On Concept Lattices for Numberings

The theory of numberings studies uniform computations for families of mathematical objects. In this area, computability-theoretic properties of at most countable families of sets S are typically classified via the corresponding Rogers upper semilattices. In most cases, a Rogers semilattice cannot be a lattice. Working within the framework of Formal Concept Analysis, we develop two new approaches to the classification of families S. Similarly to the classical theory of numberings, each of the approaches assigns to a family S its own concept lattice. The first approach captures the cardinality of a family S: if S contains more than 2 elements, then the corresponding concept lattice FC₁(S) is a modular lattice of height 3, such that the number of its atoms to the cardinality of S. Our second approach gives a much richer environment. We prove that for any countable poset P, there exists a family S such that the induced concept lattice FC2 (S) is isomorphic to the Dedekind-MacNeille completion of P. We also establish connections with the class of enumerative lattices introduced by Hoyrup and Rojas in their studies of algorithmic randomness. We show that every lattice FC2 (S) is anti-isomorphic to an enumerative lattice. In addition, every enumerative lattice is anti-isomorphic to a sublattice of the lattice FC2 (S) for some family S.