Subgroups of paratopological groups and feebly compact groups

Totally Lindelöf and totally ω-narrow paratopological groups

Given a T0 paratopological group G and a class C of continuous homomorphisms of paratopological groups, we define the C-semicompletionC[G) and C-completionC[G] of the group G that contain G as a dense subgroup, satisfy the T0-separation axiom and have certain universality properties. For special classes C, we present some necessary and sufficient conditions on G in order that the (semi)completions C[G) and C[G] be Hausdorff. Also, we give an example of a Hausdorff paratopological abelian group G whose C-semicompletion C[G) fails to be a T1-space, where C is the class of continuous homomorphisms of sequentially compact topological groups to paratopological groups. In particular, the group G contains an ω-bounded sequentially compact subgroup H such that H is a topological group but its closure in G fails to be a subgroup.

Free Abelian paratopological groups over metric spaces

Some three-space properties of semitopological and paratopological groups

On bM-ω-balancedness and [formula omitted]-factorizability of para(semi)topological groups

In this article, we introduce fuzzy paratopological group, fuzzy semitopological group and fuzzy quasitopological group with illustrative examples and properties. These new notions belong to fuzzy topological group. Here, we prove that each fuzzy regular paratopological group is completely regular by using fuzzy uniformities. Moreover, we prove some results related to fuzzy semitopological group and fuzzy quasitopological group. In addition, we provide an application in the area of decision making during robot crash by using our above stated notions and nano topology.

We define a category containing the discrete quantum groups (and hence the discrete groups and the duals of compact groups) and the compact quantum groups (and hence the compact groups and the duals of discrete groups). The dual of an object can be defined within the same category and we have a biduality theorem. This theory extends the duality between compact quantum groups and discrete quantum groups (and hence the one between compact abelian groups and discrete abelian groups). The objects in our category are multiplier Hopf algebras, with invertible antipode, admitting invariant functionals (integrals), satisfying some extra condition (to take care of the non-abelianness of the underlying algebras). If we start with a multiplier Hopf ∗-algebra with positive invariant functionals, then also the dual is a multiplier Hopf ∗-algebra with positive invariant functionals. This makes it possible to formulate this duality also within the framework of C∗-algebras.

We study Pontryagin reflexivity in the class of precompact topological Abelian groups. We find reflexive groups among precompact not pseudocompact and among pseudocompact not compact groups. Making use of Martin's axiom we give an example of a reflexive countably compact not compact Abelian group. We also prove that every pseudocompact Abelian group is a quotient of a reflexive pseudocompact group with respect to a closed reflexive pseudocompact subgroup.

In this paper the structure of all open semigroups in any compact topological group is determined. We can then extend this result to a classification of the angular semigroups of any abelian topological group H which contains a compact open subgroup K. Coupling this with the results for Euclidean n-space, we obtain an essentially complete description of the angular semigroups of an arbitrary locally compact group. For compact groups, the result is remarkably simple.

Notes on compact Discrete Analysis 2017:16, 57 pp. This is the second paper in a two-part series. The first paper, [also published in this journal](http://discreteanalysisjournal.com/article/2105-notes-on-nilspaces-algebraic-aspects), developed the algebraic theory of nilspaces, which are an abstraction both of the notion of a nilmanifold, and of that of an abelian group (both with their attendant parallelogram structures). In this paper, the author applies this algebraic theory to study compact which can be viewed as a generalization of the notion of a compact abelian group. The main motivation for this is that such compact nilspaces arise naturally when trying to attack additive combinatorial problems such as the inverse conjecture for the Gowers norms via the methods of nonstandard analysis (or ultraproducts), as was demonstrated in work of Szegedy ([On higher order Fourier analysis](https://arxiv.org/abs/1203.2260)). The algebraic theory in the first part lets one describe $k$-step compact nilspaces as abelian extensions of $k-1$-step with the extension given by a cocycle. Among other things, this can be used to assign to each nilspace a probability measure that is analogous to the Haar probability measure one can assign to a compact group. However, it is not immediately obvious that this cocycle has any good measurability or continuity properties. Much of the delicate analysis in this paper (particularly those concerning the properties of continuous systems of measures) is devoted to ensuring that such good properties can be imposed (locally at least) on the cocycle, perhaps after modifying the cocycle by a suitable coboundary that does not affect the isomorphism class of the extension. Indeed, one can view the paper as developing some basic for although the formidable machinery of modern cohomology is not really deployed here. As with the first paper in this series, the paper closely follows the previous work of Camarena and Szegedy, but with many more details provided in the arguments. (An alternative approach to this theory has also recently been worked out in a series of papers by Gutman, Manners, and Varju.) A full structural description of compact nilspaces seems to be quite difficult, but in this paper two important partial descriptions are obtained. The first is to describe every compact nilspace as an inverse limit of compact nilspaces of rank. This is somewhat analogous to how the Peter-Weyl theorem can be used to describe a compact group as the inverse limit of finite-dimensional compact Lie groups. The second is to show that every connected compact nilspace of finite rank is isomorphic to a nilmanifold. These results were previously obtained by Camarena and Szegedy (and later used by Szegedy to provide a new proof of the inverse conjecture for the Gowers uniformity norms), but the author here reproves these results with much more detailed proofs, in particular treating with some care certain steps that were only briefly discussed in the prior paper of Camarena and Szegedy.

In Pontrjagin's theory of duality for compact abelian groups, the following theorem is known : Let G be a compact abelian group, G* the dual group. Then the topological dimension of G, in the sense of Lebesgue, is equal to the rank of discrete abelian group G*. It was Prof. T. Tannaka who has called my attention to the lack of corresponding theorem in non-commutative case. I intend to give, in this note, a theorem of this kind in the following form: THEOREM A. Let G be an arbitrary compact group, G the aggregate of continuous finite dimensional representstions of G, C[ G ] the algebra over the complex numbers C generated by the coefficients of representations in G, i.e., the representative ring of G in the sense of C. Chevalley^. Then the topological dimension of G, in the sense of Lebesgue, is equal to the transcendental degree of C[G] over C.

We prove that the category M of abelian groups with a Polish cover introduced in collaboration with Bergfalk and Panagiotopoulos is the left heart of (the derived category of) the quasi-abelian category A of abelian Polish groups in the sense of Beilinson–Bernstein–Deligne and Schneiders. Thus, M is an abelian category containing A as a full subcategory such that the inclusion functor A→M is exact and finitely continuous. Furthermore, M is uniquely characterized up to equivalence by the following universal property: for every abelian category B, a functor A→B is exact and finitely continuous if and only if it extends to an exact and finitely continuous functor M→B. In particular, this provides a description of the left heart of A as a concrete category.We provide similar descriptions of the left heart of a number of categories of algebraic structures endowed with a topology, including: non-Archimedean abelian Polish groups; locally compact abelian Polish groups; totally disconnected locally compact abelian Polish groups; Polish R-modules, for a given Polish group or Polish ring R; and separable Banach spaces and separable Fréchet spaces over a separable complete non-Archimedean valued field.

Given a locally compact group G, G L U C is the largest semigroup compactification of G and G * = G L U C ∖ G . We show that (i) for every locally compact compactly generated Abelian group G and for every p , q ∈ G * , the multiplication in G L U C is discontinuous at ( p , q ) , (ii) there is a locally compact σ-compact torsion-free Abelian group G for which, assuming Martin's Axiom, there are p , q ∈ G * such that the multiplication in G L U C is continuous at ( p , q ) , and (iii) it is consistent with ZFC that for every locally compact Abelian group G and for every p , q ∈ G * , the left translation by p in G * is discontinuous at q.

Given a compact group G and r ∈ N, let sr(G) denote the least upper bound of real ϵ > 0 such that for every measurable r-coloring of G, there exists a monochrome symmetric subset of measure ≥ ϵ. A subset A ⊆ G is symmetric if there exists g ∈ G such that gA −1 g = A. We give a general picture of asymptotic behaviour of the function sr(G) for compact Abelian groups.

We study the existence of many nonclosed pure subgroups of nondiscrete locally compact abelian groups. It is shown that every nondiscrete locally compact abelian group has uncountably many nonclosed pure subgroups. This in particular solves a question of Armacost. It is also shown that, if G G is a nondiscrete locally compact abelian group and if either G G is a compact group or the torsion part t ( G ) t\left ( G \right ) of G G is nonopen, then G G has 2 c {2^c} proper dense pure subgroups, where c c denotes the power of the continuum. This in particular gives a partial answer to another question of Armacost.