chrystopher L. |
| contents | We give a practical computer algebra implementation of the Covering Lemma for finite transformation semigroups. The lemma states that given a surjective relational morphism $(X,S)\twoheadrightarrow(Y,T)$, we can establish emulation by a cascade product (subsemigroup of the wreath product): $(X,S)\hookrightarrow (Y,T)\wr (Z,U)$. The dependent component $(Z,U)$ contains the kernel of the morphism, the information lost in the map.
The implementation complements the existing tools for the holonomy decomposition algorithm. It gives an incremental method to get a coarser decomposition when computing the complete skeleton for holonomy is not feasible. Here, we describe a simplified and generalized algorithm for the lemma and compare it to the holonomy method. Incidentally, the kernel-based method could be the easiest way of understanding the hierarchical decompositions of transformation semigroups and thus the celebrated Krohn-Rhodes theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_11923 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | From Relation to Emulation and Interpretation: Computer Algebra Implementation of the Covering Lemma for Finite Transformation Semigroups Egri-Nagy, Attila Nehaniv, Chrystopher L. Group Theory 20M20, 20M35 We give a practical computer algebra implementation of the Covering Lemma for finite transformation semigroups. The lemma states that given a surjective relational morphism $(X,S)\twoheadrightarrow(Y,T)$, we can establish emulation by a cascade product (subsemigroup of the wreath product): $(X,S)\hookrightarrow (Y,T)\wr (Z,U)$. The dependent component $(Z,U)$ contains the kernel of the morphism, the information lost in the map. The implementation complements the existing tools for the holonomy decomposition algorithm. It gives an incremental method to get a coarser decomposition when computing the complete skeleton for holonomy is not feasible. Here, we describe a simplified and generalized algorithm for the lemma and compare it to the holonomy method. Incidentally, the kernel-based method could be the easiest way of understanding the hierarchical decompositions of transformation semigroups and thus the celebrated Krohn-Rhodes theory. |
| title | From Relation to Emulation and Interpretation: Computer Algebra Implementation of the Covering Lemma for Finite Transformation Semigroups |
| topic | Group Theory 20M20, 20M35 |
| url | https://arxiv.org/abs/2404.11923 |