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).