Ordered sets and duality for distributive lattices.

*(English)*Zbl 0557.06007
Orders: description and roles, Proc. Conf. Ordered sets appl., l’Arbresle/France 1982, Ann. Discrete Math. 23, 39-60 (1984).

[For the entire collection see Zbl 0539.00003.]

The category \({\mathcal D}\) of distributive lattices with 0 and 1 and 0-1- lattice morphisms is canonically dual to the category \({\mathcal P}\) of Priestley spaces (i.e., compact partially ordered spaces in which for \(x\leq \!/y\) there is a clopen \(V=\downarrow V\) with \(y\in V\) and \(x\not\in V)\) and continuous order preserving maps. This is the justly famous Priestley duality, modestly called \({\mathcal D}\)-\({\mathcal P}\) duality in this paper.

The author has written an eminently useful account of the historical background of this duality, its basic theory up to and including the most recent contributions, and of a wide range of its applications from the most classical to the most timely ones. A bibliography of 100 entries which the author places in their proper perspective in the text provides the machinery on which the survey runs. (Another comprehensive bibliography in this general area was recently compiled by R.-E. Hoffmann, D. S. Scott and the reviewer in a collection edited by R.-E. Hoffmann (and the reviewer): Continuous lattices..., Marcel Dekker (1985).)

The author has generated an important source of reference for the duality method applied to distributive lattices. A review can’t do justice to the article, but a recording of the headlines may indicate the flavor of the discourse: Basic duality for distributive lattices; equational subcategories of \({\mathcal D}\); the influence of category theory and universal algebra; duality in action: Post algebras and beyond; a miscellany of duality applications; topology and order in Priestley spaces; duality in a wider perspective; a \({\mathcal D}\)-\({\mathcal P}\) dictionary. - Duality theories are more prevalent than one is inclined to surmise at first, and many of them are general, and elegant, and useless. The classical example of a useful duality is Pontryagin duality between the category of abelian groups and compact abelian groups. The fact that Priestley duality is another example of a useful duality theory is best expressed in the author’s own words: ”The \({\mathcal D}\)-\({\mathcal P}\) duality seems to stand in a strategic position. It is general enough to exhibit many of the structural features that pervade duality theory, yet is special enough to provide a practical method of solving problems, and this method has the merit of being pictorial.”

The category \({\mathcal D}\) of distributive lattices with 0 and 1 and 0-1- lattice morphisms is canonically dual to the category \({\mathcal P}\) of Priestley spaces (i.e., compact partially ordered spaces in which for \(x\leq \!/y\) there is a clopen \(V=\downarrow V\) with \(y\in V\) and \(x\not\in V)\) and continuous order preserving maps. This is the justly famous Priestley duality, modestly called \({\mathcal D}\)-\({\mathcal P}\) duality in this paper.

The author has written an eminently useful account of the historical background of this duality, its basic theory up to and including the most recent contributions, and of a wide range of its applications from the most classical to the most timely ones. A bibliography of 100 entries which the author places in their proper perspective in the text provides the machinery on which the survey runs. (Another comprehensive bibliography in this general area was recently compiled by R.-E. Hoffmann, D. S. Scott and the reviewer in a collection edited by R.-E. Hoffmann (and the reviewer): Continuous lattices..., Marcel Dekker (1985).)

The author has generated an important source of reference for the duality method applied to distributive lattices. A review can’t do justice to the article, but a recording of the headlines may indicate the flavor of the discourse: Basic duality for distributive lattices; equational subcategories of \({\mathcal D}\); the influence of category theory and universal algebra; duality in action: Post algebras and beyond; a miscellany of duality applications; topology and order in Priestley spaces; duality in a wider perspective; a \({\mathcal D}\)-\({\mathcal P}\) dictionary. - Duality theories are more prevalent than one is inclined to surmise at first, and many of them are general, and elegant, and useless. The classical example of a useful duality is Pontryagin duality between the category of abelian groups and compact abelian groups. The fact that Priestley duality is another example of a useful duality theory is best expressed in the author’s own words: ”The \({\mathcal D}\)-\({\mathcal P}\) duality seems to stand in a strategic position. It is general enough to exhibit many of the structural features that pervade duality theory, yet is special enough to provide a practical method of solving problems, and this method has the merit of being pictorial.”

Reviewer: K.H.Hofmann

##### MSC:

06D05 | Structure and representation theory of distributive lattices |

06F30 | Ordered topological structures |

54F05 | Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces |

06D25 | Post algebras (lattice-theoretic aspects) |

06E15 | Stone spaces (Boolean spaces) and related structures |