Showing papers on "Topological semigroup published in 2017"

Categorically Closed Topological Groups

Abstract: Let C → be a category whose objects are semigroups with topology and morphisms are closed semigroup relations, in particular, continuous homomorphisms. An object X of the category C → is called C → -closed if for each morphism Φ ⊂ X × Y in the category C → the image Φ ( X ) = { y ∈ Y : ∃ x ∈ X ( x , y ) ∈ Φ } is closed in Y. In the paper we survey existing and new results on topological groups, which are C → -closed for various categories C → of topologized semigroups.

Topological properties of Taimanov semigroups

Abstract: A semigroup $T$ is called a Taimanov semigroup if $T$ contains two distinct elements $0,\infty$ such that $xy=\infty$ for any distinct points $x,y\in T\setminus\{0,\infty\}$ and $xy=0$ in all other cases. We prove that any Taimanov semigroup $T$ has the following topological properties: (i) each $T_1$-topology with continuous shifts on $T$ is discrete; (ii) $T$ is closed in each $T_1$-topological semigroup containing $T$ as a subsemigroup; (iii) every non-isomorphic homomorphic image $Z$ of $T$ is a zero-semigroup and hence $Z$ is a topological semigroup in any topology on $Z$.

Devaney chaos, Li-Yorke chaos, and multi-dimensional Li-Yorke chaos for topological dynamics

Abstract: Let π : T × X → X , written T ↷ π X , be a topological semiflow/flow on a uniform space X with T a multiplicative topological semigroup/group not necessarily discrete. We then prove: • If T ↷ π X is non-minimal topologically transitive with dense almost periodic points, then it is sensitive to initial conditions. As a result of this, Devaney chaos ⇒ Sensitivity to initial conditions, for this very general setting. Let R + ↷ π X be a C 0 -semiflow on a Polish space; then we show: • If R + ↷ π X is topologically transitive with at least one periodic point p and there is a dense orbit with no nonempty interior, then it is multi-dimensional Li–Yorke chaotic; that is, there is a uncountable set Θ ⊆ X such that for any k ≥ 2 and any distinct points x 1 , … , x k ∈ Θ , one can find two time sequences s n → ∞ , t n → ∞ with s n ( x 1 , … , x k ) → ( x 1 , … , x k ) ∈ X k and t n ( x 1 , … , x k ) → ( p , … , p ) ∈ Δ X k . Consequently, Devaney chaos ⇒ Multi-dimensional Li–Yorke chaos. Moreover, let X be a non-singleton Polish space; then we prove: • Any weakly-mixing C 0 -semiflow R + ↷ π X is densely multi-dimensional Li–Yorke chaotic. • Any minimal weakly-mixing topological flow T ↷ π X with T abelian is densely multi-dimensional Li–Yorke chaotic. • Any weakly-mixing topological flow T ↷ π X is densely Li–Yorke chaotic. We in addition construct a completely Li–Yorke chaotic minimal SL ( 2 , R ) -acting flow on the compact metric space R ∪ { ∞ } . Our various chaotic dynamics are sensitive to the choices of the topology of the phase semigroup/group T.

Involutions and trivolutions on second dual of algebras related to locally compact groups and topological semigroups

Abstract: We investigate involutions and trivolutions in the second dual of algebras related to a locally compact topological semigroup and the Fourier algebra of a locally compact group. We prove, among the other things, that for a large class of topological semigroups namely, compactly cancellative foundation $$*$$ -semigroup S when it is infinite non-discrete cancellative, $$M_a(S)^{**}$$ does not admit an involution, and $$M_a(S)^{**}$$ has a trivolution with range $$M_a(S)$$ if and only if S is discrete. We also show that when G is an amenable group, the second dual of the Fourier algebra of G admits an involution extending one of the natural involutions of A(G) if and only if G is finite. However, $$A(G)^{**}$$ always admits trivolution.

A View on Intuitionistic Smarandache Topological Semigroup Structure Spaces

Abstract: The purpose of this paper is to introduce the concepts of intuitionistic Smarandache topological semigroups, intuitionistic Smarandache topological semigroup structure spaces, intuitionistic SG exteriors and intuitionistic SG semi exteriors.

On the Topological Semigroup of Equational Classes of Finite Functions Under Composition

Abstract: We consider the set of equational classes of finite functions endowed with the operation of class composition. Thus defined, this set gains a semigroup structure. This paper is a contribution to the under-standing of this semigroup. We present several interesting properties of this semigroup. In particular, we show that it constitutes a topological semigroup that is profinite and we provide a description of its regular elements in the Boolean case.

Ideals in $\mathcal{P} _G$ and $\beta G$

Abstract: For a discrete group $G$, we use the natural correspondence between ideals in the Boolean algebra $ \mathcal{P}_G$ of subsets of $G$ and closed subsets in the Stone-$\check{C}$ech compactifi-cation $\beta G$ as a right topological semigroup to introduce and characterize some new ideals in $\beta G$. We show that if a group $G$ is either countable or Abelian then there are no closed ideals in $\beta G$ maximal in $G^*$, $G^* = \beta G \setminus G$, but this statement does not hold for the group $S_\kappa$ of all permutations of an infinite cardinal $\kappa$. We characterize the minimal closed ideal in $\beta G$ containing all idempotents of $G^*$.