Local Lie semigroups and open embeddings into global topological semigroups

The semigroup version of Lie's Third Fundamental Theorem asserts that each strictly positive cone and each finite-dimensional Lie wedge is a tangent wedge of some local semigroup. This paper investigates the question whether this local semigroup can be embedded into a global topological semigroup in such a fashion that the embedding preserves all existing products and its image is open. In the case of a strictly positive cone (in particular a finite-dimensional pointed cone) this question will be answered affirmatively. This result contrasts the situation of open embeddings into global subsemigroups of Lie groups, where counterexamples even in low-dimensional Lie groups occur. In the finite-dimensional case a homotopy-like congruence on the semigroup of conal curves induces a quotient semigroup (called conal path semigroup) which also solves the topological embedding problem.