  We investigate finite deterministic automata in sets with non-homogeneous
atoms: integers with successor. As there are uncount- ably many deterministic
finite automata in this setting, we restrict our attention to automata with
semilinear transition function. The main re- sults is a minimization procedure
for semilinear automata. The proof is subtle and refers to decidability of
existential Presburger arithmetic with divisibility predicates. Interestingly,
the minimization is not obtained by the standard partition refinement
procedure, and we demonstrate that this procedure does not necessarily
terminate for semilinear automata.
