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

• Published in: Mathematical logic quarterly Vol. 70; no. 3; pp. 267 - 274
• Main Author: Barriga, Eliana
• Format: Journal Article
• Language: English
• Published: Berlin Wiley Subscription Services, Inc 01.08.2024
• Subjects: Homomorphisms; Subgroups; Theorems
• ISSN: 0942-5616; 1521-3870
• DOI: 10.1002/malq.202300028

We state conditions for which a definable local homomorphism between two locally definable groups G$\mathcal {G}$, G′$\mathcal {G^{\prime }}$ can be uniquely extended when G$\mathcal {G}$ 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$G$ not necessarily abelian over a sufficiently saturated real closed field R$R$; namely, that the o‐minimal universal covering group G∼$\widetilde{G}$ of G$G$ is an open locally definable subgroup of H(R)0∼$\widetilde{H(R)^{0}}$ for some R$R$‐algebraic group H$H$ (Theorem 3.3). Finally, for an abelian definably connected semialgebraic group G$G$ over R$R$, we describe G∼$\widetilde{G}$ as a locally definable extension of subgroups of the o‐minimal universal covering groups of commutative R$R$‐algebraic groups (Theorem 3.4).