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ORDERINGS OF MONOMIAL IDEALS
, 2003
"... We study the set of monomial ideals in a polynomial ring as an ordered set, with the ordering given by reverse inclusion. We give a short proof of the fact that every antichain of monomial ideals is finite. Then we investigate ordinal invariants for the complexity of this ordered set. In particular ..."
Abstract

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We study the set of monomial ideals in a polynomial ring as an ordered set, with the ordering given by reverse inclusion. We give a short proof of the fact that every antichain of monomial ideals is finite. Then we investigate ordinal invariants for the complexity of this ordered set. In particular, we give an interpretation of the height function in terms of the HilbertSamuel polynomial, and we compute upper and lower bounds on the maximal order type.
KruskalFriedman Gap . . .
, 2002
"... We investigate new extensions of the KruskalFriedman theorems concerning wellquasiordering of finite trees with the gap condition. For two labelled trees s and t we say that s is embedded with gap into t if there is an injection from the vertices of s into t which maps each edge in s to a unique ..."
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We investigate new extensions of the KruskalFriedman theorems concerning wellquasiordering of finite trees with the gap condition. For two labelled trees s and t we say that s is embedded with gap into t if there is an injection from the vertices of s into t which maps each edge in s to a unique path in t with greaterorequal labels. We show that finite trees are wellquasiordered with respect to the gap embedding when the labels are taken from an arbitrary wellquasiordering and each tree path can be partitioned into k ∈ N or less comparable subpaths. This result generalizes both [Kˇrí89] and [OT87], and is also optimal in the sense that unbounded partiality over tree paths yields a counter example.