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

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.