Results 1 
4 of
4
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 MAXIMAL LINEAR EXTENSION THEOREM IN SECOND ORDER ARITHMETIC
, 2011
"... 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. ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
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.
Poset algebras over well quasiordered posets
"... A new class of partial ordertypes, class G + bqo is defined and investigated here. A poset P is in the class G + bqo iff the poset algebra F(P) is generated by a better quasiorder G that is included in L(P). The free Boolean algebra F(P) and its free distrivutive lattice L(P) were defined in [ABKR ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
A new class of partial ordertypes, class G + bqo is defined and investigated here. A poset P is in the class G + bqo iff the poset algebra F(P) is generated by a better quasiorder G that is included in L(P). The free Boolean algebra F(P) and its free distrivutive lattice L(P) were defined in [ABKR]. The free Boolean algebra F(P) contains the partial order P and is generated by it: F(P) has the following universal property. If B is any Boolean algebra and f is any orderpreserving map from P into a Boolean algebra B, then f can be extended to an homomorphism ˆ f of F(P) into B. We also define L(P) as the sublattice of F(P) generated by P. We prove that if P is any well quasiordering, then L(P) is well founded, and is a countable union of well quasiorderings. We prove that the class G + bqo is contained in the class of well quasiordered sets. We prove that G + bqo is preserved under homomorphic image, finite products, and lexicographic sum over better quasiordered index sets. We prove also that every countable well quasiordered set is in G + bqo. We do not know, however if the class of well quasiordered sets is contained in G + bqo. Additional results concern homomorphic images of posets algebras.