Commun. Korean Math. Soc. 2005 Vol. 20, No. 3, 443-456 Printed September 1, 2005

Inheung Chon Seoul Women's University

Abstract : Let $G$ be a Lie group, let $L(G)$ be its Lie algebra, and let $ \text{exp}: L(G) \to G$ denote the exponential mapping.~For $S \subseteq G$, we define the {\it tangent set} of $S$ by $L(S)=\{ X \in L(G): \text{exp}(tX) \in S \,\, \text{for all}\,\, t \ge 0 \}$.~We say that a semigroup $S$ is {\it strictly infinitesimally generated} if $S$ is the same as the semigroup generated by $\exp(L(S))$.~We find a tangent set of the semigroup of all non-singular totally positive matrices and show that the semigroup is strictly infinitesimally generated by the tangent set of the semigroup. This generalizes the familiar relationships between connected Lie subgroups of $G$ and their Lie algebras.