On a class of semigroups on 𝐸_{𝑛}
Paul S. Mostert, Allen L. Shields · Proceedings of the American Mathematical Society · 1956
Very little is known about continuous, associative multiplications with identity on a manifold unless the manifold is compact (in which case it must be a Lie group [l]).In this, and in forthcoming papers, the authors will study this problem, particularly when the manifold is E" («-dimensional Euclidean space), and with other restrictions.Here, we classify all (topological) semigroups on the half line [0, ) in which 0 and 1 play their natural roles of zero and identity, and use this as an aid in classifying semigroups 5 with identity on En, «>1, such that some (« -1)-dimensional compact connected submanifold is a subsemigroup containing the identity of 5.It turns out that only E2 and £4 will admit such a situation.In fact, we prove the following two theorems (see §1 for definitions):