The Ideal Approach to Computing Closed Subsets in Well-Quasi-orderings

Goubault-Larrecq, Jean; Halfon, Simon; Karandikar, Prateek; Kumar, K. Narayan; Schnoebelen, Philippe

doi:10.1007/978-3-030-30229-0_3

Cited by 4 publications

(5 citation statements)

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“…One can also show that reduced word-products are canonical forms for irreducible closed subsets of X * , namely that two reduced word-products denote the same irreducible closed set if and only if they are syntactically equal. The proof is identical to the corresponding result on wqos, see [16,Theorem 4.22]; but we will not make use of that fact.…”

Section: The Bounds On ||H 0v X|| Are Tightmentioning

confidence: 72%

Statures and Sobrification Ranks of Noetherian Spaces

Jean¹,

Laboureix²

2021

Preprint

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There is a rich theory of maximal order types of well-partial-orders (wpos), pioneered by de Jongh and Parikh (1977) and Schmidt (1981). Every wpo is Noetherian in its Alexandroff topology, and there are more; this prompts us to investigate an analogue of that theory in the wider context of Noetherian spaces.The notion of maximal order type does not seem to have a direct analogue in Noetherian spaces per se, but the equivalent notion of stature, investigated by Blass and Gurevich (2008) does: we define the stature ||X|| of a Noetherian space X as the ordinal rank of its poset of proper closed subsets. We obtain formulas for statures of sums, of products, of the space of words on a space X, of the space of finite multisets on X, in particular. They confirm previously known formulas on wpos, and extend them to Noetherian spaces.The proofs are, by necessity, rather different from their wpo counterparts, and rely on explicit characterizations of the sobrifications of the corresponding spaces, as obtained by Finkel and the first author (2020).We also give formulas for the statures of some natural Noetherian spaces that do not arise from wpos: spaces with the cofinite topology, Hoare powerspaces, powersets, and spaces of words on X with the socalled prefix topology.Finally, because our proofs require it, and also because of its independent interest, we give formulas for the ordinal ranks of the sobrifications of each of those spaces, which we call their dimension.

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Section: The Bounds On ||H 0v X|| Are Tightmentioning

confidence: 72%

Statures and Sobrification Ranks of Noetherian Spaces

Jean¹,

Laboureix²

2021

Preprint

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“…Consider now a proper ideal I k+1 of D k+1 : this means I k+1 ∩ E k+1 = ∅. This implies in turn ↓Post ∃ (I k+1 ) ∩ E k = ∅ by (11), thus there exists J such that I k+1 → J and J ∩ E k = ∅.…”

Section: Transitions Between Proper Idealsmentioning

confidence: 99%

“…Crucially for the applicability of our approach, effective ideal representations exist for most wqos of interest [10,11].…”

Section: Ideal Decompositionsmentioning

confidence: 99%

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The ideal view on Rackoff's coverability technique

Lazić

Schmitz

2021

Information and Computation

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“…We will see that this leads to canonical forms for transfinite products. As in [7,Theorem 4.22], and to reduce excessive pedantry related to the difference between syntax and semantics, we write A, B (resp., P, Q) to denote atoms, resp. sequences of atoms (syntax), and A, B, P , Q for their respective semantics.…”

Section: Reduced Productsmentioning

confidence: 99%