Partition Theorems for Layered Partial Semigroups

Farah, Ilijas; Hindman, Neil; McLeod, Jillian

doi:10.1006/jcta.2001.3237

Cited by 8 publications

(36 citation statements)

References 11 publications

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Section: Introductionmentioning

confidence: 93%

“…A broad generalization of Gowers' theorem has been proved by Farah, Hindman, and McLeod in [3,Theorem 3.13] in the framework, developed therein, of layered partial semigroups and layered actions. Such a result provides, in particular, a common generalization of Gowers' theorem and the Hales-Jewett theorem; see [3,Theorem 3.15]. As general as [3, Theorem 3.13] is, it nonetheless does not cover the case where one considers FIN k endowed with the multiple tetris operations described below, since these do not form a layered action in the sense of [3,Definition 3.3].…”

Section: Introductionmentioning

confidence: 93%

“…Such a result provides, in particular, a common generalization of Gowers' theorem and the Hales-Jewett theorem; see [3,Theorem 3.15]. As general as [3, Theorem 3.13] is, it nonetheless does not cover the case where one considers FIN k endowed with the multiple tetris operations described below, since these do not form a layered action in the sense of [3,Definition 3.3].…”

Section: Introductionmentioning

confidence: 99%

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Gowers' Ramsey Theorem for generalized tetris operations

Lupini

2017

Journal of Combinatorial Theory, Series A

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We prove a generalization of Gowers' theorem for FIN k where, instead of the single tetris operation T : FIN k → FIN k−1 , one considers all maps from FIN k to FINj for 0 ≤ j ≤ k arising from nondecreasing surjections f : {0, 1, . . . , k + 1} → {0, 1, . . . , j + 1}. This answers a question of Bartošová and Kwiatkowska. We also prove a common generalization of such a result and the Galvin-Glazer-Hindman theorem on finite products, in the setting of layered partial semigroups introduced by Farah, Hindman, and McLeod.

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Section: Introductionmentioning

confidence: 93%

Section: Introductionmentioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

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Gowers' Ramsey Theorem for generalized tetris operations

Lupini

2017

Journal of Combinatorial Theory, Series A

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“…A common generalization of Gowers' Ramsey theorem and the infinite Hales-Jewett theorems has been established by Farah-Hindman-McLeod in the setting of layered actions on adequate partial semigroups [10]. In a different direction, the infinite Gowers Ramsey theorem has been strengthened in [18] by considering multiple tetris operations.…”

Section: Introductionmentioning

confidence: 99%

“…and a i,d ∈ L i for d ∈ ω and i ∈ k is monochromatic.More generally, one can regard any layered action on a partial semigroup S in the sense of[10, Definition 3.3] as an action of the tree I k on S in the sense of Definition 3.1. Any such an action is automatically Ramsey, although this is not easy to see directly.…”

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confidence: 99%

Actions on semigroups and an infinitary Gowers–Hales–Jewett Ramsey theorem

Lupini

2018

Trans. Amer. Math. Soc.

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We introduce the notion of (Ramsey) action of a tree on a (filtered) semigroup. We then prove in this setting a general result providing a common generalization of the infinitary Gowers Ramsey theorem for multiple tetris operations, the infinitary Hales-Jewett theorems (for both located and nonlocated words), and the Farah-Hindman-McLeod Ramsey theorem for layered actions on partial semigroups. We also establish a polynomial version of our main result, recovering the polynomial Milliken-Taylor theorem of Bergelson-Hindman-Williams as a particular case. We present applications of our Ramsey-theoretic results to the structure of delta sets in amenable groups.2000 Mathematics Subject Classification. Primary 05D10, 54D80; Secondary 20M99, 05C05, 06A06. . 2.2.Actions of trees on compact right topological semigroups. Suppose that P is an ordered set, and X is a compact right topological semigroup.Definition 2.2. An action α of P on X is given by• an order-preserving function P → S (X), t → X t ,• a subsemigroup F α ⊂ End (X), such that for every τ ∈ F α there exists an function f τ : P → P-which we call the spine of τ -such that τ maps X t to X fτ (t) for every t ∈ P, and such that τ (x) = x for any x ∈ X t and t ∈ P such that f τ (t) = t.Given an action α of P on X we let X α be set of functions ξ : P → X such that ξ (t) ∈ X t and τ • ξ = ξ • f τ for every τ ∈ F α and t ∈ P. When X α is nonempty, we endow X α with the product topology and the entrywise operation. This turns X α into a compact right topological semigroup. Observe that an idempotent in X α is an element ξ of X α such that ξ (t) is an idempotent element of X t for every t ∈ P. We say that an idempotent ξ inSuppose now that T is a rooted tree. We regard T as an ordered set endowed with the canonical rooted tree order obtained by setting t ′ ≤ t if and only if t ′ is a descendent of t.and f maps two adjacent nodes either to the same node or to adjacent nodes.It is clear that any regressive homomorphism fixes the root, and maps every branch to itself.Definition 2.4. A Ramsey action α of T on X is given by an action of T on X in the sense of Definition 2.2 such that X α is nonempty and, for every τ ∈ F α , the corresponding spine f τ : T → T is a regressive homomorphism.A similar proof as [18, Lemma 2.1] shows the following.Proposition 2.5. Suppose that T is a rooted tree of height ≤ ω with root r. If α is a Ramsey action of T on X, then X α contains an order-preserving idempotent. Furthermore, if ξ (0) is an idempotent element of X α , then X α contains an order-preserving idempotent ξ such that ξ (r) = ξ (0) (r).Proof. Fix an idempotent element ξ (0) of X α . Let, for k ∈ ω, π k : T → T be the function that maps every node to its k-th predecessor, where we convene that the k-th predecessor of a node of height at most k is the root, and the 0-th predecessor of every node is itself. Let T k be the set of nodes in T of height at most k. We define by recursion on k ∈ ω idempotent elements ξ (k) of X α such that ξ (k) (t) + ξ (k) (t 0 ) = ξ (k) (t) whenever t 0...

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Recent Progress in the Topological Theory of Semigroups and the Algebra of βS

Hindman¹,

Strauss²

2002

Recent Progress in General Topology II

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Partition Theorems for Layered Partial Semigroups

Cited by 8 publications

References 11 publications

Gowers' Ramsey Theorem for generalized tetris operations

Gowers' Ramsey Theorem for generalized tetris operations

Actions on semigroups and an infinitary Gowers–Hales–Jewett Ramsey theorem

Recent Progress in the Topological Theory of Semigroups and the Algebra of βS

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