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Large intervals in the clone lattice

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Abstract

We give three examples of cofinal intervals in the lattice of (local) clones on an infinite set X, whose structure is on the one hand non-trivial but on the other hand reasonably well understood. Specifically, we will exhibit clones \({{\fancyscript C}_1, {\fancyscript C}_2, {\fancyscript C}_3}\) such that

  1. (1)

    the interval \({[{\fancyscript C}_1, {\fancyscript O}]}\) in the lattice of local clones is (as a lattice) isomorphic to {0, 1, 2, . . .} under the divisibility relation,

  1. (2)

    the interval \({[{\fancyscript C}_2, {\fancyscript O}]}\) in the lattice of local clones is isomorphic to the congruence lattice of an arbitrary semilattice,

  1. (3)

    the interval \({[{\fancyscript C}_3, {\fancyscript O}]}\) in the lattice of all clones is isomorphic to the lattice of all filters on X.

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Authors and Affiliations

  1. DMG/Algebra, TU Wien, Wiedner Hauptstraße 8-10/104.1, 1040, Wien, Austria

    Martin Goldstern

  2. Mathematics, Hebrew University of Jerusalem, 91904, Jerusalem, Israel

    Saharon Shelah

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  1. Martin Goldstern
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  2. Saharon Shelah
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Correspondence to Martin Goldstern.

Additional information

Presented by Á. Szendrei.

The first author is grateful to the Hebrew University of Jerusalem for the hospitality during his visit, and to the Austrian Science foundation for supporting the joint research under FWF grant P13325-MAT. The second author is supported by the German-Israeli Foundation for Scientific Research & Development Grant No. G-294.081.06/93. Publication number 747.

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Goldstern, M., Shelah, S. Large intervals in the clone lattice. Algebra Univers. 62, 367–374 (2009). https://doi.org/10.1007/s00012-010-0047-6

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  • DOI: https://doi.org/10.1007/s00012-010-0047-6

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