CHARACTERIZATIONS OF LINEAR CONGRUENCES ON A FREE MONOID

Linear congruences on a free monoid X* coincide with the congruences on X* induced by nontrivial homomorphisms into the additive group of integers. 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 the authors. These characterizations include descriptions involving maximality as prefix congruences.