Abstract
AbstractA 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 S 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 ∪ Vx) ∩ V ≠ ∅ belongs to the neighborhood U of 1.
Highlights
In this paper natural numbers are positive integers and a “space" means a “topological space"
We show that a locally compact monothetic topological monoid S 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 of S and for any element a ∈ S there exists a neighborhood V of a such that any element x ∈ S with ∩ V ≠ ∅ belongs to the neighborhood U of 1
Summary
In this paper natural numbers are positive integers and a “space" means a “topological space". Whether a counterpart of Pontrjagin’s theorem holds for locally compact monothetic topological monoids was a well-known problem, posed by Koch [4]. Remark that if such a monoid is compact, it is a (topological) group [4], [5]. It is well-known (see, for instance, [10]) that a commutative cancellative semigroup can be embedded into the group of its quotients This algebraic construction can has no obvious topological counterparts. The last property is a simple condition expressed in terms of elements and neighborhoods This notion turned out to be useful in obtaining a positive answer to the Koch problem for a partial case. A monothetic non-viscous topological monoid S is cancellative (see Proposition 3.5) and if S is locally compact, it is a compact topological group, see Theorem 3.9
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