On a class of semigroups on $E\sb n$

Paul S. Mostert, Allen L. Shields
1956 Proceedings of the American Mathematical Society  
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
more » ... ): Theorem A. Let S be the half-line [0, w ). Suppose Sis a semigroup with zero at 0 and identity at 1. Then (i) if S contains no other idempotents, its multiplication is the ordinary multiplication of real numbers on [0, a>) ; (ii) if S contains an idempotent different from 0 and 1, then it contains a largest (in the sense of the regular order of real numbers) such idempotent e. Moreover, e 1, and B a compact, connected submanifold of dimension « -1. If B is a subsemigroup containing the identity of S, then : (i) « = 2 or 4 and B is a Lie group which is S1 if « = 2 and S3 if » = 4 (where S{ denotes the i-sphere) ; (ii) there exists a subsemigroup J contained in the center of S which is iseomorphic to a semigroup of the type described in Theorem A ; (iii) the subsemigroup J meets each orbit xB = Bx of B in exactly one point, and JB = S; (iv) if 0 denotes the zero of J, then 0 is a zero for S, and (J\ {0} ) XB is iseomorphic to (J\ {0} )S = 5\ {0} in the natural way.
doi:10.1090/s0002-9939-1956-0078646-6 fatcat:3cpnk3ebuvbd7p7yho62ttu5aa