Kolmogorov complexities Kmax, Kmin on computable partially ordered sets

We introduce a machine free mathematical framework to get a natural formalization of some general notions of infinite computation in the context of Kolmogorov complexity. Namely, the classes and of functions X→D which are pointwise maximum of partial or total computable sequences of functions where D=(D,<) is some computable partially ordered set. The enumeration theorem and the invariance theorem always hold for , leading to a variant of Kolmogorov complexity. We characterize the orders D such that the enumeration theorem (resp. the invariance theorem) also holds for . It turns out that may satisfy the invariance theorem but not the enumeration theorem. Also, when satisfies the invariance theorem then the Kolmogorov complexities associated to and are equal (up to a constant).Letting , where Drev is the reverse order, we prove that either (=ct is equality up to a constant) or are ⩽ct incomparable and and . We characterize the orders leading to each case. We also show that cannot be both much smaller than KD at any point.These results are proved in a more general setting with two orders on D, one extending the other.