Representations and characterizations of polynomial functions on chains

TL;DR

This paper explores how polynomial functions on chains (totally ordered lattices) can be represented and characterized using simplices and new axioms, extending previous work with relaxed and additional conditions.

Contribution

It provides new representations and axiomatizations of lattice polynomial functions specifically on chains, including conditions like comonotonic minitivity and maxitivity.

Findings

Representations via standard simplices for polynomial functions on chains

New axiomatizations relaxing previous conditions

Inclusion of comonotonic minitivity and maxitivity conditions

Abstract

We are interested in representations and characterizations of lattice polynomial functions f:L^n -> L, where L is a given bounded distributive lattice. In companion papers [arXiv 0901.4888, arXiv 0808.2619], we investigated certain representations and provided various characterizations of these functions both as solutions of certain functional equations and in terms of necessary and sufficient conditions. In the present paper, we investigate these representations and characterizations in the special case when L is a chain, i.e., a totally ordered lattice. More precisely, we discuss representations of lattice polynomial functions given in terms of standard simplices and we present new axiomatizations of these functions by relaxing some of the conditions given in [arXiv 0901.4888, arXiv 0808.2619] and by considering further conditions, namely comonotonic minitivity and maxitivity.