Miroslav Haviar

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

A Fresh Perspective on Canonical Extensions for Bounded Lattices

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

Reconciliation of approaches to the construction of canonical extensions of bounded lattices

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.

Affine complete Stone algebras

A description is given of then-generated free algebras in the variety of modular ortholattices generated by an ortholatticeMO2 of height 2 with 4 atoms. In the subvariety lattice of orthomodular lattices, the varietyV(MO2) is the unique cover of the variety of Boolean algebras, in whichn-generated free algebras were described by G. Boole in 1854. It is shown that then-generated free algebra in the varietyV(MO2) is a product of then-generated free Boolean algebra22n and Φ(n) copies of the generatorMO2, and formula is presented for Φ(n). To achieve this result, algebraic methods of the theory of orthomodular lattices are combined with recently developed methods of natural duality theory for varieties of algebras.

Applications of Priestley duality in transferring optimal dualities

This paper presents a novel treatment of the canonical extension of a bounded lattice, in the spirit of the theory of natural dualities. At the level of objects, this can be achieved by exploiting the topological representation due to M. Ploščica, and the canonical extension can be obtained in the same manner as can be done in the distributive case by exploiting Priestley duality. To encompass both objects and morphisms the Ploščica representation is replaced by a duality due to Allwein and Hartonas, recast in the style of Ploščica’s paper. This leads to a construction of canonical extension valid for all bounded lattices, which is shown to be functorial, with the property that the canonical extension functor decomposes as the composite of two functors, each of which acts on morphisms by composition, in the manner of hom-functors.

Transferring optimal dualities : theory and practice

Traditionally in natural duality theory the algebras carry no topology and the objects on the dual side are structured Boolean spaces. Given a duality, one may ask when the topology can be swapped to the other side to yield a partner duality (or, better, a dual equivalence) between a category of topological algebras and a category of structures. A prototype for this procedure is provided by the passage from Priestley duality for bounded distributive lattices to Banaschewski duality for ordered sets. Moreover, the partnership between these two dualities yields as a spin-off a factorisation of the functor sending a bounded distributive lattice to its natural extension, alias, in this case, the canonical extension or profinite completion. The main theorem of this paper validates topology swapping as a uniform way to create new dual adjunctions and dual equivalences: we prove that, for every finite algebra of finite type, each dualising alter ego gives rise to a partner duality. We illustrate the theorem via a variety of natural dualities, some classic and some less familiar. For lattice-based algebras this leads immediately, as in the Priestley–Banaschewski example, to a concrete description of canonical extensions.

Bohr Compactifications of Algebras and Structures

We provide new insights into the relationship between different constructions of the canonical extension of a bounded lattice. This follows on from the recent construction of the canonical extension using Ploščica’s maximal partial maps into the two-element set by Craig, Haviar and Priestley (2012). We show how this complete lattice of maps is isomorphic to the stable sets of Urquhart’s representation and to the concept lattice of a specific context, and how to translate our construction to the original construction of Gehrke and Harding (2001). In addition, we identify the completely join- and completely meet-irreducible elements of the complete lattice of maximal partial maps.