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

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