Total orderization invariant maps on distributive lattices

TL;DR

This paper introduces total orderization invariant maps on distributive lattices, generalizing median-based operations and providing new characterizations for various functions on vector lattices.

Contribution

It develops a theory of total orderization invariant maps and shows how they generalize and characterize several classes of functions on vector lattices.

Findings

Generalizes median operations to n variables on distributive lattices.

Provides new characterizations for symmetric continuous positively homogeneous functions.

Reveals novel properties of maps on vector lattices.

Abstract

Given any finite subset $A$ of order $n$ of a distributive lattice and $k\in\{1,...,n\}$, there is a natural extension of the median operation to $n$ variables which generalizes the notion of the $k$th smallest element of $A$. By applying each of these operations to $A$, a totally ordered set $to(A)$ is obtained. We refer to $to(A)$ as the total orderization of $A$. After developing a brief theory of total orderization invariant maps on distributive lattices, it is shown in this paper how these functions generalize and provide new characterizations for symmetric continuous positively homogeneous functions, bounded orthosymmetric multilinear maps, and certain power sum polynomials on vector lattices. These theorems generalize several results by Bernau, Huijsmans, Kusraev, Azouzi, Boulabiar, Buskes, Boyd, Ryan, and Snigireva and in turn reveal novel properties of the various maps studied in this paper.