A. Cherubini et M. Petrich, CHARACTERIZATIONS OF LINEAR CONGRUENCES ON A FREE MONOID, Proceedings of the American Mathematical Society, 119(3), 1993, pp. 719-726
Linear congruences on a free monoid X coincide with the congruences o
n X induced by nontrivial homomorphisms into the additive group of in
tegers. For a finite X, we characterize abstractly several classes of
linear congruences on X, 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 t
he authors. These characterizations include descriptions involving max
imality as prefix congruences.