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Infinite iteration of matrix semigroups. I: Structure theorem for torsion semigroups. (English) Zbl 0584.20053
The 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.J├╝rgensen

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
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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
[4] Lallement, G, Semigroups and combinatorial applications, (1979), Wiley New York · Zbl 0421.20025
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