Extension of Locally Defined Indefinite Functions on Ordered Groups

Abstract.

We give a definition of κ-indefinite function of archimedean type, on an interval of an ordered group Ω with an archimedean point. We say that Ω has the indefinite extension property if every continuous κ-indefinite function of archimedean type, on an interval of Ω, can be extended to a continuous κ-indefinite function on the whole group Ω.

We show that if a group Γ is semi-archimedean and it has the indefinite extension property, then $\Gamma \times \mathbb{Z}$ with the lexicographic order and the product topology has the indefinite extension property. As a corollary it is obtained that the groups $\mathbb{Z}^n$ and $\mathbb{R} \times \mathbb{Z}^n ,$ with the lexicographic order and the usual topologies, have the indefinite extension property.