Affine completeness of some free binary algebras

TL;DR

This paper proves that certain free binary algebras, including binary trees and free monoids, are affine complete, meaning all congruence-preserving functions are polynomial functions, which advances understanding of their algebraic structure.

Contribution

It establishes the affine completeness of specific free binary algebras, a property not previously confirmed for these structures.

Findings

Binary trees with labeled leaves are affine complete.

Free monoids on at least two letters are affine complete.

All congruence-preserving functions in these algebras are polynomial functions.

Abstract

A function on an algebra is congruence preserving if, for any congruence, it maps pairs of congruent elements onto pairs of congruent elements. An algebra is said to be affine complete if every congruence preserving function is a polynomial function. We show that the algebra of (possibly empty) binary trees whose leaves are labeled by letters of an alphabet containing at least one letter, and the free monoid on an alphabet containing at least two letters are affine complete.