On effective presentations of formal concept lattices

We study the isomorphism types of ordered structures of concepts for computable formal contexts from the point of view of computable model theory. We show that, although these structures could have the cardinality of the continuum or – if they are countable – could have an arbitrary high hyperarithmetical complexity, they are in some sense very close to computable orderings. We give some sufficient conditions for these orderings to have computable presentations. A full description is given of the isomorphism types of discrete concept lattices. We also give some counterexamples.