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18
Finite generation of symmetric ideals
 TRANS. AMER. MATH. SOC
, 2005
"... Let A be a commutative Noetherian ring, and let R = A[X] be the polynomial ring in an infinite collection X of indeterminates over A. Let SX be the group of permutations of X. The group SX acts on R in a natural way, and this in turn gives R the structure of a left module over the left group ring R ..."
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Cited by 27 (9 self)
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Let A be a commutative Noetherian ring, and let R = A[X] be the polynomial ring in an infinite collection X of indeterminates over A. Let SX be the group of permutations of X. The group SX acts on R in a natural way, and this in turn gives R the structure of a left module over the left group ring R[SX]. We prove that all ideals of R invariant under the action of SX are finitely generated as R[SX]modules. The proof involves introducing a certain wellquasiordering on monomials and developing a theory of Gröbner bases and reduction in this setting. We also consider the concept of an invariant chain of ideals for finitedimensional polynomial rings and relate it to the finite generation result mentioned above. Finally, a motivating question from chemistry is presented, with the above framework providing a suitable context in which to study it.
Forward analysis for WSTS, part II: Complete WSTS
 In ICALP’09, volume 5556 of LNCS
, 2009
"... Abstract. We describe a simple, conceptual forward analysis procedure for ∞complete WSTS S. This computes the clover of a state s0, i.e., a finite description of the closure of the cover of s0. When S is the completion of a WSTS X, the clover in S is a finite description of the cover in X. We show ..."
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Cited by 14 (5 self)
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Abstract. We describe a simple, conceptual forward analysis procedure for ∞complete WSTS S. This computes the clover of a state s0, i.e., a finite description of the closure of the cover of s0. When S is the completion of a WSTS X, the clover in S is a finite description of the cover in X. We show that this applies exactly when X is an ω 2WSTS, a new robust class of WSTS. We show that our procedure terminates in more cases than the generalized KarpMiller procedure on extensions of Petri nets. We characterize the WSTS where our procedure terminates as those that are cloverflattable. Finally, we apply this to wellstructured counter systems. 1
Wellquasiordering infinite graphs with forbidden finite planar minor
 Transactions of the American Mathematical Society
, 1990
"... Abstract. We prove that given any sequence G\, Gi,... of graphs, where G\ is finite planar and all other G, are possibly infinite, there are indices;', j such that i < j and G ¡ is isomorphic to a minor of Gj. This generalizes results of Robertson and Seymour to infinite graphs. The restrict ..."
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Cited by 11 (2 self)
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Abstract. We prove that given any sequence G\, Gi,... of graphs, where G\ is finite planar and all other G, are possibly infinite, there are indices;', j such that i < j and G ¡ is isomorphic to a minor of Gj. This generalizes results of Robertson and Seymour to infinite graphs. The restriction on G \ cannot be omitted by our earlier result. The proof is complex and makes use of an excluded minor theorem of Robertson and Seymour, its extension to infinite graphs, NashWilliams ' theory of betterquasiordering, especially his infinite tree theorem, and its extension to something we call treestructures over QOcategories, which includes infinitary version of a wellquasiordering theorem of Friedman. 1.
On Noetherian Spaces
"... A topological space is Noetherian iff every open is compact. Our starting point is that this notion generalizes that of wellquasi order, in the sense that an Alexandroffdiscrete space is Noetherian iff its specialization quasiordering is well. For more general spaces, this opens the way to verify ..."
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A topological space is Noetherian iff every open is compact. Our starting point is that this notion generalizes that of wellquasi order, in the sense that an Alexandroffdiscrete space is Noetherian iff its specialization quasiordering is well. For more general spaces, this opens the way to verifying infinite transition systems based on nonwell quasi ordered sets, but where the preimage operator satisfies an additional continuity assumption. The technical development rests heavily on techniques arising from topology and domain theory, including sobriety and the de Groot dual of a stably compact space. We show that the category Nthr of Noetherian spaces is finitely complete and finitely cocomplete. Finally, we note that if X is a Noetherian space, then the set of all (even infinite) subsets of X is again Noetherian, a result that fails for wellquasi orders. 1.
Finiteness theorems in stochastic integer programming
, 2006
"... We study Graver test sets for families of linear multistage stochastic integer programs with varying number of scenarios. We show that these test sets can be decomposed into finitely many “building blocks”, independent of the number of scenarios, and we give an effective procedure to compute them. ..."
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Cited by 5 (2 self)
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We study Graver test sets for families of linear multistage stochastic integer programs with varying number of scenarios. We show that these test sets can be decomposed into finitely many “building blocks”, independent of the number of scenarios, and we give an effective procedure to compute them. The paper includes an introduction to NashWilliams ’ theory of betterquasiorderings, which is used to show termination of our algorithm. We also apply this theory to finiteness results for Hilbert functions.
On BetterQuasiOrdering Countable SeriesParallel Orders
"... We prove that any infinite sequence of countable seriesparallel orders contains an increasing (with respect to embedding) infinite subsequence. This result generalizes Laver's and Corominas' theorems concerning betterquasiorder of the classes of countable chains and trees. ..."
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We prove that any infinite sequence of countable seriesparallel orders contains an increasing (with respect to embedding) infinite subsequence. This result generalizes Laver's and Corominas' theorems concerning betterquasiorder of the classes of countable chains and trees.
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 5 (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.
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 ..."
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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.
From Well to Better, the Space of Ideals
"... On the one hand, the ideals of a well quasiorder (wqo) naturally form a compact topological space into which the wqo embeds. On the other hand, NashWilliams ’ barriers are given a uniform structure by embedding them into the Cantor space. We prove that every map from a barrier into a wqo restricts ..."
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On the one hand, the ideals of a well quasiorder (wqo) naturally form a compact topological space into which the wqo embeds. On the other hand, NashWilliams ’ barriers are given a uniform structure by embedding them into the Cantor space. We prove that every map from a barrier into a wqo restricts on a barrier to a uniformly continuous map, and therefore extends to a continuous map from a countable closed subset of the Cantor space into the space of ideals of the wqo. We then prove that, by shrinking further, any such continuous map admits a canonical form with regard to the points whose image is not isolated. As a consequence, we obtain a simple proof of a result on better quasiorders (bqo); namely, a wqo whose set of non principal ideals is bqo is actually bqo.