Associative polynomial functions over bounded distributive lattices

TL;DR

This paper explores generalized associativity for polynomial functions over bounded distributive lattices, providing explicit descriptions and showing the equivalence of two generalizations.

Contribution

It introduces and characterizes two generalizations of associativity for polynomial functions over bounded distributive lattices, proving their equivalence.

Findings

Explicit descriptions of associative polynomial functions

Both generalizations of associativity are essentially the same

Provides foundational understanding for polynomial functions in lattice theory

Abstract

The associativity property, usually defined for binary functions, can be generalized to functions of a given fixed arity n>=1 as well as to functions of multiple arities. In this paper, we investigate these two generalizations in the case of polynomial functions over bounded distributive lattices and present explicit descriptions of the corresponding associative functions. We also show that, in this case, both generalizations of associativity are essentially the same.