Linear congruences on a free monoid X ∗ {X^{\ast }} coincide with the congruences on X ∗ {X^{\ast }} induced by nontrivial homomorphisms into the additive group of integers. For a finite X X , we characterize abstractly several classes of linear congruences on X ∗ {X^{\ast }} , in particular, π \pi -linear congruences, called p p -linear and determined by Reis, ξ \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.