A dichotomy result for strictly increasing bisymmetric maps

TL;DR

This paper proves that strictly increasing bisymmetric maps on intervals are continuous and quasi-arithmetic means, and refines the symmetry condition needed for this result.

Contribution

It introduces a refined condition where symmetry only for a pair of points suffices to ensure continuity and quasi-arithmetic mean properties.

Findings

Strictly increasing bisymmetric maps are continuous.

Such maps are quasi-arithmetic means.

Symmetry can be weakened to a pair of points.

Abstract

In this paper we show some remarkable consequences of the method which proves that every bisymmetric, symmetric, reflexive, strictly monotonic binary map on a proper interval is continuous, in particular it is a quasi-arithmetic mean. Now we demonstrate that this result can be refined in the way that the symmetry condition can be weakened by assuming symmetry only for a pair of distinct points of an interval.