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 (Formula presented.), (Formula presented.) can be uniquely extended when (Formula presented.) 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 (Formula presented.) not necessarily abelian over a sufficiently saturated real closed field (Formula presented.); namely, that the o-minimal universal covering group (Formula presented.) of (Formula presented.) is an open locally definable subgroup of (Formula presented.) for some (Formula presented.) -algebraic group (Formula presented.) (Theorem 3.3). Finally, for an abelian definably connected semialgebraic group (Formula presented.) over (Formula presented.), we describe (Formula presented.) as a locally definable extension of subgroups of the o-minimal universal covering groups of commutative (Formula presented.) -algebraic groups (Theorem 3.4).