2. Natural Dualities
3. Strong Dualities
4. Examples of Strong Dualities
5. Sample Applications
6. What Makes a Duality Useful?
7. Piggyback Dualities
8. Optimal Dualities and Entailment
9. Completeness Theorems for Entailment
10. Dualisable Algebras
Appendix A: Algebra
Appendix B: Boolean Spaces
Bibliography
Notation
Index
In 1936 Marshall Stone published a truly novel theorem. Stone wanted to understand Boolean algebras. Like many other classes of algebras, Boolean algebras could be described in two different ways. They are normally defined abstractly as the class of all models of a familiar set of axioms. Alternatively, a class of algebras is often described by giving some concrete way to represent its members. For example, Stone was aware of the work of Boole and Tarski who had shown that certain Boolean algebras are representable as a subalgebra of the Boolean algebra of subsets of some set. Interesting as they are, these descriptions do not always provide a direct and usable insight into the structure of the individual algebras themselves. What Stone discovered was a representation for all Boolean algebras which gave algebraists a usable understanding of their structure. Using topological spaces to construct the representations, he proved that every Boolean algebra is isomorphic to the algebra of all clopen subsets of some totally disconnected compact space.
In fact, Stone [Sto36] proved much more than a representation theorem. He showed that the homomorphisms between Boolean algebras correspond precisely to the continuous maps between the corresponding spaces. In modern language, he proved that the category of Boolean algebras is dually equivalent to the category of Boolean spaces. Two similar results appeared almost simultaneously. Birkhoff [Bir37] showed that every finite distributive lattice is isomorphic to the lattice of all decreasing subsets of a finite ordered set, and established a close correspondence between finite distributive lattices and finite ordered sets. Looking at the category of abelian groups, L. Pontryagin [Pon34a, Pon34b] discovered that it was dually equivalent to the category of compact abelian groups. The beauty of these theorems is that they translate algebraic problems, normally stated in abstract symbolic language, into dual, topological problems where our geometric intuitions can be brought to bear.
These three seminal results planted the seeds for a considerable assortment of new dualities. Hu [Hu71] realised that Stone's theorem extended to the variety generated by any primal algebra. Still the most widely used of these new dualities is Hilary Priestley's duality [Pri70, Pri72] which combined the theorems of Stone and Birkhoff into a dual equivalence between the variety of all bounded distributive lattices and the category of all totally order-disconnected compact spaces. This discovery initiated a plethora of parallel results: Hofmann, Mislove and Stralka [HMS74] (semilattices with 1), Davey [Dav76] (relative Stone algebras), Cornish and Fowler [CF77] (de Morgan algebras), Isbell [Isb80] and Werner [Wer81] (median algebras), Davey [Dav82] (Stone and double Stone algebras) and Davey and Werner [DW83] (Kleene algebras, abelian groups, quasi-primal algebras). During this period, Davey [Dav78] began to use the methods of universal algebra to formulate a general theory of duality. The early eighties saw this method crystallised in the papers of Davey and Werner [DW83] and subsequently, Clark and Krauss [CK84].
Out of these foundations has evolved a powerful theory of what have come to be known as natural dualities. A class of algebras is a candidate for a natural duality provided that it is the quasi-variety generated by some finite algebra. A natural duality for a quasi-variety gives us a uniform method to represent each algebra in the quasi-variety as the algebra of all continuous homomorphisms over some structured Boolean space. When we are lucky, the association between algebras and corresponding structured spaces forms a dual equivalence between the quasi-variety of algebras and the dual category of structured Boolean spaces. If this happens in a special canonical way we say that we have a strong duality. This book presents a detailed exposition of natural dualities up to the point that the theory had developed by mid-1997. It is primarily intended for the working algebraist who, like Marshall Stone, wants to understand some special finitely generated quasi-variety.
Two appendices summarise the prerequisite background in universal algebra and topology. We rely heavily on foundational results of Birkhoff, Jonsson and Pixley. In Chapter 1 we give a bare-hands proof of Priestley's duality for distributive lattices, and then draw on it in the rest of the chapter to formulate the general concepts out of which natural dualities arise. In Chapter 2 we develop the method of Davey and Werner [DW83] for identifying a category of structured Boolean spaces dual to a given quasi-variety. Chapter 3 investigates strong dualities and culminates in two major theorems which tell exactly how, if the generating algebra satisfies certain conditions, we can directly construct a strong duality. The NU Strong Duality Theorem utilises Baker and Pixley [BP75] to generalise Priestley [Pri72], while the Two-for-One Strong Duality Theorem builds on the theorem of Pontryagin [Pon34a, Pon34b]. Each of the above listed strong dualities is a direct consequence of one of these two theorems, and Stone's theorem is a consequence of both. This fact is established in Chapter 4 where we give a vast assortment of examples of strong dualities for many familiar quasi-varieties. Chapters 2 and 3 might be viewed as the theoretical core of the subject. They have their roots in Davey and Werner [DW83] and Clark and Krauss [CK84] and their joint extension, Clark and Davey [CD95]. We expect that many readers will choose to scan much of the initial material and jump to the theorems in the last section of Chapter 3. The examples in Chapter 4 illustrate how these theorems can be directly applied with minimal reference to the lengthy development needed to establish them.
It is our underlying contention that natural dualities, and particularly strong dualities, can provide a very useful way to understand a quasi-variety. We substantiate this claim in Chapter 5, where we give a diverse selection of applications of our dualities. In each case, we show how one of the dualities obtained in the previous chapter can be applied to solve a particular algebraic problem that arises, quite independently of duality theory, within the given quasi-variety. We refine the method of Adams and Clark [AC90] to determine endomorphism monoid uniqueness, the method of Clark [Cla89] and Clark and Schmid [CS96] to characterise algebraic and existential closure, the method of Davey [Dav76] to characterise injectives and its variations, and the method of Davey and Werner [DW83] to characterise category equivalence.
Our success in Chapter 5 depends on using dualities in which the dual of an algebra is a structured space which is in some tangible sense simpler or more tractable than the original algebra. For example, finite distributive lattices are dual under Birkhoff--Priestley duality to finite ordered sets while rings in the quasi-variety generated by Z_6 are equivalent to Boolean rectangular bands. On the other hand, dualities frequently arise in which the structured spaces are much more complex than the associated algebras. In general, such a duality is unlikely to be any use to the algebraist. In Chapter 6, an updated version of [CD96], we identify a number of different criteria which can make a strong duality useful, and in each case we give a characterisation of those quasi-varieties which admit a strong duality satisfying that criterion. Chapters 8 and 9 address the problem of complexity of the dual structures from a different point of view. Given a natural duality, how can we simplify the type of the dual category to obtain a more useful natural duality? In Chapter 8, based on Davey and Priestley [DP96a], we present an algorithmic method for reducing the type of a dualising category to a type that is, in an appropriate sense, minimal among all dualising categories. Chapter 9 presents a different method of optimising dualities which, though not algorithmic, is generally more convenient to use when the example is small. The results of this chapter first appeared in Davey, Haviar and Priestley [DHP95a].
The notions of natural duality and strong duality apply to quasi-varieties generated by a single finite algebra. In Chapter 7 we show how Davey and Priestley [DP87] broadened the scope of these notions to include quasi-varieties generated by finitely many finite algebras. Given a finite set of finite algebras which have distributive lattice reducts, we describe the method of constructing a broadened duality which rides piggyback on the underlying Priestley duality as it evolved from Davey and Werner [DW85, DW86] to Davey [Dav93] and Priestley [Pri95a]. We exhibit applications of the piggyback technique taken from Davey and Priestley [DP93a], Davey [Dav96] and Davey and Pitkethly [DP00].
Which finite algebras generate a dualisable quasi-variety? This question is at the forefront of research in the theory of natural dualities. In Chapter 10 we review what is known. As the previous chapters have dealt with finite algebras that generate dualisable quasi-varieties, we focus here on methods of showing that a particular quasi-variety is not dualisable. A central result is the Big NU Obstacle Theorem 10.2.2 of Davey, Heindorf and McKenzie [DHM95] which asserts that near-unanimity is necessary for dualisability in a congruence join-semidistributive quasi-variety. Several methods of demonstrating non-dualisability are presented: one based on Heindorf [Hei93], one first used in Davey and Werner [DW83] and one based on Clark and Krauss [CK84]. We conclude by determining exactly which clones on a two-element set generate a dualisable quasi-variety.
The tome now before you might easily convey the misimpression that it contains a complete account of the theory of natural dualities. As an account of the part of that theory that is known to date, it is indeed quite complete. But in absolute terms our present knowledge is very limited. Outside of congruence distributive varieties, we know little of which finite algebras generate a dualisable quasi-variety. We do not even yet know which finite groups, finite rings or finite semigroups generate a dualisable quasi-variety. Even within congruence distributive varieties we have no general method to determine whether or not a particular finite algebra generates a dualisable quasi-variety. We can extend many dualities to strong dualities, but we do not know if this is always possible. There still remain many fundamental questions that we cannot yet answer.
It is our hope that working algebraists will be able to use the strong duality theorems of Chapter 3 to produce new dualities for the quasi-varieties they study, to use the reduction methods of Chapters 2, 8 and 9 to simplify those dualities, and then to invoke the resulting dualities to help solve real problems in their chosen quasi-varieties. We further hope that some particularly hard working algebraists will be undaunted when the theorems of Chapter 2, 3 or 7 do not directly apply, and will continue to find new methods of producing dualities and strong dualities. Given success in these endeavours, we can then hope that some of the very hardest working algebraists will join our effort to resolve some of the still outstanding problems in the theory of natural dualities itself.