The arity gap of polynomial functions over bounded distributive lattices

TL;DR

This paper classifies the arity gap of polynomial functions over bounded distributive lattices, revealing that most have an arity gap of 1, except for truncated median functions which have an arity gap of 2.

Contribution

It provides a complete classification of polynomial functions over bounded distributive lattices based on their arity gap, including a characterization of essential arguments.

Findings

Almost all lattice polynomial functions have arity gap 1.

Truncated median functions have arity gap 2.

The paper offers a characterization of essential arguments for these functions.

Abstract

Let A and B be arbitrary sets with at least two elements. The arity gap of a function f: A^n \to B is the minimum decrease in its essential arity when essential arguments of f are identified. In this paper we study the arity gap of polynomial functions over bounded distributive lattices and present a complete classification of such functions in terms of their arity gap. To this extent, we present a characterization of the essential arguments of polynomial functions, which we then use to show that almost all lattice polynomial functions have arity gap 1, with the exception of truncated median functions, whose arity gap is 2.