Countably compact sequential topological semigroup with two-sides cancellations is a topological group

We prove that the existence of a selective ultrafilter implies the existence of a countably compact Hausdorff group topology on the free Abelian group of size continuum. As a consequence, we show that the existence of a selective ultrafilter implies the existence of a Wallace semigroup (i.e., a countably compact both-sided cancellative topological semigroup which is not a topological group).

A semitopological group (topological group) is a group endowed with a topology for which multiplication is separately continuous (multiplication is jointly continuous and inversion is continuous). In this paper we answer (Arhangel’skii et al., Math Maced 8:1–19, 2010, Problem 10.4), by showing that if (G, ·, τ) is a semitopological group and (G, τ) is homeomorphic to a product of Cech-complete spaces, then (G, ·, τ) is a topological group.

A semigroup S is called an absolute coretract if for any continuous homomorphism f from a compact Hausdorff right topological semigroup T onto a compact Hausdorff right topological semigroup containing S algebraically there exists a homomorphism g \colon S→ T such that f\circ g=idS. The semigroup β\ben contains isomorphic copies of any countable absolute coretract. In this article we define a class C of semigroups of idempotents each of which is a decreasing chain of rectangular semigroups. It is proved that every semigroup from C is an absolute coretract and every finite semigroup of idempotents, which is an absolute coretract, belongs to C .

Introduction. Limit properties of the convolution sequence of a regular measure on a compact topological semigroup are examined in this paper. Similar questions, as they arise in the case of a compact group, were examined byKAWADA & ITO [4], Recently BELLMAN [1] and GRENANDER [3] considered special limit theorems for products of independent identically distributed random operators. Such problems are closely related to those in this paper. It should be noted that similar questions arise when considering the structure of stationary stochastic processes [7]. Various results on compact semigroups are used in characterizing the class of limit measures [5], [6], [8].

We investigate involutions and trivolutions in the second dual of algebras related to a locally compact topological semigroup and the Fourier algebra of a locally compact group. We prove, among the other things, that for a large class of topological semigroups namely, compactly cancellative foundation $$*$$ -semigroup S when it is infinite non-discrete cancellative, $$M_a(S)^{**}$$ does not admit an involution, and $$M_a(S)^{**}$$ has a trivolution with range $$M_a(S)$$ if and only if S is discrete. We also show that when G is an amenable group, the second dual of the Fourier algebra of G admits an involution extending one of the natural involutions of A(G) if and only if G is finite. However, $$A(G)^{**}$$ always admits trivolution.

In this note, we firstly discuss some properties of spaces which are countably sieve-complete, densely q-complete and strongly Baire. By some known conclusions, we finally show that if G is a regular countably sieve-complete semitopological group then G is a topological group. If a regular semitopological group G has a dense subgroup which is countably sieve-complete (densely q-complete), then G is a topological group. If G is a regular countably sieve-complete semitopological group then G is a D-space if and only if G is paracompact. We point out that some conditions in Theorem 2.14 and Corollary 2.15 in [17] are not essential.

The known theory for a discrete oid \(T\) shows that how to find a subset \(T^{\infty}\) of \(\beta T\) which is a compact right topological semigroup (see section 2 for details).In this paper we try to find an analogue of almost periodic functions for oids. We discover, new compact semigroups by using a certain subspace of functions \(u^{\infty}(T)\) of \(C(T)\) for an oid \(T\) for which \(f\beta\) is continuous on \(T^{\infty}\times(T\cup T^{\infty}\cup TT^{\infty})\),where \((T\cup T^{\infty}\cup TT^{\infty})\) is a suitable subspace of \(\beta T\) for a wide range.

Totally Lindelöf and totally ω-narrow paratopological groups

ABSTRACT. Some results of Mukherjea and Tserpes are generalized by showing that any sequentially compact, cancellative topological semigroup is a topological group. Hence, any countably compact, cancellative topological semigroup with any additional condition that would imply sequential compactness is also a topological group. Finally, it is shown that any w‐bounded, cancellative topological semigroup is also a topological group.

were discussed in [5]. Let Z(v) be the support (or spectrum) of V. One can just as well take S as the closure of Un (,(V))n = S(V) since all the convolutions v(i) are concentrated on S(v). It was shown that limn zv, =IL exists in the sense that limn,f ffdVn =ffd,u for every continuous f on S. Further, ,u is regular with support the kernel K (minimal two-sided ideal) of S(v) and is idempotent, that is,

In this paper we present a number of sufficient conditions on a sequence of probability measuresµ n on a locally compact (second countable) Hausdorff topological semigroupS that guarantee the weak convergence of the sequence of convolution productsµ k,n ≡µ k + 1 *···*µ n (k<n), asn→∞, for allk≥0.

LetS be a uniform semigroup (this includes all topological groups and affine semigroups). We show that a certain space of uniformly continuous functions onS has a left invariant mean iffS has the fixed point property for uniformly continuous affine actions ofS on compact convex sets. This is closely related to but independent of the results of T. Mitchell in [13] and A. Lau in [10]. Interesting examples and consequences are given for the special cases of topological groups and affine convolution semigroups of probability measures on a locally compact semigroup or group.

In this paper, it is shown that every compact Hausdorff $K$-space has countable tightness. This result gives a positive answer to a problem posed by Malykhin and Tironi [‘Weakly Fréchet–Urysohn and Pytkeev spaces’, Topology Appl.104 (2000), 181–190]. We show that a semitopological group $G$ that is a $K$-space is first countable if and only if $G$ is of point-countable type. It is proved that if a topological group $G$ is a $K$-space and has a locally paracompact remainder in some Hausdorff compactification, then $G$ is metrisable.

The Ellis theorem and continuity in groups

M. Sion and T. Traynor investigated ([15]-[17]), measures and integrals having values in topological groups or semigroups. Their definition of integrability was a modification of Phillips-Rickart bilinear vector integrals, in locally convex topological vector spaces.The purpose of this paper is to develop a good notion of an integration process in partially ordered groups, based on their order structure. The results obtained generalize some of the results of J. D. M. Wright ([19]-[22]) where the measurable functions are real-valued and the measures take values in partially ordered vector spaces.Let if be a σ-algebra of subsets of T, X a lattice group, Y, Z partially ordered groups and m : H → F a F-valued measure on H. By F(T, X), M(T, X), E(T, X) and S(T, X) are denoted the lattice group of functions with domain T and with range X, the lattice group of (H, m)-measurable functions of F(T, X) and the lattice group of (H, m)-elementary measurable functions of F(T, X) and the lattice group of (H, m)-simple measurable functions of F(T, X) respectively.