Results 1 
2 of
2
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

Cited by 4 (1 self)
 Add to MetaCart
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.
THE STRENGTH OF JULLIEN’S INDECOMPOSABILITY THEOREM
"... Abstract. Jullien’s indecomposability theorem states that if a scattered countable linear order is indecomposable, then it is either indecomposable to the left, or indecomposable to the right. The theorem was shown by Montalbán to be a theorem of hyperarithmetic analysis. We identify the strength of ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
Abstract. Jullien’s indecomposability theorem states that if a scattered countable linear order is indecomposable, then it is either indecomposable to the left, or indecomposable to the right. The theorem was shown by Montalbán to be a theorem of hyperarithmetic analysis. We identify the strength of the theorem relative to standard reverse mathematics markers. We show that it lies strictly between weak Σ 1 1 choice and ∆11 comprehension. §1. Introduction. A linear order (U;<U) (denoted simply U below) is scattered if it does not contain a copy of the order of rational numbers. A cut in the order U is a pair 〈L,R 〉 so that L ∩ R = ∅, L ∪ R = U, L is closed downward (or leftward) in <U, and R is closed upward (or rightward). The cut is a decomposition of U if U does not embed into either one of L, R. The order U