In this thesis three combinatorial problems are studied in four papers. In Paper 1 we study the structure of the k-assignment polytope, whose vertices are the m× n (0,1)-matrices with exactly k 1:s and at most one 1 in each row and each column. This is a natural generalisation of the Birkhoff polytope and many of the known properties of the Birkhoff polytope are generalised. A representation of...

A join-completion of a poset is a completion for which each element is obtainable as a supremum, or join, of elements from the original poset. It is well known that the join-completions of a poset are in one-to-one correspondence with the closure systems on the lattice of up-sets of the poset. A ∆1-completion of a poset is a completion for which, simultaneously, each element is obtainable as a ...

A join-completion of a poset is a completion for which each element is obtainable as a supremum, or join, of elements from the original poset. It is well known that the join-completions of a poset are in one-to-one correspondence with the closure systems on the lattice of up-sets of the poset. A ∆1-completion of a poset is a completion for which, simultaneously, each element is obtainable as a ...

Let D = (V (D), A(D)) be a digraph. The competition graph of D, is the graph with vertex set V (D) and edge set {uv ∈ `V (D) 2 ́ : ∃w ∈ V (D),−→ uw,−→ vw ∈ A(D)}. The double competition graph of D, is the graph with vertex set V (D) and edge set {uv ∈ `V (D) 2 ́ : ∃w1, w2 ∈ V (D),−−→ uw1,−−→ vw1,−−→ w2u,−−→ w2v ∈ A(D)}. A poset of dimension at most two is a digraph whose vertices are some points ...

(1) Let R be a division ring. Prove that every module over R is free. You will need to use Zorn’s lemma: Recall that a partially order set (=poset) S is a set with a relation x ≤ y defined between some pairs of elements x, y ∈ S, such that: (i) x ≤ x ; (ii) x ≤ y and y ≤ x implies x = y ; (iii) x ≤ y , y ≤ z ⇒ x ≤ z . A chain in S is a subset T ⊂ S such that for all t, t ′ in T , either t ≤ t ′...

Let a poset P be called chain-complete when every chain, including the empty chain, has a sup in P. Many authors have investigated properties of posets satisfying some sort of chain-completeness condition (see [,11, [-31, [6], I-71, [17], [,181, ['191, [,211, [,221), and used them in a variety of applications. In this paper we study the notion of chain-completeness and demonstrate its usefulnes...

We define a new object, called a signed poset, that bears the same relation to the hyperoctahedral group B n (i.e., signed permutations on n letters), as do posets to the symmetric group S n. We then prove hyperoctahedral analogues of the following results: (1) the generating function results from the theory of P-partitions; (2) the fundamental theorem of finite distributive lattices (or Birkho...

A binary poset code of codimension m (of cardinality 2n−m , where n is the code length) can correct maximum m errors. All possible poset metrics that allow codes of codimension m to be m-, (m − 1)-, or (m − 2)-perfect are described. Some general conditions on a poset which guarantee the nonexistence of perfect poset codes are derived; as examples, we prove the nonexistence of r-perfect poset co...

We study a poset N on the free monoid X∗ on a countable alphabet X. This poset is determined by the fact that its total extensions are precisely the standard term orders on X ∗. We also investigate the poset classifying degree-compatible standard term orders, and the poset classifying sorted term orders. For the latter poset, we give a Galois coconnection with the Young lattice.

S. Janson [Poset limits and exchangeable random posets, Combinatorica 31 (2011), 529–563] defined limits of finite posets in parallel to the emerging theory of limits of dense graphs. We prove that each poset limit can be represented as a kernel on the unit interval with the standard order, thus answering an open question of Janson. We provide two proofs: real-analytic and combinatorial. The co...