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

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