Infix congruences on a free monoid

Abstract:

A congruence $\rho$ on a free monoid ${X^{\ast }}$ is said to be infix if each class $C$ of $\rho$ satisfies $u \in C$ and $xuy \in C$ imply $xy = 1$. The main purpose of this paper is a characterization of commutative maximal infix congruences. These turn out to be congruences induced by homomorphisms $\tau$ from ${X^{\ast }}$ to ${{\mathbf {N}}^0}$, the monoid of nonnegative integers under addition, with ${\tau ^{ - 1}}(0) = 1$.