On open subsemigroups of connected groups

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 ∈ . 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 ∈ and such that the uniqueness assertion fails. The author gratefully acknowledges the support he received from the Alexander von Humboldt Foundation during the time he prepared this paper.