Hilary A. Priestley

The Syntax and Semantics of Entailment in Duality Theory

Both syntactic and semantic solutions are given for the entailment problem of duality theory. The test algebra theorem provides both a syntactic solution to the entailment problem in terms of primitive positive formulae and a new derivation of the corresponding result in clone theory, viz. the syntactic description of Inv(Pol(R)) for a given set R of finitary relations on a finite set. The semantic solution to the entailment problem follows from the syntactic one, or can be given in the form of an algorithm. It shows, in the special case of a purely relational type, that duality-theoretic entailment is describable in terms of five constructs, namely trivial relations, intersection, repetition removal, product, and refractive projection. All except the last are concrete, in the sense that they are described by a quantifier-free formula. It is proved that if the finite algebra M generates a congruence-distributive variety and all subalgebras of M are subdirectly irreducible, then concrete constructs suffice to describe entailment. The concept of entailment appropriate to strong dualities is also introduced, and described in terms of coordinate projections, restriction of domains, and composition of partial functions. ?

Endoprimal distributive lattices are endodualisable

AbstractL. Márki and R. Pöschel have characterised the endoprimal distributive lattices as those which are not relatively complemented. The theory of natural dualities implies that any finite algebraA on which the endomorphisms of A yield a duality on the quasivariety

Optimal natural dualities. II: General theory

\mathbb{I}\mathbb{S}\mathbb{P}(A)

On Priestley Spaces of Lattice-Ordered Algebraic Structures

is necessarily endoprimal. This note investigates endodualisability for finite distributive lattices, and shows, in a manner which elucidates Márki and Pöschels proof, that it is equivalent to endoprimality.

Catalytic distributive lattices and compact zero-dimensional topological lattices

The techniques of natural duality theory are applied to certain finitely generated varieties of Heyting algebras to obtain optimal dualities for these varieties, and thereby to address algebraic questions about them. In particular, a complete characterisation is given of the endodualisable finite subdirectly irreducible Heyting algebras. The procedures involved rely heavily on Priestley duality for Heyting algebras.

Ordered Sets and Complete Lattices

A general theory of optimal natural dualities is presented, built on the test algebra technique introduced in an earlier paper. Given that a set R of finitary algebraic relations yields a duality on a class of algebras A = ISP(M), those subsets R′ of R which yield optimal dualities are characterised. Further, the manner in which the relations in R are constructed from those in R′ is revealed in the important special case that M generates a congruence-distributive variety and is such that each of its subalgebras is subdirectly irreducible. These results are obtained by studying a certain algebraic closure operator, called entailment, definable on any set of algebraic relations on M . Applied, by way of illustration, to the variety of Kleene algebras and to the proper subvarieties Bn of pseudocomplemented distributive lattices, the theory improves upon and illuminates previous results.

Finitely generated free modular ortholattices. II

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.

Intrinsic Spectral Topologies

The laws defining many important varieties of lattice-ordered algebras, such as linear Heyting algebras, MV-algebras and l-groups, can be cast in a form which allows dual representations to be derived in a very direct, and semi-automatic, way. This is achieved by developing a new duality theory for implicative lattices, which encompass all the varieries above. The approach focuses on distinguished subsets of the prime lattice filters of an implicative lattice, ordered as usual by inclusion. A decomposition theorem is proved, and the extent to which the order on the prime lattice filters determines the implicative structure is thereby revealed.