  Looking at some monoids and (semi)rings (natural numbers, integers and p-adic
integers), and more generally, residually finite algebras (in a strong sense),
we prove the equivalence of two ways for a function on such an algebra to
behave like the operations of the algebra. The first way is to preserve
congruences or stable preorders. The second way is to demand that preimages of
recognizable sets belong to the lattice or the Boolean algebra generated by the
preimages of recognizable sets by derived unary operation of the algebra (such
as translations, quotients,. . . ).
