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Functions on Distributive Lattices with the Congruence Substitution Property: Some Problems of Grätzer from 1964

Authors
  • Farley, Jonathan David
Type
Published Article
Journal
Advances in Mathematics
Publication Date
Jan 01, 2000
Accepted Date
Mar 09, 1999
Volume
149
Issue
2
Pages
193–213
Identifiers
DOI: 10.1006/aima.1999.1854
Source
Elsevier
Keywords
License
Unknown

Abstract

Let L be a bounded distributive lattice. For k⩾1, let S k ( L) be the lattice of k-ary functions on L with the congruence substitution property (Boolean functions); let S( L) be the lattice of all Boolean functions. The lattices that can arise as S k ( L) or S( L) for some bounded distributive lattice L are characterized in terms of their Priestley spaces of prime ideals. For bounded distributive lattices L and M, it is shown that S 1( L)≅ S 1( M) implies S k ( L)≅ S k ( M). If L and M are finite, then S k ( L)≅ S k ( M) implies L≅ M. Some problems of Grätzer dating to 1964 are thus solved.

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