## On Noetherian Spaces

Citations: | 10 - 5 self |

### BibTeX

@MISC{Goubault-larrecq_onnoetherian,

author = {Jean Goubault-larrecq},

title = {On Noetherian Spaces},

year = {}

}

### OpenURL

### Abstract

A topological space is Noetherian iff every open is compact. Our starting point is that this notion generalizes that of well-quasi order, in the sense that an Alexandroff-discrete space is Noetherian iff its specialization quasi-ordering is well. For more general spaces, this opens the way to verifying infinite transition systems based on non-well 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 well-quasi orders. 1.

### Citations

456 | Domain Theory
- Abramsky, Jung
- 1994
(Show Context)
Citation Context ...ologies, of sober space, of sobrification of a space, and of stably compact spaces are central to our work. Topology and domain theory form another wonderful piece of mathematics, and one may consult =-=[12, 7, 18, 21]-=-. Last but not least, Noetherian spaces arise from algebraic geometry [13]: we discuss this briefly in Section 8. 2. Preliminaries I: Order and Topology A topology O on a set X is a collection of subs... |

321 |
A compendium on continuous lattices
- GIERZ
- 1980
(Show Context)
Citation Context ...ologies, of sober space, of sobrification of a space, and of stably compact spaces are central to our work. Topology and domain theory form another wonderful piece of mathematics, and one may consult =-=[12, 7, 18, 21]-=-. Last but not least, Noetherian spaces arise from algebraic geometry [13]: we discuss this briefly in Section 8. 2. Preliminaries I: Order and Topology A topology O on a set X is a collection of subs... |

192 | Well-structured transition systems everywhere! Theor
- Schnoebelen
- 2001
(Show Context)
Citation Context ...ounded, i.e., has no infinite descending chain, but also has no infinite antichain (a set of incomparable elements). One use of well quasi-orderings is in verifying well-structured transition systems =-=[2, 4, 11, 14]-=-. These are transition systems, usually infinite-state, with two ingredients. First, a well quasi-ordering ≤ on the set X of states. Second, the transition relation δ commutes with ≤, i.e., if x δ y a... |

175 | B.: Verifying programs with unreliable channels
- Abdulla, Jonsson
- 1996
(Show Context)
Citation Context ...commutes with ≤, i.e., if x δ y and x ≤ x ′ , then there is a state y ′ such that x ′ δ y ′ and y ≤ y ′ : δ x ≤ �� x ′ δ �� �� y ≤ �� ′ y Examples include Petri nets, VASS [15], lossy channel systems =-=[3]-=-, timed Petri nets [6] to cite a few. ∗ Partially supported by the INRIA ARC ProNoBis. (1) For any subset A of X, let Pre ∃ δ(A) be the preimage {x ∈ X|∃y ∈ A · x δ y}. The commutation property ensure... |

56 | Y.K.: Algorithmic analysis of programs with well quasi-ordered domains
- Abdulla, Čerāns, et al.
- 2000
(Show Context)
Citation Context ...ounded, i.e., has no infinite descending chain, but also has no infinite antichain (a set of incomparable elements). One use of well quasi-orderings is in verifying well-structured transition systems =-=[2, 4, 11, 14]-=-. These are transition systems, usually infinite-state, with two ingredients. First, a well quasi-ordering ≤ on the set X of states. Second, the transition relation δ commutes with ≤, i.e., if x δ y a... |

53 |
On the reachability problem for 5-dimensional vector addition systems
- Hopcroft, Pansiot
- 1979
(Show Context)
Citation Context ..., the transition relation δ commutes with ≤, i.e., if x δ y and x ≤ x ′ , then there is a state y ′ such that x ′ δ y ′ and y ≤ y ′ : δ x ≤ �� x ′ δ �� �� y ≤ �� ′ y Examples include Petri nets, VASS =-=[15]-=-, lossy channel systems [3], timed Petri nets [6] to cite a few. ∗ Partially supported by the INRIA ARC ProNoBis. (1) For any subset A of X, let Pre ∃ δ(A) be the preimage {x ∈ X|∃y ∈ A · x δ y}. The ... |

45 |
Timed petri nets and BQOs
- Abdulla, Jonsson
- 2001
(Show Context)
Citation Context ... if x δ y and x ≤ x ′ , then there is a state y ′ such that x ′ δ y ′ and y ≤ y ′ : δ x ≤ �� x ′ δ �� �� y ≤ �� ′ y Examples include Petri nets, VASS [15], lossy channel systems [3], timed Petri nets =-=[6]-=- to cite a few. ∗ Partially supported by the INRIA ARC ProNoBis. (1) For any subset A of X, let Pre ∃ δ(A) be the preimage {x ∈ X|∃y ∈ A · x δ y}. The commutation property ensures that the preimage Pr... |

44 | A classification of symbolic transition systems
- Henzinger, Maujumdar, et al.
- 2005
(Show Context)
Citation Context ...ounded, i.e., has no infinite descending chain, but also has no infinite antichain (a set of incomparable elements). One use of well quasi-orderings is in verifying well-structured transition systems =-=[2, 4, 11, 14]-=-. These are transition systems, usually infinite-state, with two ingredients. First, a well quasi-ordering ≤ on the set X of states. Second, the transition relation δ commutes with ≤, i.e., if x δ y a... |

42 | R.: Symbolic algorithms for infinite-state games
- Alfaro, Henzinger, et al.
(Show Context)
Citation Context ...re ∃∗ δℓ(V ) is a special case of the above evaluation scheme for formulae: Pre ∃∗ δℓ(V ) = I �µX · A ∨ 〈ℓ〉X�δ ρ, where ρ is arbitrary and UA = V . One may also evaluate some forms of monotonic games =-=[1, 9]-=-: reading δℓ1 as the transition relation for player 1, and δℓ2 as that for player 2, the formula µX · A ∨ 〈ℓ1〉(B ∧ [ℓ2]X) is true exactly at those states x0 such that player 1 has a strategy to win (r... |

28 | Algegraic simplification - Buchberger, Loos - 1982 |

28 |
Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudoné) : IV. Étude locale des schémas et des morphismes de schémas, Troisième partie”. Publications Mathématiques de l’IHÉS
- Grothendieck
- 1966
(Show Context)
Citation Context ...e central to our work. Topology and domain theory form another wonderful piece of mathematics, and one may consult [12, 7, 18, 21]. Last but not least, Noetherian spaces arise from algebraic geometry =-=[13]-=-: we discuss this briefly in Section 8. 2. Preliminaries I: Order and Topology A topology O on a set X is a collection of subsets (the opens) of X that is closed under arbitrary unions and finite inte... |

26 | Cartesian Closed Categories of Domains - Jung - 1998 |

24 | Comparing Cartesian closed categories of (core) compactly generated spaces - Escardó, Lawson, et al. - 2004 |

24 |
On better-quasi-ordering transfinite sequences
- Nash-Williams
- 1968
(Show Context)
Citation Context ...x ∈ A such that x ≤ y. It is well-known that ≤ ♯ needs not be well even when ≤ is well. This is a shortcoming, among others, of the theory of well quasi-orderings. Such shortcomings led Nash-Williams =-=[23]-=- to invent better quasi-orderings (bqos). Bqos have a rather unintuitive definition but a wonderful theory, see [19]. The only application of bqos we know of to verification problems is by Abdulla and... |

23 | Polynomial Constants are Decidable
- Müller-Olm, Seidl
- 2002
(Show Context)
Citation Context ... ′ )∈δ (E) ∩ {P ◦ (P1, . . . , Pk) |P ∈ I}, and (E) is the ideal generated by E. This is easily computed using Gröbner bases when K = Q. This was explored by Müller-Olm and Seidl in a very nice paper =-=[22]-=-, and we refer the reader to this for missing details and a gentler introduction. We leave it as future research to explore the application of other Noetherian rings to computer verification problems;... |

21 | Better is better than well: On efficient verification of infinite-state systems
- Abdulla, Nylen
- 2000
(Show Context)
Citation Context ...ent better quasi-orderings (bqos). Bqos have a rather unintuitive definition but a wonderful theory, see [19]. The only application of bqos we know of to verification problems is by Abdulla and Nylén =-=[5]-=-, where it is used to show the termination of the backward reachability iteration, using disjunctive constraints. This paper is not on bqos, and in fact not specifically on well quasi-orderings. While... |

18 |
domain theory and theoretical computer science
- Topology
- 1998
(Show Context)
Citation Context ...ologies, of sober space, of sobrification of a space, and of stably compact spaces are central to our work. Topology and domain theory form another wonderful piece of mathematics, and one may consult =-=[12, 7, 18, 21]-=-. Last but not least, Noetherian spaces arise from algebraic geometry [13]: we discuss this briefly in Section 8. 2. Preliminaries I: Order and Topology A topology O on a set X is a collection of subs... |

16 | Ensuring Completeness of Symbolic Verification Methods for Infinite-State Systems
- Abdulla, Jonsson
(Show Context)
Citation Context |

16 |
Partial well-ordering of sets of vectors
- Rado
- 1954
(Show Context)
Citation Context ...-ordering on X. P(X) and P ∗ (X), with the upper topology of ≤ ♭ , are Noetherian. This is remarkable: in general ≤♭ is not a well quasiordering on P(X). The standard counterexample is Rado’s example =-=[24]-=-. Let XRado be the set {(m, n) ∈ N2 |m < n}, ordered by ≤Rado: (m, n) ≤Rado (m ′ , n ′ ) iff m = m ′ and n ≤ n ′ , or n < m ′ . It is well-known that ≤Rado is a well quasi-ordering. However, H(XRado) ... |

11 |
Foundations of BQO theory
- Marcone
(Show Context)
Citation Context ... others, of the theory of well quasi-orderings. Such shortcomings led Nash-Williams [23] to invent better quasi-orderings (bqos). Bqos have a rather unintuitive definition but a wonderful theory, see =-=[19]-=-. The only application of bqos we know of to verification problems is by Abdulla and Nylén [5], where it is used to show the termination of the backward reachability iteration, using disjunctive const... |

9 |
Deciding monotonic games
- Abdulla, Bouajjani, et al.
- 2003
(Show Context)
Citation Context ...re ∃∗ δℓ(V ) is a special case of the above evaluation scheme for formulae: Pre ∃∗ δℓ(V ) = I �µX · A ∨ 〈ℓ〉X�δ ρ, where ρ is arbitrary and UA = V . One may also evaluate some forms of monotonic games =-=[1, 9]-=-: reading δℓ1 as the transition relation for player 1, and δℓ2 as that for player 2, the formula µX · A ∨ 〈ℓ1〉(B ∧ [ℓ2]X) is true exactly at those states x0 such that player 1 has a strategy to win (r... |

9 |
Stably compact spaces and the probabilistic powerspace construction
- Jung
- 2004
(Show Context)
Citation Context |

8 |
Algebraic posets, algebraic cpo’s and models of concurrency
- Mislove
- 1981
(Show Context)
Citation Context ...ose topology is the upper topology of a wellfounded partial order that has properties W and T. When X is Alexandroff-discrete, S(X) is isomorphic to the ideal completion of X, with its Scott topology =-=[20]-=-. This shows first that, when X is well-quasi ordered, and equipped with its Alexandroff topology, then the upper topology on S(X) is just the familiar Scott topology. Second, this gives a more concre... |

6 |
A new proof of Hilbert’s Nullstellensatz
- Zariski
- 1947
(Show Context)
Citation Context ...hat radical ideals, hence also the closed subsets FI, are in one-to-one correspondence with affine varieties, i.e., with sets of common zeroes of polynomials over K: this is Hilbert’s Nullstellensatz =-=[25]-=-. Now consider polynomial automata over K, i.e., pairs A = (Q, δ), where Q is a finite set of states, R = K[X1, . . .,Xk], and δ ⊆ Q × Pfin(R) × R k × Q is the transition relation. The intent is to mo... |

5 | A note on well quasi-orderings for powersets
- Jančar
- 1999
(Show Context)
Citation Context ...ology of ≤ ♯ , is Noetherian.sAgain, this contrasts with the theory of well-quasi orderings. Rado’s example shows that ≤ ♯ is in general not a well quasi-ordering on Pfin(X). It is when ≤ is ω 2 -wqo =-=[16]-=-. 8. Ring Spectra, and the Zariski Topology Let R be a commutative ring. The spectrum Spec(R) of R is the set of all prime ideals of R. It is equipped with the Zariski topology, whose closed subsets a... |