In this paper we answer several questions of Dikran Dikranjan about
algebraically determined topologies on the group of (finitely supported)
permutations of a set X.Comment: 10 page
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.
A topological semigroup is monothetic provided it contains a dense cyclic subsemigroup. The Koch problem asks whether every locally compact monothetic monoid is compact. This problem was opened for more than sixty years, till in 2018 Zelenyuk obtained a negative answer. In this paper we obtain a positive answer for Koch's problem for some special classes of topological monoids. Namely, we show that a locally compact monothetic topological monoid is a compact topological group if and only if S is a submonoid of a quasitopological group if and only if S has open shifts if and only if S is non-viscous in the sense of Averbukh. The last condition means that any neighborhood U of the identity 1 of S and for any element a ∈ S there exists a neighborhood V of a such that any element x ∈ S with (xV ∪ V x) ∩ V = ∅ belongs to the neighborhood U of 1.
We prove that a Hausdorff paratopological group G is meager if and only if there are a nowhere dense subset A ⊂ G and a countable set C ⊂ G such that CA = G = AC.2010 MSC: 22A05, 22A30.
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