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Natural dualities. (English) Zbl 0839.06009

Baker, K. A. (ed.) et al., Lattice theory and its applications. In celebration of Garrett Birkhoff’s 80th birthday. Lemgo: Heldermann Verlag. Res. Expo. Math. 23, 185-209 (1995).
This is a state of the art survey of the duality theory for distributive-lattice-ordered algebras: Starting with a quasi-primitive class \({\mathcal A}= \text{ISP} (\underline {P})\) of algebras, one is interested in finding on the underlying set \(P\) of the generating algebra \(\underline {P}\) a topological relational structure \(\underset \sim P= (P; \tau, R)\) that induces an equivalence between \({\mathcal A}\) and a suitable category of topological relational structures of the same type as \(\underset\sim P\). The goal is to have a representation of any \(A\in {\mathcal A}\) as the algebra of continuous \(R\)-preserving maps from the subspace \({\mathcal A} (A,\underline P)\) of \(\underset \sim P^A\) into \(\underset \sim P\).
The author reviews carefully the relevant literature, especially the work of B. A. Davey and H. Werner. She provides the interested reader with numerous examples, simplifications and corrections as well as an overview and outlook of work in progress by people in this field.
For the entire collection see [Zbl 0824.00024].
Reviewer: K.Kaiser (Houston)

MSC:

06D05 Structure and representation theory of distributive lattices
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