Semigroups with identity on a manifold

Abstract

The principal question with which this paper is concerned is the following: if a manifold M admits a continuous associative multiplication with identity and no other idempotents, is it a group? We are able to show that in the case Al is the line or plane, the answer is in the affirmative. However, the general problem remains open. We show that in any case, there is a maximal, connected, open subgroup G of M containing the identity. This then determines a subspace L, the boundary of G in M, which we show is an ideal of G- (the closure of G). Thus, a study of M necessarily involves a thorough study of L. We conjecture, for example, that L always contains an idempotent (we prove this in case L is the line or plane, or when M is two dimensional and L is a regular boundary). Clearly this would imply an affirmative answer to the question raised above for arbitrary dimensions. The existence of the open subgroup G gives us as corollaries two results of A. D. Wallace [1; 2], both of which are encompassed in the following more general result: if a compact manifold S with boundary B is a semigroup with identity, then the set of elements with inverses is contained in B or else is all of S. We consider also the following problem: if the closed right half plane is a semigroup in which the open half plane is a group G, how many possibilities are there for multiplication on the y-axis (=L)? We show that if G is isomorphic to the two dimensional vector group, there are exactly four possibilities, and examples are given of each of them. If G is the nonabelian group, we can identify two possibilities, but we are unable to prove these are all. (It might be interesting to note that even in the plane, we are unable to decide whether L need be a regular boundary for G or not.)