On Two Types of Concept Lattices in the Theory of Numberings

Abstract

The theory of numberings studies uniform computations for families of mathematical objects. A large body of literature is devoted to investigations of Rogers semilattices for computable families S, i.e. uniformly enumerable families of computably enumerable subsets of the set of natural numbers $$\omega $$ . Working within the framework of Formal Concept Analysis, we introduce two approaches to classification of at most countable families $$S \subset P (\omega )$$ . Similarly to the classical theory of numberings, each of the approaches assigns to a family S its own concept lattice. Our first approach captures the cardinality of a family S. We prove the following: if S contains at least two elements, then the corresponding concept lattice is a modular lattice of height 3 such that the number of its atoms equals the cardinality of S. Our second approach provides a much richer environment: we show that any countable complete lattice can be obtained as the concept lattice induced by an appropriate family S. In addition, we employ the index sets technique, and consider the following isomorphism problem: given two computable families S and T, how hard is it to determine whether the corresponding concept lattices are isomorphic? The isomorphism problem for the first approach is a $$\varPi ^0_3$$ -complete set, and the isomorphism problem for the second approach is $$\varSigma ^1_1$$ -hard.