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 M. J. Canfell for permission to quote from his unpublished thesis.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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The context for this paper is a class of distributive lattice expansions, called double quasioperator algebras (DQAs). The distinctive feature of these algebras is that their operations preserve or reverse both join and meet in each coordinate. Algebras of this type provide algebraic semantics for certain non-classical propositional logics. In particular, MV-algebras, which model the Łukasiewicz infinite-valued logic, are DQAs.Varieties of DQAs are here studied through their canonical extensions. A variety of this type having additional operations of arity at least 2 may fail to be canonical; it is already known, for example, that the variety of MV-algebras is not. Non-canonicity occurs when basic operations have two distinct canonical extensions and both are necessary to capture the structure of the original algebra. This obstruction to canonicity is different in nature from that customarily found in other settings. A generalized notion of canonicity is introduced which is shown to circumvent the problem. In addition, generalized canonicity allows one to capture on the canonical extensions of DQAs the algebraic operations in such a way that the laws that these obey may be translated into first-order conditions on suitable frames. This correspondence may be seen as the algebraic component of duality, in a way which is made precise.In many cases of interest, binary residuated operations are present. An operation h which, coordinatewise, preserves ∨ and 0 lifts to an operation which is residuated, even when h is not. If h also preserves binary meet then the upper adjoints behave in a functional way on the frames.

This paper is a study of duality in the absence of canonicity. Specifically it concerns double quasioperator algebras, a class of distributive lattice expansions in which, coordinatewise, each operation either preserves both join and meet or reverses them. A variety of DQAs need not be canonical, but as has been shown in a companion paper, it is canonical in a generalized sense and an algebraic correspondence theorem is available. For very many varieties, canonicity (as traditionally defined) and correspondence lead on to topological dualities in which the topological and correspondence components are quite separate. It is shown that, for DQAs, generalized canonicity is sufficient to yield, in a uniform way, topological dualities in the same style as those for canonical varieties. However topology and correspondence are no longer separable in the same way.

This paper studies finitely generated quasivarieties of Sugihara algebras. These quasivarieties provide complete algebraic semantics for certain propositional logics associated with the relevant logic R-mingle. The motivation for the paper comes from the study of admissible rules. Recent earlier work by the present authors, jointly with Freisberg and Metcalfe, laid the theoretical foundations for a feasible approach to this problem for a range of logics-the Test Spaces Method. The method, based on natural duality theory, provides an algorithm to obtain the algebra of minimum size on which admissibility of sets of rules can be tested. (In the most general case a set of such algebras may be needed rather than just one.) The method enables us to identify this 'admissibility algebra' for each quasivariety of Sugihara algebras which is generated by an algebra whose underlying lattice is a finite chain. To achieve our goals, it was first necessary to develop a (strong) duality for each of these quasivarieties. The dualities promise also to also provide a valuable new tool for studying the structure of Sugihara algebras more widely.

This paper presents a systematic study of coproducts. This is carried out principally, but not exclusively, for finitely generated quasivarieties A that admit a (term) reduct in the variety D of bounded distributive lattices. In this setting we present necessary and sufficient conditions on A for the forgetful functor U A from A to D to preserve coproducts. We also investigate the possible behaviours of U A as regards coproducts in A under weaker assumptions. Depending on the properties exhibited by the functor, different procedures are then available for describing these coproducts. We classify a selection of well-known varieties within our scheme, thereby unifying earlier results and obtaining some new ones.The paper's methodology draws heavily on duality theory. We use Priestley duality as a tool and our descriptions of coproducts are given in terms of this duality. We also exploit natural duality theory, specifically multisorted piggyback dualities, in our analysis of the behaviour of the forgetful functor into D. In the opposite direction, we reveal that the type of natural duality that the class A can possess is governed by properties of coproducts in A and the way in which the classes A and U A (A) interact. Take, as above, A to be a D-based and finitely generated quasivariety. In this section we outline the results on which our analysis of coproducts in A will rest. Underlying our strategy throughout will be duality theory, in two forms. Our main results are obtained by operating with these two forms in tandem, and toggling between them. First we briefly discuss the role played by Priestley duality as a platform on top of which dualities for classes of Dbased algebras can be built. For many such classes, this technique gives a valuable, set-based, representation theory. However, although isolated results exist in the literature for particular classes, such representations do not lend themselves well in general to the description of coproducts. Secondly, we venture a little way into the theory of (multisorted) natural dualities as it applies to finitely generated D-based quasivarieties, since such dualities, in common with Priestley duality itself, have good categorical properties. The key theorem on which we shall rely is the Multisorted Piggyback Duality Theorem. We present the bare minimum of the theory necessary to state it (Theorem 2.1). It allows us immediate access to coproducts, via dual structures which are (multisorted) cartesian products. This would be of little assistance were it not possible directly to retrieve the coproduct, or at least the Priestley dual space of its D-reduct, from the dual structure. Theorem 2.3 provides exactly the translation tool we need. (We remark that the usefulness of Theorem 2.3 potentially extends well beyond the applications to coproducts given in this paper.)Priestley duality for (bounded) distributive lattices establishes a dual equivalence between the category D and the category P of Priestley spaces, that is, compact totally order-disconnected spaces with con...