This new edition of Introduction to Lattices and Order presents a radical reorganization and updating, though its primary aim is unchanged. The explosive development of theoretical computer science in recent years has, in particular, influenced the book's evolution: a fresh treatment of fixpoints testifies to this and Galois connections now feature prominently. An early presentation of concept analysis gives both a concrete foundation for the subsequent theory of complete lattices and a glimpse of a methodology for data analysis that is of commercial value in social science. Classroom experience has led to numerous pedagogical improvements and many new exercises have been added. As before, exposure to elementary abstract algebra and the notation of set theory are the only prerequisites, making the book suitable for advanced undergraduates and beginning graduate students. It will also be a valuable resource for anyone who meets ordered structures.

Introduction Stone,in [8], developed for distributive lattices a representation theory generalizing that for Boolean algebras. This he achieved by topologizing the set X of prime ideals of a distributive lattice A (with a zero element) by taking as a base {P a : aeA} (where P a denotes the set of prime ideals of A not containing a), and by showing that the map a i-> P a is an isomorphism representing A as the lattice of all open compact subsets of its dual space X.The topological spaces which arise as duals of Boolean algebras may be characterized as those which are compact and totally disconnected (i.e. the Stone spaces); the corresponding purely topological characterization of the duals of distributive lattices obtained by Stone is less satisfactory. In the present paper we show that a much simpler characterization in terms of ordered topological spaces is possible. The representation theorem itself, and much of the duality theory consequent on it [8,6], becomes more natural in this new setting, and certain results not previously known can be obtained. It is hoped to give in a later paper a more detailed exposition of those aspects of the theory barely mentioned here.I should like to thank my supervisor, Dr. D. A. Edwards, for some helpful suggestions and also Dr. M. J. Canfell for permission to quote from his unpublished thesis.

Proc. London Math. Soc. (3) 24 (1972J 507-530 508 H. A. PRIESTLEY space. Throughout, where topological properties of the ordered space {X,ST, ^) are mentioned without explicit reference to which topology is intended, that topology is y.We recall (Bonsall ([3])) that X is said to be monotonically separated if, given x, y e X, x ^ y, there exist disjoint sets U e °U, L e J£, such that x e U, y e L. LEMMA 1. (Nachbin ([15] p. 26).) Let {X,$~, ^) be an ordered space. Then if X is monotonically separated, the graph G = {(x, y)\ x < y) of î s closed in the product topology of XxX.Conversely, if 0 is closed, then, for each xeX, i{x) and d(x) are closed. Also, if&~is compact and ^ a partial order, then X is monotonically separated.

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