The algebra of complete binary trees is affine complete

TL;DR

This paper demonstrates that the algebra of complete binary trees with at least three labels is affine complete, with congruence-preserving functions characterized as polynomial, providing a novel example of a non-commutative, non-associative affine complete algebra.

Contribution

It introduces the first known example of a non-commutative, non-associative affine complete algebra based on complete binary trees.

Findings

Congruence-preserving functions are exactly the polynomial functions.

The algebra of complete binary trees is affine complete.

First known example of such an algebra.

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 complete 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. This exhibits an example of a non commutative and non associative affine complete algebra. As far as we know, it is the first example of such an algebra.