The clone generated by the median functions

TL;DR

This paper investigates the structure of clones generated by median functions on linearly ordered sets, revealing that they either generate all median functions or none, highlighting a unique property of median functions in clone theory.

Contribution

It establishes that on any linearly ordered set, median functions either generate all median functions or none, clarifying their role in clone generation.

Findings

Median functions generate each other on linearly ordered sets.

Clones contain either no median functions or all median functions.

The result applies to both finite and infinite ordered sets.

Abstract

Let X be a linearly ordered set of arbitrary size (finite or infinite). Natural functions on such a set one can define using the linear order include maximum, minimum and median functions. While it is clear what the clone generated by the maximum or the minimum looks like, this is not obvious for median functions. We show that every clone on X contains either no median function or all median functions, that is, the median functions generate each other.