Dualities and dual pairs in Heyting algebras

We extract the abstract core of finite homomorphism dualities using the techniques of Heyting algebras and (combinatorial) categories.Comment: 17 pages; v2: minor correction

On infinite-finite duality pairs of directed graphs

The (A,D) duality pairs play crucial role in the theory of general relational structures and in the Constraint Satisfaction Problem. The case where both classes are finite is fully characterized. The case when both side are infinite seems to be very complex. It is also known that no finite-infinite duality pair is possible if we make the additional restriction that both classes are antichains. In this paper (which is the first one of a series) we start the detailed study of the infinite-finite case. Here we concentrate on directed graphs. We prove some elementary properties of the infinite-finite duality pairs, including lower and upper bounds on the size of D, and show that the elements of A must be equivalent to forests if A is an antichain. Then we construct instructive examples, where the elements of A are paths or trees. Note that the existence of infinite-finite antichain dualities was not previously known

Dualities and Dual Pairs in Heyting Algebras

We extract the abstract core of finite homomorphism dualities using the techniques of Heyting algebras and (combinatorial) categorie

Homomorphisms and Structural Properties of Relational Systems

Two main topics are considered: The characterisation of finite homomorphism dualities for relational structures, and the splitting property of maximal antichains in the homomorphism order.Comment: PhD Thesis, 77 pages, 14 figure

Splitting finite antichains in the homomorphism order

A structural condition is given for finite maximal antichains in the homomorphism order of relational structures to have the splitting property. It turns out that non-splitting antichains appear only at the bottom of the order. Moreover, we examine looseness and finite antichain extension property for some subclasses of the homomorphism poset. Finally, we take a look at cut-points in this order

Regular families of forests, antichains and duality pairs of relational structures

Homomorphism duality pairs play crucial role in the theory of relational structures and in the Constraint Satisfaction Problem. The case where both classes are finite is fully characterized. The case when both side are infinite seems to be very complex. It is also known that no finite-infinite duality pair is possible if we make the additional restriction that both classes are antichains. In this paper we characterize the infinite-finite antichain dualities and infinite-finite dualities with trees or forest on the left hand side. This work builds on our earlier papers that gave several examples of infinite-finite antichain duality pairs of directed graphs and a complete characterization for caterpillar dualities