No Finite–Infinite Antichain Duality in the Homomorphism Poset of Directed Graphs

Abstract

D denotes the homomorphism poset of finite directed graphs. An antichain duality is a pair F, D of antichains of D such that (F→) ∪ (→D) = D is a partition. A generalized duality pair in D is an antichain duality F, D with finite F and D. We give a simplified proof of the Foniok-Nešetřil-Tardif theorem for the special case D, which gave full description of the generalized duality pairs in D. Although there are many antichain dualities F, D with infinite D and F, we can show that there is no antichain duality F, D with finite F and infinite D.

On Infinite–finite Duality Pairs of Directed Graphs

“…This result can be proved from the Directed Sparse Incomparability Lemma, see [1,2]. We present a self contained proof instead.…”

“…This leaves open the question of finding or characterizing duality pairs with one side finite while the other an infinite antichain. Erdős and Soukup [2] proved that no such duality pair exists with the left side finite.…”

“…Research supported in part by the Hungarian NSF, under contract NK 78439 and K 682622 Research supported by grants from NSERC and ARP 3. Research supported in part by the NSERC grant 329527 and by the Hungarian NSF grants T-046234, AT048826 and NK-62321…”

Dualities and Dual Pairs in Heyting Algebras

“…A characterization of finite dualities in the category of relational structures is provided in [35,14]. The relationship between finite dualities and duality pairs has recently been reproved in the special case of digraphs [11] using the Directed Sparse Incomparability Lemma. Here we would like to point out that sparse incomparability is not necessary to achieve these results; in fact, much weaker assumptions suffice.…”

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

“…However the antichains with the splitting property have a lot of structure. In [5] it is shown that in the case of directed graphs, such an antichain A = O ∪ D cannot have O finite and D infinite. The question of the existence of antichains with the splitting property A = O ∪ D with O infinite and D finite is answered positively in [6].…”