Infinite iteration of matrix semigroups. I: Structure theorem for torsion semigroups.

*(English)*Zbl 0584.20053The main result of this paper is a representation theorem for countable torsion monoids, that is, countable monoids in which every element is of finite order. Every such monoid T is the homomorphic image \(\sigma\) (S) of a torsion monoid S with S and \(\sigma\) having several special properties some of which can be roughly described as follows: S is obtained by an infinitely iterated sandwich matrix type of construction based on certain cyclic submonoids of T and on translations of finite subsets of T. The morphism \(\sigma\) is of a restricted kind, defined along with the matrix construction. The maximal subgroups of S are finite cyclic groups and the restriction of \(\sigma\) to these is injective. S can be characterized as an elementary projective limit of finitely iterated matrix semigroups.

Reviewer: H.Jürgensen

##### MSC:

20M10 | General structure theory for semigroups |

20M20 | Semigroups of transformations, relations, partitions, etc. |

20M15 | Mappings of semigroups |

20M30 | Representation of semigroups; actions of semigroups on sets |

##### Keywords:

representation theorem; countable torsion monoids; sandwich matrix; projective limit of finitely iterated matrix semigroups
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##### References:

[1] | Clifford, A.H; Preston, G.B, () |

[2] | Eilenberg, S, () |

[3] | Krohn, K; Rhodes, J; Tilson, B, (), Chapters 1, and 5-9 |

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