, Volume 50, Issue 1, pp 57-81

Extension of Locally Defined Indefinite Functions on Ordered Groups

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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.

To Professor José R. León, for his kind encouragement.
Both authors were supported in part by the CDCH of the Univ. Central de Venezuela and by CONICIT grant G-97000668. Both authors were visitors at IVIC during the realization of this paper.
Submitted: August 8, 2002 Revised: January 30, 2003