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Program termination and well partial orderings
, 2006
"... The following known observation is useful in establishing program termination: if a transitive relation R is covered by finitely many wellfounded relations U1,..., Un then R is wellfounded. A question arises how to bound the ordinal height R  of the relation R in terms of the ordinals αi = Ui. ..."
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The following known observation is useful in establishing program termination: if a transitive relation R is covered by finitely many wellfounded relations U1,..., Un then R is wellfounded. A question arises how to bound the ordinal height R  of the relation R in terms of the ordinals αi = Ui. We introduce the notion of the stature ‖P ‖ of a well partial ordering P and show that R  ≤ ‖α1 × · · · × αn ‖ and that this bound is tight. The notion of stature is of considerable independent interest. We define ‖P ‖ as the ordinal height of the forest of nonempty bad sequences of P, but it has many other natural and equivalent definitions. In particular, ‖P ‖ is the supremum, and in fact the maximum, of the lengths of linearizations of P. And ‖α1 × · · · × αn ‖ is equal to the natural product α1 ⊗ · · · ⊗ αn.
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 ..."
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Cited by 4 (1 self)
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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.
COMPUTABLE LINEARIZATIONS OF WELLPARTIALORDERINGS
, 2007
"... We analyze results on wellpartialorderings from the viewpoint of computability theory, and we answer a question posed by Diana Schmidt. We obtain the following results. De Jongh and Parikh showed that every wellpartialorder has a linearization of maximal order type. We show that such a lineariz ..."
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Cited by 4 (1 self)
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We analyze results on wellpartialorderings from the viewpoint of computability theory, and we answer a question posed by Diana Schmidt. We obtain the following results. De Jongh and Parikh showed that every wellpartialorder has a linearization of maximal order type. We show that such a linearization can be found computably. We also show that the process of finding such a linearization is not computably uniform, not even hyperarithmetically.
THE MAXIMAL LINEAR EXTENSION THEOREM IN SECOND ORDER ARITHMETIC
"... Abstract. We show that the maximal linear extension theorem for well partial orders is equivalent over RCA0 to ATR0. Analogously, the maximal chain theorem for well partial orders is equivalent to ATR0 over RCA0. 1. ..."
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Abstract. We show that the maximal linear extension theorem for well partial orders is equivalent over RCA0 to ATR0. Analogously, the maximal chain theorem for well partial orders is equivalent to ATR0 over RCA0. 1.