Characterizations of Linear Congruences on a Free Monoid

Abstract

Linear congruences on a free monoid ${X^{\ast }}$ coincide with the congruences on ${X^{\ast }}$ induced by nontrivial homomorphisms into the additive group of integers. For a finite $X$, we characterize abstractly several classes of linear congruences on ${X^{\ast }}$, in particular, $\pi$-linear congruences, called $p$-linear and determined by Reis, $\xi$-linear congruences, introduced by Petrich and Thierrin, and general linear congruences, introduced by the authors. These characterizations include descriptions involving maximality as prefix congruences.