Well-Partial Orderings and their Maximal Order Types

Schmidt, Diana

doi:10.1007/978-3-030-30229-0_13

Cited by 18 publications

(29 citation statements)

References 44 publications

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“…Maximal Order Types. The maximal order types of M (Q), Q <ω , and T (Q) have been studied by Weiermann [2009] and Schmidt [1979]; see also Van der Meeren [2015, Sec. 1.2] for a nice exposition of these results.…”

Section: Computing the Invariants Of Common Wqosmentioning

confidence: 99%

“…5]. For finite sequences with subsequence embedding, we recall the following result by Schmidt [1979].…”

Section: Computing the Invariants Of Common Wqosmentioning

confidence: 99%

“…The case of finite trees is actually a particular case of the results of Schmidt [1979] on embeddings in structured trees. Her results were originally stated using Schütte's Klammer symbols, but can be translated in terms of the ϑ functions of Rathjen and Weiermann [1993].…”

Section: Computing the Invariants Of Common Wqosmentioning

confidence: 99%

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On Ordinal Invariants in Well Quasi Orders and Finite Antichain Orders

2020

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We investigate the ordinal invariants height, length, and width of well quasi orders (WQO), with particular emphasis on width, an invariant of interest for the larger class of orders with finite antichain condition (FAC). We show that the width in the class of FAC orders is completely determined by the width in the class of WQOs, in the sense that if we know how to calculate the width of any WQO then we have a procedure to calculate the width of any given FAC order. We show how the width of WQO orders obtained via some classical constructions can sometimes be computed in a compositional way. In particular this allows proving that every ordinal can be obtained as the width of some WQO poset. One of the difficult questions is to give a complete formula for the width of Cartesian products of WQOs. Even the width of the product of two ordinals is only known through a complex recursive formula. Although we have not given a complete answer to this question we have advanced the state of knowledge by considering some more complex special cases and in particular by calculating the width of certain products containing three factors. In the course of writing the paper we have discovered that some of the relevant literature was written on cross-purposes and some of the notions re-discovered several times. Therefore we also use the occasion to give a unified presentation of the known results.

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Section: Computing the Invariants Of Common Wqosmentioning

confidence: 99%

“…5]. For finite sequences with subsequence embedding, we recall the following result by Schmidt [1979].…”

Section: Computing the Invariants Of Common Wqosmentioning

confidence: 99%

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On Ordinal Invariants in Well Quasi Orders and Finite Antichain Orders

2020

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“…The computation of o(P ) for natural WPOs turned out an interesting and challenging task. D. Scmidt [91] computed the maximal order type of the Higman's WPO (k * ; ≤ * ) and gave upper bounds on the maximal order types of some other important WPOs including that of Kruskal's.…”

Section: Well Quasiordersmentioning

confidence: 99%

“…To our knowledge, the maximal order types of the other above-mentioned concrete WPOs are still unknown. Also the problem of relating rk(P ) and o(P ) discussed in [91] seems still to be open. In particular, there is no known characterization of pairs of ordinals (α, β) such that α = rk(P ) and β = o(P ) for some WPO P [73].…”

Section: Well Quasiordersmentioning

confidence: 99%

Well-Quasi Orders and Hierarchy Theory

Selivanov

2020

Trends in Logic

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We discuss some applications of WQOs to several fields were hierarchies and reducibilities are the principal classification tools, notably to Descriptive Set Theory, Computability theory and Automata Theory. While the classical hierarchies of sets usually degenerate to structures very close to ordinals, the extension of them to functions requires more complicated WQOs, and the same applies to reducibilities. We survey some results obtained so far and discuss open problems and possible research directions.

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Upper Bounds on the Graph Minor Theorem

Krombholz

Rathjen

2020

Trends in Logic

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Lower bounds on the proof-theoretic strength of the graph minor theorem were found over 30 years ago by Friedman, Robertson, and Seymour (1987), but upper bounds have always been elusive. We present recently found upper bounds on the graph minor theorem and other theorems appearing in the Graph Minors series. Further, we give some ideas as to how the lower bounds on some of these theorems might be improved.

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Well-Partial Orderings and their Maximal Order Types

Cited by 18 publications

References 44 publications

On Ordinal Invariants in Well Quasi Orders and Finite Antichain Orders

On Ordinal Invariants in Well Quasi Orders and Finite Antichain Orders

Well-Quasi Orders and Hierarchy Theory

Upper Bounds on the Graph Minor Theorem

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