Showing papers on "Topological semigroup published in 2001"
TL;DR: In this article, the authors studied the bicompletion of the quasi-uniformity induced in a natural way on a topological semigroup which has a neutral element.
Abstract: We study the bicompletion of the quasi-uniformities that are induced in a natural way on a topological semigroup which has a neutral element. In particular, we show that if X is a topological semigroup, with neutral element, for which the left translations are open, then the bicompletion of the left quasi-uniformity of X can be considered a topological semigroup which contains the topological space X as a sup-dense subsemigroup. The bicompletion in the case that the left translations are not necessarily open is also discussed. In particular, both Abelian and left-cancellable topological semigroups are considered. For semigroups which are (left-)cancellable or which are locally totally bounded, theorems similar to those known from the classical theory of (para)topological groups are established.
14 citations
TL;DR: In this article, it was proved that every semigroup from C is an absolute coretract and every finite semigroup of idempotents, which is a decreasing chain of rectangular semigroups, belongs to C.
Abstract: A semigroup S is called an absolute coretract if for any continuous homomorphism f from a compact Hausdorff right topological semigroup T onto a compact Hausdorff right topological semigroup containing S algebraically there exists a homomorphism g \colon S→ T such that f\circ g=idS. The semigroup β\ben contains isomorphic copies of any countable absolute coretract. In this article we define a class C of semigroups of idempotents each of which is a decreasing chain of rectangular semigroups. It is proved that every semigroup from C is an absolute coretract and every finite semigroup of idempotents, which is an absolute coretract, belongs to C .
11 citations
TL;DR: In this paper, a topological proof of the infinitude of prime numbers is given, not depending on the additive structure of the underlying integers or commutative rings with identity.
Abstract: The late J. Knopfmacher and the author [12] have studied some ties between
arithmetic properties of the multiplicative structure of commutative rings
with identity and the topologies induced by some coset classes. In the present
communication it is shown that the ideas used there are capable of a further
extension. Namely, replacing the ideal structure of commutative rings by generalized
ideal systems, the so called x-ideals, conditions implying the existence of
infinitely many prime x-ideals are found using topologies induced by cosets
of x-ideals. This leads to new variants of Furstenberg topological proof of
the infinitude of prime numbers not depending on the additive structure of
the underlying integers or commutative rings with identity. as a byproduct
we give new proofs of the infinitude of primes based on tools taken from commutative
algebra. Mathematics Subject Classification (1991): 11N80, 11N25, 11A41, 11T99,
13A15, 20M25 Keywords: x-ideal, topological semigroup, ideal topology, infinitude
of primes, generalized primes and integers, distribution, integers, specified
multiplicative constraints, primes, ideals, multiplicative ideal theory, semigroup
rings, multiplicative semigroups of rings, multiplicative arithmetical semigroup,
semigroup, John Knopfmacher, zeta function, abscissa, convergence, commutative,
rings, ring, identities, algebra
Quaestiones Mathematicae 24(3) 2001, 373-391
8 citations
TL;DR: In this article, a partition of disjoint isomorphic subgroups of some of the subsets of the enveloping semigroup is obtained, where the set of all minimal right ideals of E(X) is denoted by Min(E(X)).
Abstract: We deal with a-minimal sets instead of minimal right ideals of the enveloping semigroup and obtain a partition of disjoint isomorphic subgroups of some of its subsets. We also give some generalizations of almost periodicity and distality in the transformation semigroups and obtain similar results. 2000 Mathematics Subject Classification. Primary 54H15. 1. Preliminaries. By a transformation semigroup (X,S,ρ) (or simply (X, S) )we mean a compact Hausdorff topological space X, a discrete topological semigroup S with identity e, and a continuous map ρ : X × S → X (ρ(x, s) = xs ∀x ∈ X,∀s ∈ S), such that (1) xe = x∀x ∈ X; (2) x(st) = (xs)t∀x ∈ X, ∀s,t ∈ S. In the transformation semigroup (X, S), for each s ∈ S define π s : X → X by π s (x) = xs (∀x ∈ X). We assume the semigroup S acts effectively on X, that is, for each s,t ∈ S, s ≠ t if and only if π s ≠ π t . The closure of {π s | s ∈ S} in X X (with pointwise convergence topology) is called the enveloping semigroup (or Ellis semigroup) of (X, S) and is denoted by E(X, S) (or simply E(X)). The enveloping semigroup E(X) has a semigroup structure (1). A nonempty subset I of E(X) is called a right ideal of E(X), if IE(X) ⊆ I, moreover, if the right ideal I of E(X) does not have any proper subset which is a right ideal of E(X), then I is called a minimal right ideal of E(X), the set of all minimal right ideals of E(X) is denoted by Min(E(X)). An element u of E(X) is called idempotent if u 2 = u. For p ∈ E(X) and a ∈ X, the maps Lp :E (X) → E(X) and θa :E (X) → X defined by Lp(q) = pq and θa(q) = aq (q ∈ E(X)), respectively, are
7 citations
01 Jan 2001
TL;DR: In this article, a transformation semigroup (X,S,?X ) is defined, where S is a finite union of its subsemigroups and X is a compact Hausdorff topological space.
Abstract: In this note we study the transformation semigroup (X,S), where S is a finite union of its subsemigroups. 2000 AMS Classification Subject: 54H15 By a transformation semigroup (X,S,?X ) (or simply (X,S)) we mean a compact Hausdorff topological space X, a discrete topological semigroup S with identity e and a continuous map ?σ: X×S→ X (?σ(x, s) = xs (∀ x∈ X, ∀ s∈ S)) such that: •∀ x∈ X xe =x, •∀ x∈ X ∀ s, t∈ S x (st) = (xs) t. In the transformation semigroup (X,S) we have the following definitions: 1. For each s∈ S, define the continuous map ?⌡ s : X→ X by x?⌡ s =xs (∀ x∈ X), we used to write s instead of ? s . The closure of {? s │s∈S} in X X with pointwise convergence, is called the enveloping sermigroup (or Ellis semigroup) of (X,S) and it is written by E(X,S) or simply E(X). E(X,S) has a semigroup structure (Ellis, 1969, Chapter 3), a nonempty subset K of E(X,S) is called a right ideal if KE(X,S) ⊆ K, and it is called a minimal right ideal if none of the right ideals of E(X,S) is a proper subset of K. 2. A nonempty subset Z of X is called invariant if ZS ⊆ Z, moreover it is called minimal if it is closed and none of the closed invariant subsets
4 citations
Journal Article•
TL;DR: In this article, it was shown that the asymptotic behavior of almost-orbits of a self-mapping family of non-empty closed covex subsets of a uniformly convex Banach space can be characterized by the following properties: (i) the set of all common fixed points of Tt, t∈e S is nonempty; (ii) t ∈E S is locally bounded; (iii) limt||Ttc-c|| = 0; (iv) c∈E F
Abstract: In this paper, let C be a nonempty closed covex subset of a uniformly convex Banach space E and r = {Tt: t∈ S} be a self-mapping family of asymptotically nonexpansive type on C such that for each t∈E S, Tt: C - C is continuous, where S is a commutative topological semigroup with identity. Let {u(t): t∈ S} be an almost- orbit ofГ . It is shown that if Г is asymptotically regular at the asymptotic center c∈ C of {u(t): t ∈ S} with respect to C, then the following statements are equivalent: (i) the set F(Г) of all common fixed points of Tt, t∈e S is nonempty; (ii) {u(t): t∈E S} is locally bounded; (iii) limt||Ttc-c|| = 0; (iv) c∈E F(Г). As an application we establish the result on the asymptotic behavior of almost-orbits of asymptotically nonexpansive families.
3 citations