arXiv subjects
Publications and source records attributed to Olena Hryniv.
Given two elements $x,y$ of a semigroup $X$ we write $x\lesssim y$ if for every homomorphism $χ:X\to\{0,1\}$ we have $χ(x)\leχ(y)$. The quasiorder $\lesssim$ is called the $binary$ $quasiorder$ on $X$. It induces the equivalence relation $\Updownarrow$ that coincides with the least semilattice congruence on $X$. In the paper we discuss some known and new properties of the binary quasiorder on semigroups.
For any metric space $X$ endowed with the action of a group $G$, and two $n$-gons $\vec x=(x_1,\dots,x_n)\in X^n$ and $\vec y=(y_1,\dots,y_n)\in X^n$ in $X$, we introduce the $G$-deviation $d(G\vec x,\vec y\,)$ of $\vec x$ from $\vec y$ as the distance in $X^n$ from $\vec y$ to the $G$-orbit $G\vec x$ of $\vec x$ in the $n$-th power $X^n$ of $X$. For some groups $G$ of affine transformations of the complex plane, we deduce simple-to-apply formulas for calculating the $G$-derviation between $n$-gons on the complex plane. We apply these formulas for defining new measures of asymmetry of triangles. These new measures can be applied in Electric Power Engineering for evaluating the quality of 3-phase electric power. One of such measures, namely the affine deviation, is espressible via the unbalance degree, which is a standard characteristic of quality of three-phase electric power.
We present several known and new results on the Baire category properties in topological groups. In particular, we prove that a Baire topological group $X$ is metrizable if and only if $X$ is point-cosmic if and only if $X$ is a $σ$-space. A topological group $X$ is Choquet if and only if its Raikov completion $\bar X$ is Choquet and $X$ is $G_δ$-dense in $\bar X$. A topological group $X$ is complete-metrizable if and only if $X$ is a point-cosmic Choquet space if and only if $X$ is a Choquet $σ$-space. Finally, we pose several open problem, in particular, if each Choquet topological group is strong Choquet.
Answering a problem posed by the second author on Mathoverflow, we prove that the connected countable Hausdorff spaces constructed by Bing and Ritter are topologically homogeneous.
Two non-empty sets $A,B$ of a metric space $(X,d)$ are called parallel if $d(a,B)=d(A,B)=d(A,b)$ for any points $a\in A$ and $b\in B$. Answering a question posed on Mathoverflow, we prove that for a cover $\mathcal C$ of a metrizable space $X$ the following conditions are equivalent: (i) the topology of $X$ is generated by a metric $d$ such that any two sets $A,B\in\mathcal C$ are parallel; (ii) the cover $\mathcal C$ is disjoint, lower semicontinuous and upper semicontinuous.
We study algebraic and topological properties of the convolution semigroups of probability measures on a topological groups and show that a compact Clifford topological semigroup $S$ embeds into the convolution semigroup $P(G)$ over some topological group $G$ if and only if $S$ embeds into the semigroup $\exp(G)$ of compact subsets of $G$ if and only if $S$ is an inverse semigroup and has zero-dimensional maximal semilattice. We also show that such a Clifford semigroup $S$ embeds into the functor-semigroup $F(G)$ over a suitable compact topological group $G$ for each weakly normal monadic functor $F$ in the category of compacta such that $F(G)$ contains a $G$-invariant element (which is an analogue of the Haar measure on $G$).
We study algebraic and topological properties of subsemigroups of the hyperspace exp(G) of non-empty compact subsets of a topological group G endowed with the Vietoris topology and the natural semigroup operation. On this base we prove that a compact Clifford topological semigroup S is topologically isomorphic to a subsemigroup of exp(G) for a suitable topological group G if and only if S is a topological inverse semigroup with zero-dimensional idempotent semilattice.
For a topological monoid S the dual inverse monoid is the topological monoid of all identity preserving homomorphisms from S to the circle with attached zero. A topological monoid S is defined to be reflexive if the canonical homomorphism from S to its second dual inverse monoid is a topological isomorphism. We prove that a (compact or discrete) topological inverse monoid S is reflexive (if and) only if S is abelian and the idempotent semilattice of S is zero-dimensional. For a discrete (resp. compact) topological monoid its dual inverse monoid is compact (resp. discrete). These results unify the Pontryagin-van Kampen Duality Theorem for abelian groups and the Hofmann-Mislove-Stralka Duality Theorem for zero-dimensional topological semilattices.
We prove that for a complete quasivariety $K$ of topological $E$-algebras of countable discrete signature $E$ and each submetrizable $ANR(k_ω)$-space $X$ its free topological $E$-algebra $F_K(X)$ in the class $K$ is a submetrizable $ANR(k_ω)$-space.