Dwight Duffus

An oriented k-uniform hypergraph (a family of ordered k-sets) has the ordering property (or Property O) if, for every linear order of the vertex set, there is some edge oriented consistently with the linear order. We find bounds on the minimum number of edges in a hypergraph with Property O.

An oriented k-uniform hypergraph (a family of ordered k-sets) has the ordering property (or Property O) if for every linear order of the vertex set, there is some edge oriented consistently with the linear order. We find bounds on the minimum number of edges in a hypergraph with Property O.

The collection CL(T) of nonempty convex sublattices of a lattice T ordered by bi-domination is a lattice. We say that T has the fixed point property for convex sublattices (CLFPP for short) if every order preserving map f from T to CL(T) has a fixed point, that is x > f(x) for some x > T. We examine which lattices may have CLFPP. We introduce the selection property for convex sublattices (CLSP); we observe that a complete lattice with CLSP must have CLFPP, and that this property implies that CL(T) is complete. We show that for a lattice T, the fact that CL(T) is complete is equivalent to the fact that T is complete and the lattice of all subsets of a countable set, ordered by containment, is not order embeddable into T. We show that for the lattice T = I(P) of initial segments of a poset P, the implications above are equivalences and that these properties are equivalent to the fact that P has no infinite antichain. A crucial part of this proof is a straightforward application of a wonderful Hausdor type result due to Abraham, Bonnet, Cummings, Dzamondja and Thompson [2010].

The collection CL(T) of nonempty convex sublattices of a lattice T ordered by bi-domination is a lattice. We say that T has the fixed point property for convex sublattices (CLFPP for short) if every order preserving map f from T to CL(T) has a fixed point, that is x > f(x) for some x > T. We examine which lattices may have CLFPP. We introduce the selection property for convex sublattices (CLSP); we observe that a complete lattice with CLSP must have CLFPP, and that this property implies that CL(T) is complete. We show that for a lattice T, the fact that CL(T) is complete is equivalent to the fact that T is complete and the lattice of all subsets of a countable set, ordered by containment, is not order embeddable into T. We show that for the lattice T = I(P) of initial segments of a poset P, the implications above are equivalences and that these properties are equivalent to the fact that P has no infinite antichain. A crucial part of this proof is a straightforward application of a wonderful Hausdorff? type result due to Abraham, Bonnet, Cummings, Dzamondja and Thompson [2010]. Key words and phrases. posets, lattices, convex sublattice, retracts, fixed point property

R. Canfield has conjectured that for all subgroups G of the automorphism 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 powers of disjoint cycles. The symmetric chain decompositions of Greene and Kleitman provide the basis for partitions of these quotients.