The algebra of binary trees is affine complete

TL;DR

This paper characterizes congruence-preserving functions on the algebra of binary trees with at least three labels, proving they are exactly the polynomial functions, thus revealing an algebraic structure of these functions.

Contribution

It establishes that on the algebra of binary trees with at least three labels, congruence-preserving functions are precisely the polynomial functions, providing a complete algebraic characterization.

Findings

Congruence-preserving functions are polynomial on the algebra of binary trees.

The result applies to trees with at least three labeled leaves.

It links algebraic properties with polynomial functions in tree algebras.

Abstract

A function on an algebra is congruence preserving if, for any congruence, it maps pairs of congruent elements onto pairs of congruent elements. We show that on the algebra of binary trees whose leaves are labeled by letters of an alphabet containing at least three letters, a function is congruence preserving if and only if it is polynomial.