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.

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.

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An algebra A is endoprimal if, for all k N, the only maps from A k to A which preserve the endomorphisms of A are its term functions. One method for finding finite endoprimal algebras is via the theory of natural dualities since an endodualisable algebra is necessarily endoprimal. General results on endoprimality and endodualisability are proved and then applied to the varieties of sets, vector spaces, distributive lattices, Boolean algebras, Stone algebras, Heyting algebras, semilattices and abelian groups. In many classes the finite endoprimal algebras turn out to be endodualisable. We show that this fails in general by proving that 2 2 1, regarded as either a bounded semilattice or upper-bounded semilattice is dualisable, endoprimal but not endodualisable.

We solve the dualisability problem in the class of three-element unary algebras. Our aim in tackling this class is to demonstrate the difficulty of the general dualisability problem. We also want to investigate the extent to which the dualisability of a finite algebra is a finiteness condition on the quasi-variety it generates.

While every finite lattice-based algebra is dualisable, the same is not true of semilattice-based algebras. We show that a finite semilattice-based algebra is dualisable if all its operations are compatible with the semilattice operation. We also give examples of infinite semilattice-based algebras that are dualisable. In contrast, we present a general condition that guarantees the inherent non-dualisability of a finite semilattice-based algebra. We combine our results to characterise dualisability amongst the finite algebras in the classes of flat extensions of partial algebras and closure semilattices. Throughout, we emphasise the connection between the dualisability of an algebra and the residual character of the variety it generates.

We characterise the strongly dualisable three-element unary algebras and show that every fully dualisable three-element unary algebra is strongly dualisable. It follows from the characterisation that, for dualisable three-element unary algebras, strong dualisability is equivalent to a weak form of injectivity.2000 Mathematics subject classification: primary 08C15,08A60; secondary 18A40. Keywords and phrases: natural duality, strong duality, full duality, unary algebras.The theory of natural dualities is a study of quasi-varieties of the form 0 §P(M), where M is a finite algebra. We aim to set up a natural dual equivalence between the category &Z := 0 §P(M) and a category X of structured topological spaces. This duality can often provide a practical representation of the algebras in si in terms of simpler objects. Priestley's duality for the quasi-variety of distributive lattices is a prime example of a very useful duality (see [9]). As well as finding and using practical dualities, natural-duality theoreticians tackle more esoteric problems. We are interested in understanding which finite algebras M allow us to set up a natural duality for BSP (M), and what the existence (or non-existence) of this duality can tell us about the quasi-variety 0 §IP(M).The theory of natural dualities is well developed and contains some powerful theorems for creating dualities. Nevertheless, our understanding of what makes an algebra dualisable, fully dualisable or strongly dualisable is rather limited. In this paper, we aim to gain some insight into strong and full dualisability by investigating threeelement unary algebras. Unary algebras, especially three-element unary algebras, may seem very simple. But, from the point of view of natural-duality theory, they are rather complicated. This study complements the paper [4], by Clark, Davey andThe author wishes to thank Brian Davey for supervising this project.