Topologizations of a set endowed with an action of a monoid

Банах, Тарас; Протасов, Ігор; Sipacheva, Olga

doi:10.1016/j.topol.2014.02.041

Cited by 9 publications

(8 citation statements)

References 18 publications

(20 reference statements)

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“…By (2) there is a cofinal subsequence { V γ δ ,1 } δ<cf(κ) of the corresponding γ's. As we already know it is a descending sequence of neighbourhoods of d. By (5) each V γ δ ,1 is not a singleton.…”

Section: Then a Is Topologizable And The Pseudocharacter Of This Topomentioning

confidence: 99%

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Topologizable structures and Zariski topology

Dutka

Ivanov

2018

Algebra Univers.

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Section: Then a Is Topologizable And The Pseudocharacter Of This Topomentioning

confidence: 99%

“…At the moment this topic has become well established, having well-known achievements (for example [6,8]). Paper [2,4] have nice descriptions of the topic and rich bibliography.…”

Section: Introductionmentioning

confidence: 99%

Topologizable structures and Zariski topology

Dutka

Ivanov

2018

Algebra Univers.

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“…That countable algebras admit Hausdorff topologizations iff their Zariski topologies are non-discrete, was shown for unoids (i.e. algebras with arbitrary families of unary operations) in [45]. In [46] this fact was announced for all universal algebras; the proof was provided in [32] and completed in [47].…”

Section: Topologizations Of Algebrasmentioning

confidence: 99%

Hindman's finite sums theorem and its application to topologizations of algebras

Saveliev¹

2018

Preprint

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The first part of the paper is a brief overview of Hindman's finite sums theorem, its prehistory and a few of its further generalizations, and a modern technique used in proving these and similar results, which is based on idempotent ultrafilters in ultrafilter extensions of semigroups. The second, main part of the paper is devoted to the topologizability problem of a wide class of algebraic structures called polyrings; this class includes Abelian groups, rings, modules, algebras over a ring, differential rings, and others. We show that the Zariski topology on such an algebra is always non-discrete. Actually, a much stronger fact holds: if K is an infinite polyring, n a natural number, and a map F of K n into K is defined by a term in n variables, then F is a closed nowhere dense subset of the space K n`1 with its Zariski topology. In particular, K n is a closed nowhere dense subset of K n`1 . The proof essentially uses a multidimensional version of Hindman's finite sums theorem established by Bergelson and Hindman. The third part of the paper lists some problems concerning topologization of various algebraic structures, their Zariski topologies, and some related questions.This paper is an extended version of the lecture at Journées sur les Arithmétiques Faibles 36: à l'occasion du 70ème anniversaire de Yuri Matiyasevich, delivered on 7th July, 2017, in Saint Petersburg.

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“…The semi-Zariski topology Z s on a semigroup S is the topology generated by the sub-base consisting of the sets {x ∈ S : axb = c} and {x ∈ S : axb = cxd} where a, b, c, d ∈ S 1 and S 1 = S ∪ {1} stands for the semigroup S with attached external unit 1 / ∈ S (i.e., an element 1 such that 1x = x1 = x for all x ∈ S 1 ). The semi-Zariski topology Z s is a particular case of algebraically determined topologies on G-acts, considered in [2].…”

Section: The Topological Structure Of Supt-perfect and Perfectly Suppmentioning

confidence: 99%

Perfectly supportable semigroups are σ-discrete in each Hausdorff shift-invariant topology

Банах

Guran

2013

Topological Algebra and Its Applications

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In this paper we introduce perfectly supportable semigroups and prove that they are σ-discrete in each Hausdorff shift-invariant topology. The class of perfectly supportable semigroups includes each semigroup S such that FSym(X) ⊂ S ⊂ FRel(X) where FRel(X) is the semigroup of finitely supported relations on an infinite set X and FSym(X) is the group of finitely supported permutations of X. a∈S supt(a) ⊂ X is called the support of S. A typical example of a supt-semigroup is the group Sym(X) of all bijections f : X → X of a set X, endowed with the support map supt : f → {x ∈ X : f (x) = x}. In Section 4 we shall describe another supt-semigroup Rel(X), which contains Sym(X) (and many other semigroups) as a supt-subsemigroup.Definition 2.1 implies the following proposition-definition.1991 Mathematics Subject Classification. 20M20, 20B35, 22A15, 22A05, 54D45. Key words and phrases. Semi-Zariski topology, supt-perfect semigroup, σ-discrete space, the group of finitely supported permutations, the semigroup of finitely supported relations.

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Topologizations of a set endowed with an action of a monoid

Cited by 9 publications

References 18 publications

Topologizable structures and Zariski topology

Topologizable structures and Zariski topology

Hindman's finite sums theorem and its application to topologizations of algebras

Perfectly supportable semigroups are σ-discrete in each Hausdorff shift-invariant topology

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