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.

A topological quasi-variety Q + T (M ∼ ) := IScP + M ∼ generated by a finite algebra M ∼with the discrete topology is said to be standard if it admits a canonical axiomatic description. Drawing on the formal language notion of syntactic congruences, we prove that Q + T (M ∼ ) is standard provided that the algebraic quasi-variety generated by M ∼ is a variety, and that syntactic congruences in that variety are determined by a finite set of terms. We give equivalent semantic and syntactic conditions for a variety to have Finitely Determined Syntactic Congruences (FDSC), show that FDSC is equivalent to a natural generalisation of Definable Principle Congruences (DPC) which we call Term Finite Principle Congruences (TFPC), and exhibit many familiar algebras M ∼ that our method reveals to be standard. As an application of our results we show, for example, that every Boolean topological lattice belonging to a finitely generated variety of lattices is profinite and that every Boolean topological group, semigroup, and ring is profinite. While the latter results are well known, the result on lattices was previously known only in the distributive case. Background, motivation and overview of resultsAn algebra M = M ; F with finite underlying set M and operations F generates an (algebraic) quasi-variety Q(M) := ISP M consisting of all isomorphic copies of subalgebras of direct powers of M. Similarly a structure M ∼ = M ; G, H, R, T with finite underlying set M , operations G, partial operations H, relations R and discrete topology T generates a topological quasi-variety Q + T (M ∼ ) := IS c P + M ∼ consisting of all isomorphic copies of topologically closed substructures of non-zero direct powers, with the product topology, of M ∼ . Interest in topological quasi-varieties stems from the fact that they arise as the duals to algebraic quasi-varieties under natural dualities. The general theory of natural dualities provides methods to Presented by R. W. Quackenbush.

We investigate first-order axiomatic descriptions of naturally occurring classes of Boolean topological structures (these structures can have operations and relations, and carry a compatible compact Hausdorff topology with a basis of clopen sets). Our methods utilize inverse limits and ultraproducts of finite structures. We illustrate the range of possible axiomatizations of these classes with applications of our methods to Boolean topological lattices, graphs, ordered structures, unary algebras and semigroups. For example, whereas the class of all k-colorable graphs is known to be axiomatizable by universal Horn sentences, we find the class of continuously k-colorable Boolean topological graphs is not even first-order axiomatizable.

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Let A be the variety generated by some finite distributivelattice-ordered algebra P , say a Heyting algebra. By restricting Priestley's duality for the class D of distributive lattices to A we obtain a topological representation for the algebras in A ; in fact this yields a dual category-equivalence between A and a category U of ordered topological spaces.Unfortunately the category jj has certain drawbacks. Our main result, The Piggyback Duality Theorem, shows that by riding piggyback on the given duality for the reduct D we can obtain a more natural dual category X for A in which products are cartesian with the dual equivalence given by natural hom-functors A(-, P) and X(-, £) , properties not shared by the category u •The Piggyback Duality Theorem is applied here to yield dualities, some new and some known, for varieties of Ockham algebras, distributive pseudocomplemented lattices, double Stone algebras, Heyting algebras and relatively pseudocomplemented semilattices, the last example riding piggyback on the Hofmann-Mislove-Stralka duality for semilattices.Received 3 December 198U. A short version of this paper in English but without proofs can be found in Davey and Werner [7D.

We introduce a new Galois connection for partial operations on a finite set, which induces a natural quasi-order on the collection of all partial algebras on this set. The quasi-order is compatible with the basic concepts of natural duality theory, and we use it to turn the set of all alter egos of a given finite algebra into a doubly algebraic lattice. The Galois connection provides a framework for us to develop further the theory of natural dualities for partial algebras. The development unifies several fundamental concepts from duality theory and reveals a new understanding of full dualities, particularly at the finite level.

ABSTRACT. This paper focuses on the equational class S" of Brouwerian algebras and the equational class L" of Heyting algebras generated by an »-element chain. Firstly, duality theories are developed for these classes. Next, the projectives in the dual categories are determined, and then, by applying the dualities, the injectives and absolute subretracts in Sn and L" are characterized. Finally, free products and the finitely generated free algebras in S" and L" are described.Recently there has been considerable interest in distributive pseudocomplemented lattices, Brouwerian algebras and Heyting algebras. In particular, activity has centered around the equational subclasses ([8] focused upon the equational class S" of Brouwerian algebras and the equational class Ln of Heyting algebras generated by an n-element chain. Firstly, a duality theory is developed for each of these classes, the dual of an algebra being a Boolean space endowed with a continuous action of the endomorphism monoid of the n-element chain. Next, the projectives in the dual categories are determined, and then, by applying the dualities, the injectives and absolute subretracts in S" and L" are characterized. Finally, free products and the finitely generated free algebras in S" and Ly, are described.1. The categories. Our standard references on category theory, universal algebra, and lattice theory are S. Mac Lane [37], G. Grätzer [17], and G. Gra'tzer [18] respectively; for our general topological requirements we refer to J. Dugundji [13] and for a discussion of Boolean a spaces we call on P. R. Halmos [23].