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TSINGHUA SCIENCE AND TECHNOLOGY 2024年第6期29卷 1642-1650页

作者： Bazhenov, Nikolay Mustafa, Manat Nurakunov, Anvar Sobolev Inst Math Lab Computabil Theory & Appl Log Novosibirsk 630090 Russia Nazarbayev Univ Sch Sci & Humanities Dept Math Astana 010000 Kazakhstan Natl Acad Sci Inst Math Bishkek 720071 Kyrgyzstan

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.

关键词： theory of numberings concept lattice index set complete lattice enumerative lattice Formal Concept Analysis

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