Showing papers on "Topological semigroup published in 2011"

Topological monoids of monotone injective partial selfmaps of $\mathbb{N}$ with cofinite domain and image

Abstract: In this paper we study the semigroup $\mathscr{I}_{\infty}^{ earrow}(\mathbb{N})$ of partial cofinal monotone bijective transformations of the set of positive integers $\mathbb{N}$. We show that the semigroup $\mathscr{I}_{\infty}^{ earrow}(\mathbb{N})$ 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. We also prove that every locally compact topology $\tau$ on $\mathscr{I}_{\infty}^{ earrow}(\mathbb{N})$ such that $(\mathscr{I}_{\infty}^{ earrow}(\mathbb{N}),\tau)$ is a topological inverse semigroup, is discrete. Finally, we describe the closure of $(\mathscr{I}_{\infty}^{ earrow}(\mathbb{N}),\tau)$ in a topological semigroup.

On the closure of the extended bicyclic semigroup

Abstract: In the paper we study the semigroup CZ which is a generalization of the bicyclic semigroup. We describe main algebraic properties of the semigroup CZ and prove that every non-trivial congruence C on the semigroup CZ is a group congruence, and moreover the quotient semigroup CZ=C is isomorphic to a cyclic group. Also we show that the semigroup CZ as a Hausdor semitopological semigroup admits only the discrete topology. Next we study the closure clT (CZ) of the semigroup CZ in a topological semigroup T. We show that the non-empty remainder of CZ in a topological inverse semigroup T consists of a group of units H(1T) of T and a two-sided ideal I of T in the case when H(1T) 6 ? and I 6 ?. In the case when T is a locally compact topological inverse semigroup and I 6 ? we prove that an ideal I is topologically isomorphic to the discrete additive group of integers and describe the topology on the subsemigroup CZ [ I. Also we show that if the group of units H(1T) of the semigroup T is non-empty, then H(1T) is either singleton or H(1T) is topologically isomorphic to the discrete additive group of integers.

Topological monoids of monotone injective partial selfmaps of ℕ with cofinite domain and image

Abstract: In this paper we study the semigroup ℐ ∞↗ (ℕ) of partial cofinal monotone bijective transformations of the set of positive integers ℕ. We show that the semigroup ℐ ∞↗ (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. We also prove that every locally compact topology τ on ℐ ∞↗ (ℕ) such that (ℐ ∞↗ (ℕ); τ) is a topological inverse semigroup, is discrete. Finally, we describe the closure of (ℐ ∞↗ (ℕ); τ) in a topological semigroup.

On monoids of injective partial selfmaps almost everywhere the identity

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

Characterizing compact Clifford semigroups that embed into convolution and functor-semigroups

Abstract: We study algebraic and topological properties of the convolution semigroup 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).

On the closure of the extended bicyclic semigroup

Abstract: In the paper we study the semigroup $\mathscr{C}_{\mathbb{Z}}$ which is a generalization of the bicyclic semigroup. We describe main algebraic properties of the semigroup $\mathscr{C}_{\mathbb{Z}}$ and prove that every non-trivial congruence $\mathfrak{C}$ on the semigroup $\mathscr{C}_{\mathbb{Z}}$ is a group congruence, and moreover the quotient semigroup $\mathscr{C}_{\mathbb{Z}}/\mathfrak{C}$ is isomorphic to a cyclic group. Also we show that the semigroup $\mathscr{C}_{\mathbb{Z}}$ as a Hausdorff semitopological semigroup admits only the discrete topology. Next we study the closure $\operatorname{cl}_T(\mathscr{C}_{\mathbb{Z}})$ of the semigroup $\mathscr{C}_{\mathbb{Z}}$ in a topological semigroup $T$. We show that the non-empty remainder of $\mathscr{C}_{\mathbb{Z}}$ in a topological inverse semigroup $T$ consists of a group of units $H(1_T)$ of $T$ and a two-sided ideal $I$ of $T$ in the case when $H(1_T) eq\varnothing$ and $I eq\varnothing$. In the case when $T$ is a locally compact topological inverse semigroup and $I eq\varnothing$ we prove that an ideal $I$ is topologically isomorphic to the discrete additive group of integers and describe the topology on the subsemigroup $\mathscr{C}_{\mathbb{Z}}\cup I$. Also we show that if the group of units $H(1_T)$ of the semigroup $T$ is non-empty, then $H(1_T)$ is either singleton or $H(1_T)$ is topologically isomorphic to the discrete additive group of integers.

Function spaces on tensor product of semigroups

Abstract: In this paper, we characterize the function space and
-space of the [topological] tensor product of [topological]semigroups. As a consequence, for arbitrary [topological] groups and , it will be shown that    is anextension of   by a proper normal subgroup N i,e.    .

Limits of Convolution Sequences of Measures on a Compact Topological Semigroup

Abstract: 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 by KaWaDa & 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].

Metrizability of Clifford topological semigroups

Abstract: We prove that a topological Clifford semigroup $S$ is metrizable if and only if $S$ is an $M$-space and the set $E=\{e\in S:ee=e\}$ of idempotents of $S$ is a metrizable $G_\delta$-set in $S$. The same metrization criterion holds also for any countably compact Clifford topological semigroup $S$.

P-Amenable Locally Compact Hypergroups

Abstract: Let K be a locally compact hypergroup with left Haar measure and let P (K) = {f ∈ L(K) : f ≥ 0, ‖f‖1 = 1 }. Then P (K) is a topological semigroup under the convolution product of L(K) induced in P (K). We say that K is P-amenable if there exists a left invariant mean on C(P (K)), the space of all bounded continuous functions on P (K). In this note, we consider the Pamenability of hypergroups. The P-amenability of hypergroup joins K = H ∨ J where H is a compact hypergroup and J is a discrete hypergroup with H∩J = {e} is characterized. It is also shown that Z-hypergroups are P-amenable if Z(K) ∩G(K) is compact.

Rosenblatt’s Contributions to Random Walks on Compact Semigroups

Abstract: Murray Rosenblatt’s interest in random walks on compact semigroups probably came from his work on representations of stationary processes as shifts of functions of independent random variables described in [11], where products of matrices were studied. His papers [12],[5],[13],[14] generalized the work of Levy [2] on random walks on the circle and the work of Kawada and Ito [4] on random walks on compact groups. In [14], he also completely characterized the structure of the limit measures for the special case of compact semigrougs of n ´ n stochastic matrices.