Abstract
We study into a semilattice of numberings generated by a given fixed numbering via operations of completion and taking least upper bounds. It is proved that, except for the trivial cases, this semilattice is an infinite distributive lattice every principal ideal in which is finite. The least upper and the greatest lower bounds in the semilattice are invariant under extensions in the semilattice of all numberings. Isomorphism types for the semilattices in question are in one-to-one correspondence with pairs of cardinals the first component of which is equal to the cardinality of a set of non-special elements, and the second — to the cardinality of a set of special elements, of the initial numbering.
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References
V. L. Selivanov, “On the structure of degrees of generalized index sets,” Algebra Logika, 21, No. 4, 472–491 (1982).
V. L. Selivanov, “Boolean hierarchies of partitions over a reducible base,” Algebra Logika, 43, No. 1, 77–109 (2004).
D. Spreen, “Strong reducibility of partial numberings,” Arch. Math. Log., 44, No. 2, 209–217 (2005).
S. A. Badaev, S. S. Goncharov, and A. Sorbi, “Completeness and universality of arithmetical numberings,” in Computability and Models, S. B. Cooper and S. S. Goncharov (eds.), Kluwer Academic/Plenum Publishers, New York (2003), pp. 11–44.
Yu. L. Ershov, Numeration Theory [in Russian], Nauka, Moscow (1977).
Z. G. Khisamiev, “Subfamilies of special elements of complete numberings,” Algebra Logika, 45, No. 6, 758–764 (2006).
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Supported by INTAS grant No. 00-429.
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Translated from Algebra i Logika, Vol. 46, No. 1, pp. 83–102, January–February, 2007.
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Khisamiev, Z.G. Distributive lattices of numberings. Algebr Logic 46, 50–61 (2007). https://doi.org/10.1007/s10469-007-0006-2
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DOI: https://doi.org/10.1007/s10469-007-0006-2