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Embedding of regular semigroups in wreath products. (English) Zbl 0572.20045
A great deal of the author’s previous papers have addressed the problem of describing the structure of various types of regular semigroups. The present paper is a major step towards a more uniform and simplified treatment of this question. The main tool used in proving these structure theorems is that of embedding (regular) semigroups in wreath products of ”simpler” semigroups. The classes of regular semigroups for which structure descriptions are obtained in this way include the following: \({\mathcal H}\)-compatible inverse; inverse; \({\mathcal R}\)-unipotent; natural \({\mathcal R}\)-unipotent; generalized \({\mathcal L}\)-unipotent; left natural; \(\omega^ n\)-bisimple; \(\omega^ nI\)-bisimple, simple I-regular; \(\omega\) Y-inverse; \(\omega\) Y-\({\mathcal R}\)-unipotent; \(\omega\)-\({\mathcal R}\)- unipotent; generalized \(\omega\)-\({\mathcal L}\)-unipotent bisimple; orthodox; bisimple inverse; standard regular; and a few others.
This paper demonstrates the power of wreath product embeddings as a tool. It indicates a major step in understanding the structure of regular semigroups. On the other hand, its results do not provide a general uniform structure theory for regular semigroups yet. It seems to be one of the natural questions to ask for future research, how Nambooripad’s structure theory of regular semigroups translates into this framework.
Reviewer: H.Jürgensen

MSC:
20M10 General structure theory for semigroups
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