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 $G$, $G^{\prime }$ can be uniquely extended when $G$ is simply connected (Theorem 2.1). As an application of this result we obtain an easy proof of [3, Theorem 9.1] (cf. Corollary 2.3). We also prove that [3, Theorem 10.2] 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 $\stackrel{\sim }{G}$ of $G$ is an open locally definable subgroup of $\stackrel{\sim }{H{\left(R\right)}^0}$ for some $R$-algebraic group $H$ (Theorem 3.3). Finally, for an abelian definably connected semialgebraic group $G$ over $R$, we describe $\stackrel{\sim }{G}$ as a locally definable extension of subgroups of the o-minimal universal covering groups of commutative $R$-algebraic groups (Theorem 3.4).