Congruence Preserving Functions on Free Monoids

TL;DR

This paper characterizes congruence preserving functions on free monoids with at least three generators, showing they have a specific form and establishing the free monoid as a noncommutative affine complete algebra.

Contribution

It provides a complete description of congruence preserving functions on free monoids with at least three generators and proves such monoids are noncommutative affine complete algebras.

Findings

Congruence preserving functions have a specific form involving concatenation with fixed words.

Free monoids with ≥3 generators are noncommutative affine complete algebras.

First nontrivial example of a noncommutative affine complete algebra.

Abstract

A function on an algebra is congruence preserving if, for any congruence, it maps congruent elements to congruent elements. We show that, on a free monoid generated by at least 3 letters, a function from the free monoid into itself is congruence preserving %nonmonogenic if and only if it is of the form $x \mapsto w_0 x w_1 \cdots w_{n-1} x w_n$ for some finite sequence of words $w_0,\ldots,w_n$. We generalize this result to functions of arbitrary arity. This shows that a free monoid with at least three generators is a (noncommutative) affine complete algebra. Up to our knowledge, it is the first (nontrivial) case of a noncommutative affine complete algebra.