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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 14 (8 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.
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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Cited by 10 (5 self)
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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.
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 8 (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 restriction on G ..."
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Cited by 8 (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.
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.
Finiteness theorems in stochastic integer programming
 Foundations of Computational Mathematics
, 2003
"... Abstract. 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 comp ..."
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Cited by 3 (0 self)
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Abstract. 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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Cited by 3 (0 self)
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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.
ON SCATTERED POSETS WITH FINITE DIMENSION
, 812
"... Abstract. We discuss a possible characterization, by means of forbidden configurations, of posets which are embeddable in a product of finitely many scattered chains. Introduction and presentation of the results A fundamental result, due to Szpilrajn [26], states that every order on a set is the int ..."
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Abstract. We discuss a possible characterization, by means of forbidden configurations, of posets which are embeddable in a product of finitely many scattered chains. Introduction and presentation of the results A fundamental result, due to Szpilrajn [26], states that every order on a set is the intersection of a family of linear orders on this set. The dimension of the order, also called the dimension of the ordered set, is then defined as the minimum cardinality of such a family (Dushnik, Miller [11]). Specialization of Szpilrajn’s result to several