Maximal subsemigroups of Lie groups that are total

Jimmie D. Lawson
1987 Proceedings of the Edinburgh Mathematical Society  
In this section we develop the algebraic machinery necessary for the later developments. Definition 3.1. A subsemigroup M of a group G is called a maximal subsemigroup of G if (i) the only subsemigroups containing M are M and G, and (ii) M is not a subgroup. (Condition (ii) is a technical convenience, insuring the existence of the maximal ideal M*=M\H(M).) Remark 3.2. If M is a maximal subsemigroup of G, then eeM (otherwise consider {e}uM). Lemma 33. Let M be a maximal subsemigroup of G, and T
more » ... igroup of G, and T a submonoid with . If MT~x £ T" l M, then T~lM = G. Proof. We have T-l MT' x M^T~lT-l MM^T-l M, so T~lM is a subsemigroup containing T~1 (since eeM) and M. By maximality of M, G = T~ 1 M. • Although elementary in nature, the next proposition is crucial in the theory of
doi:10.1017/s0013091500026870 fatcat:3v2ofco73jadjpzovvhf6qjf4a