Extensions of definable local homomorphisms in o-minimal structures and semialgebraic groups

We state conditions for which a definable local homomorphism between two locally definable groups $\mathcal{G}$, $\mathcal{G^{\prime}}$ can be uniquely extended when $\mathcal{G}$ is simply connected (Theorem 2.1). As an application of this result we obtain an easy proof of [3, Thm. 9.1] (see Corollary 2.2). We also prove that Theorem 10.2 in [3] also holds for any definably connected definably compact semialgebraic group $G$ not necessarily abelian over a sufficiently saturated real closed field $R$; namely, that the o-minimal universal covering group $\widetilde{G}$ of $G$ is an open locally definable subgroup of $\widetilde{H\left(R\right)^{0}}$ for some $R$-algebraic group $H$ (Thm. 3.3). Finally, for an abelian definably connected semialgebraic group $G$ over $R$, we describe $\widetilde{G}$ as a locally definable extension of subgroups of the o-minimal universal covering groups of commutative $R$-algebraic groups (Theorem 3.4)