Yevhen Zelenyuk

Weak projectives of finite semigroups

Abstract It is described weak projectives in the category of finite semigroups. These are precisely finite weak projectives in the category of compact right topological semigroups.

Closed Left Ideal Decompositions of U(G)

Let G be an infinite discrete group and let βG be the Stone–Čech compactification of G. We take the points of βG to be the ultrafilters on G, identifying the principal ultrafilters with the points of G. The set U (G) of uniform ultrafilters on G is a closed two-sided ideal of βG. For every p ∈ U (G), define Ip ⊆ βG by Ip = ⋂ A∈p cl(GU (A)), where U (A) = {p ∈ U (G) : A ∈ p}. We show that if |G| is a regular cardinal, then {Ip : p ∈ U (G)} is the finest decomposition of U (G) into closed left ideals of βG such that the corresponding quotient space of U (G) is Hausdorff.

Local homomorphisms of topological groups

A mapping f:GS from a left topological group G into a semigroup S is a local homomorphism if for every xGe , there is a neighborhood Ux of e such that f(xy)=f(x)f(y) for all yUxe . A local homomorphism f:GS is onto if for every neighborhood U of e, f(Ue)=S . We show that every countable regular left topological group containing a discrete subset with exactly one accumulation point admits a local homomorphism onto ; it is consistent that every countable topological group containing a discrete subset with exactly one accumulation point admits a local homomorphism onto any countable semigroup; it is consistent that every countable nondiscrete maximally almost periodic topological group admits a local homomorphism onto the countably infinite right zero semigroup.

On group operations on homogeneous spaces

It is proved that every countably infinite homogeneous regular space admits a structure of any countably infinite group with continuous left shifts.

On topologizing groups

Abstract We prove that every infinite group admits a non-discrete zero-dimensional Hausdorff topology with continuous translations and inversion.

Transformations and Colorings of Groups

Let G be a compact topological group and let f : G → G be a continuous transformation of G. Define f �: G → G by f �(x) = f(x 1)x and let µ = µG be Haar measure on G. Assume that H = Im fis a subgroup of G and for every measurable C ⊆ H, µG(( f �) 1(C)) = µH(C). Then for every measurable C ⊆ G, there exist S ⊆ C and g ∈ G such that f(Sg 1 ) ⊆ Cg 1 and µ(S) ≥ (µ(C)) 2 .

Large rectangular semigroups in Stone-Cech compactifications

We show that large rectangular semigroups can be found in certain Stone-Cech compactifications. In particular, there are copies of the 2 c × 2 c rectangular semigroup in the smallest ideal of (βN, +), and so, a semigroup consisting of idempotents can be embedded in the smallest ideal of (βN, +) if and only if it is a subsemigroup of the 2 c x 2 c rectangular semigroup. In fact, we show that for any ordinal λ with cardinality at most c, βN contains a semigroup of idempotents whose rectangular components are all copies of the 2 c × 2 c rectangular semigroup and form a decreasing chain indexed by λ + 1, with the minimum component contained in the smallest ideal of βN. As a fortuitous corollary we obtain the fact that there are ≤L-chains of idempotents of length c in βN. We show also that there are copies of the direct product of the 2 c x 2 c rectangular semigroup with the free group on 2 c generators contained in the smallest ideal of βN.

COUNTING SYMMETRIC BRACELETS

An \(r\)-ary necklace (bracelet) of length \(n\) is an equivalence class of \(r\)-colourings of vertices of a regular \(n\)-gon, taking all rotations (rotations and reflections) as equivalent. A necklace (bracelet) is symmetric if a corresponding colouring is invariant under some reflection. We show that the number of symmetric \(r\)-ary necklaces (bracelets) of length \(n\) is \(\frac{1}{2}(r+1)r^\frac{n}{2}\) if \(n\) is even, and \(r^\frac{n+1}{2}\) if \(n\) is odd. DOI: 10.1017/S0004972713000701

The number of minimal right ideals of beta G

Let G be an infinite Abelian group of cardinality k and let βG denote the Stone-Cech compactification of G as a discrete semigroup. We show that βG contains 2 2k many minimal right ideals.

Symmetric colorings of the dihedral group

ABSTRACT Let G be a finite group and let r∈ℕ. An r-coloring of G is any mapping χ:G→{1,…,r}. Colorings χ and ψ are equivalent if there exists g∈G such that χ(xg−1) = ψ(x) for every x∈G. A coloring χ is symmetric if there exists g∈G such that χ(gx−1g) = χ(x) for every x∈G. Let Sr(G) denote the number of symmetric r-colorings of G and sr(G) the number of equivalence classes of symmetric r-colorings of G. We count Sr(G) and sr(G) in the case where G is the dihedral group Dn.