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Denote by G and D, respectively, the the homomorphism poset of the finite undirected and directed graphs, respectively. A maximal antichain A in a poset P splits if A has a partition (B, C) such that for each p ∈ P either b ≤ P p for some b ∈ B or p ≤ p c for some c ∈ C. We construct both splitting and non-splitting infinite maximal antichains in G and in(More)
Canfield and Mason have conjectured that for all subgroups G of the automor-phism group of the Boolean lattice B n (which can be regarded as the symmetric group S n) the quotient order B n /G is a symmetric chain order. We provide a straightforward proof of a generalization of a result of K. K. Jordan: namely, B n /G is an SCO whenever G is generated by(More)
The following equivalent results in the Boolean lattice 2 are proven. (a) Every fibre of 2 contains a maximal chain. (b) Every cutset of 2 contains a maximal antichain. (c) Every red-blue colouring of the vertices of 2 produces either a red maximal chain or a blue maximal antichain. (d) Given any n antichains in 2 there is a disjoint maximal antichain.(More)
There are two types of graphs commonly associated with finite (partially) ordered sets: the comparability graph and the covering graph. While the first type has been characterized, only partial descriptions of the second are known. We prove that the covering graphs of distributive lattices are precisely those graphs which are retracts of hypercubes. Much of(More)
We discuss Hedetniemi's conjecture in the context of categories of relational structures under homomorphisms. In this language Hedetniemi's conjecture says that if there are no homomor-phisms from the graphs G and H to the complete graph on n vertices then there is no homomorphism from G x H to the complete graph. If an object in some category has just this(More)