Construction of right topological compactifications for discrete versions of subsemigroups of compact groups

Helmer, D.; Işık, Nilgün

doi:10.1007/bf02573285

Cited by 4 publications

(10 citation statements)

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“…PYM [49] showed that they were the maximal ideal spaces of the C*-algebra of bounded functions f on G which have the property that osc (/, x) (the oscillation of / at x) exceeds any ε > 0 at only a finite number of points. The simple construction in Example 4.7 is due to HELMER and ISIK [19]. They also point out a curious fact: the construction is only of interest for metrizable G, since if G is not metrizable the only compactification of G \ {1} is G itself.…”

Section: Theorem 44 Let S Be a Locally Compact Right Topological Sementioning

confidence: 96%

Compact semigroups with one-sided continuity

Pym¹

1990

The Analytical and Topological Theory of Semigroups

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This survey must begin by drawing attention to competing accounts of the subject. First, there is WOLFGANG RUPPERT'S book [55], This advertises itself as being about separately continuous semigroups but in fact contains a lot of material about semigroups with multiplication continuous on one side only. Then there is NEIL HINDMAN'S survey [24]. This, even more improbably for an article which contains a substantial amount on compact right topological semigroups, claims to be about ultrafilters and Ramsey theory. Finally there is the book by JOHN BERGLUND, HUGO JUNGHENN and PAUL MILNES -not the familiar volume of 1978 in the Lecture Notes series, but the entirely new book which will appear in 1989. This offers a superb elegant account of the whole theory of topological semigroups and their compactifications, not as deep as RUPPERT'S masterful summary of his special area, but providing a real headache for anyone attempting to produce an alternative over-view of the field. I shall, of course, try to say a little about all aspects of the subject, but in order to avoid too much overlap I shall concentrate on those areas which are more familiar to me and about which I may, at this moment, have more information than other authors.Any reviewer of this field faces another, much more irritating problem: terminology. I shall be dealing with semigroups in which the multiplication (s,t) i-> st is continuous in one of the variables. Obviously it does not matter which is chosen, and different authors have made different choices. If we decide on continuity in s, we might describe this as 'left-continuity' because s is the left-hand variable. But we might equally observe that it is the operation of translation on the right by the element t which is continuous and use the term 'right-continuity'. Again, individual authors have followed their own inclinations. What is worse, if you find a writer's terminology unpalatable, you cannot simply ask your word processor to go through a paper interchanging 'left' and 'right', because everyone agrees what (for example) a left ideal is. It is obviously time that a decision was made on a uniform terminology. In spite of the consequences for some of my own preferences, I hereby declare that BERGLUND, JUNGHENN and MILNES [6] will become the standard reference work in this area; I shall follow their terminology and notation and I recommend all other writers to do the same.There is one further aspect of notation about which an innocent reader should be warned. In this subject, commutative semigroups can be dense in non-commutative Unauthenticated Download Date | 6/15/16 5:45 PM 198 John S. Pymones. For this reason the practice has grown up of sometimes using the expression 5 +1 to denote a non-commutative product of s and t, and we shall do this when convenient. An addition sign does not necessarily imply commutativity.

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Section: Theorem 44 Let S Be a Locally Compact Right Topological Sementioning

confidence: 96%

Compact semigroups with one-sided continuity

Pym¹

1990

The Analytical and Topological Theory of Semigroups

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“…A and B be as in (2)(3). For each infinite subset U <=A let U, U V U 2 ,U 3 ,... and U', U{, U' 2 , U' 3 ,... be the partition of N. Then in order to distinguish the results of choosing different subsets U, we shall call S v the semigroup produced by the procedure of Section 2, and {pu\, {q v } its minimal left ideals.…”

Section: Definition 3 1 (I) Letmentioning

confidence: 99%

“…Pym [10] gave a generalization of this construction with 2 C idempotents, but this was not so transparent. Recently, Helmer and I §ik [3] have simplified the construction in [11] by producing a straightforward construction of a compact right topological semigroup S in which the topological centre A(S) = {seS: t->s + t is continuous on S} is dense, S = A(S) \JL where L is a left ideal, L has 2 C idempotents and every maximal group in L is dense.…”

Section: Introductionmentioning

confidence: 99%

A construction of a compact right topological semigroup

El-Mabhouh¹

1994

Math. Proc. Camb. Phil. Soc.

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“…Compactifications by filters of zero sets 365 (ii) From [4], theorem 4 and [5], we have fi(G\{x}) = G so that any continuous function on G\{x} has a continuous extension to G. Let A =/~1({0}) for some continuous function f:G\{x}->M, and l e t / be the continuous extension of/ to G.…”

Section: Maximal Prime Z-filters and Z-ultrafiltersmentioning

confidence: 99%

“…The semigroup structure of the space of prime z filters In this section we define a semigroup operation © on the set of prime z-filters of a compact metrizable weakly cancellative semitopological semigroup S in such a way that # becomes a compact right topological semigroup (that is a semigroup in which addition is continuous in the left hand variable) and we investigate algebraic and topological properties of # and the relations between &P and S. We finally show that when S is a compact group then # is algebraically and topologically isomorphic with the group constructed by Helmer and Isik in [4]. Although we use + for the semigroup multiplication, it need not be commutative.…”

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confidence: 96%

Compactifications of discrete versions of semitopological semigroups by filters of zero sets

Budak

1991

Math. Proc. Camb. Phil. Soc.

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The maximal proper prime filters together with the ultrafilters of zero sets of any metrizable compact topological space are shown to have a compact Hausdorff topology in which the ultrafilters form a discrete, dense subspace. This gives a general theory of compactifications of discrete versions of compact metrizable topological spaces and some of the already known constructions of compact right topological semigroups are special cases of the general theory. In this way, simpler and more elegant proofs for these constructions are obtained.In [8], Pym constructed compactifications for discrete semigroups which can be densely embedded in a compact group. His techniques made extensive use of function algebras. In [4] Helmer and Isik obtained the same compactifications by using the existence of Stone ech compactifications. The aim of this paper is to present a general theory of compactifications of semitopological semigroups so that Helmer and Isik's results in [4] are a simple consequence. Our proofs are different and are based on filters which provide a natural way of getting compactifications. Moreover we present new insights by emphasizing maximal proper primes which are not ultrafilters.We start by defining filters of zero sets (called z-filters) on a given topological space X, and their convergence. In the case of compact metrizable topological spaces, we establish the connections between proper maximal prime z-filters on X and zultrafilters in β(X\{x})\(X\{x}) where β(X\{x}) is the Stone-ech compactification of X\{x}. We then define a topology on the set of all prime z-filters on X such that the subspace of all proper maximal primes is compact Hausdorff. We denote by the set of all proper maximal prime z-filters on X together with the z-ultrafilters and show that when X is a compact metrizable cancellative semitopological semigroup, is a compact right topological semigroup with dense topological centre. Also, when is considered for a compact Hausdorff metrizable group, the semigroup obtained is exactly the same (algebraically and topologically) as the semigroup obtained in [4]. Hence the result in [4] is just a consequence of the general theory presented in this paper.

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Construction of right topological compactifications for discrete versions of subsemigroups of compact groups

Cited by 4 publications

References 5 publications

Compact semigroups with one-sided continuity

Compact semigroups with one-sided continuity

A construction of a compact right topological semigroup

Compactifications of discrete versions of semitopological semigroups by filters of zero sets

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