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On Priestley duals of products. (English) Zbl 0774.06006

The authors investigate in this well-written paper the Priestley dual \(P(K)\) of a Cartesian product \(K\) of bounded, distributive lattices \(K_ i\). The space \(P(K)\) always contains the topological sum \(Q\) of the spaces \(P(K_ i)\) and, in the Boolean case, \(P(K)\) is the Stone-Čech compactification of \(Q\). In the general case, \(P(K)\) is some Priestley compactification of \(Q\), i.e., \(Q\) is a dense subspace of \(P(K)\) which induces the order on \(Q\). The authors investigate such compactifications in general and present interesting results for direct products, ultraproducts and double \(p\)-algebras.
Reviewer: K.Kaiser (Houston)

MSC:

06D05 Structure and representation theory of distributive lattices
18B30 Categories of topological spaces and continuous mappings (MSC2010)
06E15 Stone spaces (Boolean spaces) and related structures
06D15 Pseudocomplemented lattices
06D20 Heyting algebras (lattice-theoretic aspects)
54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
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References:

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