Showing papers on "Topological semigroup published in 2010"

Openly factorizable spaces and compact extensions of topological semigroups

Abstract: We prove that the semigroup operation of a topological semigroup $S$ extends to a continuous semigroup operation on its Stone-Cech compactification $\beta S$ provided $S$ is a pseudocompact openly factorizable space, which means that each map $f:S\to Y$ to a second countable space $Y$ can be written as the composition $f=g\circ p$ of an open map $p:X\to Z$ onto a second countable space $Z$ and a map $g:Z\to Y$. We present a spectral characterization of openly factorizable spaces and establish some properties of such spaces.

Symmetric inverse topological semigroups of finite rank ≤ n

Abstract: We establish topological properties of the symmetric inverse topological semigroup of finite transformations of the rank ≤ n. We show that the topological inverse semigroup is algebraically h -closed in the class of topological inverse semigroups. Also we prove that a topological semigroup S with countably compact square S×S does not contain the semigroup for infinite cardinal λ and show that the Bohr compactification of an infinite topological symmetric inverse semigroup of finite transformations of the rank ≤ n is the trivial semigroup.

On the Brandt λ0-extensions of monoids with zero

Abstract: We study algebraic properties of the Brandt λ0-extensions of monoids with zero and non-trivial homomorphisms between the Brandt λ0-extensions of monoids with zero. We introduce finite, compact topological Brandt λ0-extensions of topological semigroups and countably compact topological Brandt λ0-extensions of topological inverse semigroups in the class of topological inverse semigroups and establish the structure of such extensions and non-trivial continuous homomorphisms between such topological Brandt λ0-extensions of topological monoids with zero. We also describe a category whose objects are ingredients in the constructions of finite (compact, countably compact) topological Brandt λ0-extensions of topological monoids with zeros.

Algebra in superextensions of semilattices

Abstract: Given a semilattice X we study the algebraic properties of the semigroup �(X) of upfamilies on X. The semigroup �(X) contains the Stone-y Cech extension �(X), the superextension �(X), and the space of filters '(X) on X as closed subsemigroups. We prove that �(X) is a semilattice iff �(X) is a semilattice iff '(X) is a semilattice iff the semilattice X is finite and linearly ordered. We prove that the semigroup �(X) is a band if and only if X has no infinite antichains, and the semigroup �(X) is commutative if and only if X is a bush with finite branches. topological semigroup. The algebraic properties of this semigroup (for example, the existence of idempotents or minimal left ideals) have important consequences in combinatorics of numbers, see (HS), (P). In (G2) it was observed that the binary operationextends not only to �(X) but also to the space �(X) of all upfamilies on X. By definition, a family F of non-empty subsets of a discrete space X is called an upfamily if for any sets ABX the inclusion A 2 F implies B 2 F. The space �(X) is a closed subspace of the double power-set P(P(X)) enodwed with the compact Hausdorff topology of the Tychonoff power f0,1g P(X) . In the papers (G1), (G2), (BGN)-(BG4) the space �(X) was denoted by G(X) and its elements were called inclusion hyperspaces 1 . The extension of a binary operationfrom X to �(X) can be defined in the same way as for ultrafilters, i.e., by the formula (1) applied to any two upfamilies U,V 2 �(X). If X is a semigroup, then �(X) is a compact Hausdorff right-topological semigroup containing �(X) as closed subsemigroups. The algebraic properties of this semigroups were studied in details in (G2). The space �(X) of upfamilies over a discrete space X contains many interesting subspaces. First we recall some definitions. An upfamily A 2 �(X) is called to be � a filter if A1 \ A2 2 A for all sets A1,A2 2 A;

Topological monoids of almost monotone injective co-finite partial selfmaps of positive integers

Abstract: In this paper we study the semigroup $I_\infty^\dnearrow(N)$ of partial co-finite almost monotone bijective transformations of the set of positive integers $\mathbb{N}$. We show that the semigroup $I_\infty^\dnearrow(N)$ has algebraic properties similar to the bicyclic semigroup: it is bisimple and all of its non-trivial group homomorphisms are either isomorphisms or group homomorphisms. Also we prove that every Baire topology $\tau$ on $I_\infty^\dnearrow(N)$ such that $(I_\infty^\dnearrow(N),\tau)$ is a semitopological semigroup is discrete, describe the closure of $(I_\infty^\dnearrow(N),\tau)$ in a topological semigroup and construct non-discrete Hausdorff semigroup topologies on $I_\infty^\dnearrow(N)$.

Johnson amenability for topological semigroups

Abstract: –A notion of amenability for topological semigroups is introduced. A topological semigroup S iscalled Johnson amenable if for every Banach S -bimodule E , every bounded crossed homomorphism fromS to E* is principal. In this paper it is shown that a discrete semigroup S is Johnson amenable if and only if1(S) is an amenable Banach algebra. Also, we show that if a topological semigroup S is Johnson amenable,then it is amenable, but the converse is not true.

Embedding topological semigroups into the hyperspaces over topological groups

Abstract: 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 topolog- ically 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.

Extending binary operations to funtor-spaces

Abstract: Given a continuous monadic functor T in the category of Tychonov spaces for each discrete topological semigroup X we extend the semigroup operation of X to a right-topological semigroup operation on TX whose topological center contains the dense subsemigroup of all elements of TX that have finite support.

On monoids of injective partial selfmaps almost everywhere the identity

Abstract: In this paper we study the semigroup $\mathscr{I}^{\infty}_\lambda$ of injective partial selfmaps almost everywhere the identity of a set of infinite cardinality $\lambda$ We describe the Green relations on $\mathscr{I}^{\infty}_\lambda$, all (two-sided) ideals and all congruences of the semigroup $\mathscr{I}^{\infty}_\lambda$ We prove that every Hausdorff hereditary Baire topology $\tau$ on $\mathscr{I}^{\infty}_\omega$ such that $(\mathscr{I}^{\infty}_\omega,\tau)$ is a semitopological semigroup is discrete and describe the closure of the discrete semigroup $\mathscr{I}^{\infty}_\lambda$ in a topological semigroup Also we show that for an infinite cardinal $\lambda$ the discrete semigroup $\mathscr{I}^{\infty}_\lambda$ does not embed into a compact topological semigroup and construct two non-discrete Hausdorff topologies turning $\mathscr{I}^{\infty}_\lambda$ into a topological inverse semigroup

On Maximal Ideals of Compact Connected Topological Semigroups

Abstract: Several results concerning ideals of a compact topological semigroup 𝑆 with 𝑆2=𝑆 can be found in the literature. In this paper, we further investigate in a compact connected topological semigroup 𝑆 how the conditions 𝑆2=𝑆 and 𝑆2≠𝑆 affect the structure of ideals of 𝑆, especially the maximal ideals.

Embedding topological semigroups into the hyperspaces over topological groups

Abstract: 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.

Subsemigroups of s containing the idempotents

Abstract: Let S be a discrete semigroup and let P(S) be the set of points p in the Stone- ˇ Cech compactification, S , of S with the property that every neighborhood of p contains arbitrarily large finite sum sets or finite product sets, (de- pending on whether the operation in S is denoted by + or ·). Then P(S) contains all of the idempotents of S , where the operation on S extends that on S making S into a right topological semigroup with S contained in its topological cen- ter. If S is commutative, then P(S) is a compact subsemi- group of S. Responding to a question of Vitaly Bergelson, we show that if S is any semigroup which can be embedded in a compact topological group, then P(S) is not the smallest closed semigroup containing the idempotents of S and the closure of the semigroup generated by the idempotents of S is not a semigroup.

On monoids of almost identity injective partial selfmaps

Abstract: In the paper we study the semigroup I 1 of almost identity injective partial selfmaps of the set of cardinality . We describe the Green relations on I 1 , all (two-sided) ideals and all congruences of the semigroup I 1 . We prove that every Hausdor hereditary Baire topology on I 1 such that (I 1 ; ) is a semitopological semigroup is discrete and describe the closure of the discrete semigroup I 1 in a topological semigroup. Also we show that the discrete semigroup I 1 does not embed into a compact topological semigroup and construct two non-discrete Hausdor topologies turning I 1 into a topological inverse semigroup.

Common idempotents in compact left topological left semirings

Abstract: A classical result of topological algebra states that any compact left topological semigroup has an idempotent. We refine this by showing that any compact left topological left semiring has a common, i.e. additive and multiplicative simultaneously, idempotent. As an application, we partially answer a question related to algebraic properties of ultrafilters over natural numbers. Finally, we observe that similar arguments establish the existence of common idempotents in much more general, non-associative universal algebras.