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On open subsemigroups of connected groups

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Abstract

It is well known that a cancellative semigroup can be embedded into a group if it satisfies “Ore’s condition” of being either left or right reversible. However Ore’s condition is by no means necessary, so it is natural to ask which subsemigroups of a group are left or right reversible, or satisfy a condition of a similar type. In the present paper we study this question on open subsemigroups of connected locally compact groups; we also show how to use concepts related with reversibility to prove assertions like the following: Suppose thatS is an open subsemigroup of a connected Lie groupG such that 1 ∈\(\bar S\). IfG is solvable or ifS is invariant thenS is connected andS determinesG uniquely; that is to say, ifS can be embedded as an open subsemigroup into a connected Lie groupG’ thenG’ is isomorphic withG. Examples show that there are non-connected open subsemigroupsS of Sl(2,R) with 1 ∈\(\bar S\) and such that the uniqueness assertion fails.

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Authors and Affiliations

  1. Institut für Mathematik und angewandte Statistik, Universität für Bodenkultur, Gregor Mendelstr. 33, A-1180, Wien, Austria

    Wolfgang A. F. Ruppert

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  1. Wolfgang A. F. Ruppert
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Communicated by Karl H. Hofmann

The author gratefully acknowledges the support he received from the Alexander von Humboldt Foundation during the time he prepared this paper.

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Ruppert, W.A.F. On open subsemigroups of connected groups. Semigroup Forum 39, 347–362 (1989). https://doi.org/10.1007/BF02573307

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  • DOI: https://doi.org/10.1007/BF02573307

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