Semigroups with commuting threads
Let \(\hat S\) denote a compact topological semigroup satisfying the following two conditions: (1) (\(\forall x\in \hat S\), \(n\in {\mathbb{N}})\) (\(\exists !y\in \hat S)\) \(y^ n=x\) (unique divisibility), and (2) if A and B denote one-parameter semigroups in \(\hat S\) then \(AB=BA\) (commuting thread property). Then there is a zero 0 and \(S=\hat S\setminus \{0\}\) is (isomorphic to) a closed subsemigroup of a Lie group G. Moreover, G is almost abelian, and S is a Lie semialgebra. Specifically, a Lie algebra \({\mathfrak g}\) is almost abelian if every proper vector subspace is a subalgebra iff \({\mathfrak g}\) is either abelian or the split extension of some \({\mathbb{R}}^ n\) by R acting via scalar multiplication. A connected Lie group is almost abelian if its Lie algebra is almost abelian. These Lie groups are exponential. A Lie semialgebra in an almost abelian Lie group G is the diffeomorphic image, under exp, of a closed convex cone in \({\mathfrak g}\). (For background on the Lie theory of semigroups, see \textit{J. Hilgert}, \textit{K. H. Hofmann}, and \textit{J. D. Lawson}, Lie groups, convex cones, and semigroups [Oxford Univ. Press, 1989, \(xxxiii+646 pp.]\). Preprints of the paper under review circulated in 1981.)
scientific article; zbMATH DE number 4099573
scientific article