Quasi-multipliers on topological semigroups and their Stone–Čech compactification

In this paper, we introduce and study the notion of quasi-multipliers on a semi-topological semigroup [Formula: see text]. The set of all quasi-multipliers on [Formula: see text] is denoted by [Formula: see text]. First, we study the problem of extension of quasi-multipliers on topological semigroups to its Stone–Čech compactification. Indeed, we prove if [Formula: see text] is a topological semigroup such that [Formula: see text] is pseudocompact, then [Formula: see text] can be regarded as a subset of [Formula: see text] Moreover, with an extra condition we describe [Formula: see text] as a quotient subsemigroup of [Formula: see text] Finally, we investigate quasi-multipliers on topological semigroups, its relationship with multipliers and give some concrete examples.

Closed subsets of compact-like topological spaces

<p>We investigate closed subsets (subsemigroups, resp.) of compact-like topological spaces (semigroups, resp.). We show that each Hausdorff topological space is a closed subspace of some Hausdorff ω-bounded pracompact topological space and describe open dense subspaces of<br />countably pracompact topological spaces. We construct a pseudocompact topological semigroup which contains the bicyclic monoid as a closed subsemigroup. This example provides an affirmative answer to a question posed by Banakh, Dimitrova, and Gutik in [4]. Also, we show that the semigroup of ω×ω-matrix units cannot be embedded into a Hausdorff topological semigroup whose space is weakly H-closed.</p>

Commutative Topological Semigroups Embedded into Topological Abelian Groups

In this paper, we give conditions under which a commutative topological semigroup can be embedded algebraically and topologically into a compact topological Abelian group. We prove that every feebly compact regular first countable cancellative commutative topological semigroup with open shifts is a topological group, as well as every connected locally compact Hausdorff cancellative commutative topological monoid with open shifts. Finally, we use these results to give sufficient conditions on a commutative topological semigroup that guarantee it to have countable cellularity.

Positive answers to Koch’s problem in special cases

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 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.

The Levi-Civita equation in function classes

Let G be an Abelian topological semigroup with unit. By a classical result (called Theorem A), if V is a finite dimensional translation invariant linear space of complex valued continuous functions defined on G, then every element of V is an exponential polynomial. More precisely, every element of V is of the form $$\sum _{i=1}^np_i \cdot m_i$$ ∑ i = 1 n p i · m i , where $$m_1 ,\ldots ,m_n$$ m 1 , … , m n are exponentials belonging to V, and $$p_1 ,\ldots ,p_n$$ p 1 , … , p n are polynomials of continuous additive functions. We generalize this statement by replacing the set of continuous functions by any algebra $${{\mathcal {A}}}$$ A of complex valued functions such that whenever an exponential m belongs to $${{\mathcal {A}}}$$ A , then $$m^{-1}\in {{\mathcal {A}}}$$ m - 1 ∈ A . As special cases we find that Theorem A remains valid even if the topology on G is not compatible with the operation on G, or if the set of continuous functions is replaced by the set of measurable functions with respect to an arbitrary $$\sigma $$ σ -algebra. We give two proofs of the result. The first is based on Theorem A. The second proof is independent, and seems to be more elementary than the existing proofs of Theorem A.

Topological semigroups and their prequantale models

In this paper, we introduce a condition (?) on topological semigroups, and prove that every T1 topological semigroup satisfying condition (?) has a bounded complete algebraic prequantale model. On the basis of this result, we also show that every T0 topological semigroup satisfying condition (?) can be embedded into a compact and locally compact sober topological semigroup.