In this paper we determine the structure of the minimal ideal in the enveloping semigroup for the... more In this paper we determine the structure of the minimal ideal in the enveloping semigroup for the natural action of a connected semisimple Lie group on its maximal compact subgroup. In particular, if G = KAN is an Iwasawa decomposition of the group G, then the group in the minimal left ideal is isomorphic both algebraically and topologically with the normalizer M of AN in K. Complete descriptions are given for the enveloping semigroups in the cases G = SL(2, C) and G = SL(2, R).
For many semigroups (S, +) it is possible to prove the existence of extremal left translation inv... more For many semigroups (S, +) it is possible to prove the existence of extremal left translation invariant and extremal left translation invariant zero dimensional topologies. In this paper the origin of such topologies and their relation to idempotents in /3S, the Stone-tech compactification of S, for semigroups with identity is investigated.
ABSTRACT The paper begins by presenting a construction of the largest semigroup compactification ... more ABSTRACT The paper begins by presenting a construction of the largest semigroup compactification GLUC of a locally compact group as a quotient of the Stone-Cech compactification of the discrete group βGd. This presentation is used in a proof of the local structure theorem for GLUC, which gives a topological description of neighbourhoods of each point, and some new extensions of this result. These immediately imply Veech's Theorem. Finally a result is given which extends Veech's Theorem: for σ-compact groups the map g → gx is injective for all x ∈ GLUC on a set larger than G.
The equivalence between oids and semigroups which satisfy a condition involving finite sums is es... more The equivalence between oids and semigroups which satisfy a condition involving finite sums is established. Some of the already known results on the structure of Stone-Čech compactifications of discrete semigroups are obtained as immediate consequences. It is also shown that most commutative semigroups contain oids so that oid theory has applications to the Stone-Čech compactifications of many semigroups.
The existence of non-fixed, almost translation invariant ultrafilters on any infinite semigroupS ... more The existence of non-fixed, almost translation invariant ultrafilters on any infinite semigroupS satisfying some algebraic properties is established using an ultrafilter approach. The structure of the Stone-Čech compactification of any discrete semigroup is investigated using filters and closed subsets ofßS.
Proceedings of the American Mathematical Society, 1989
We present a short ultrafilter proof of a result which has applications in combinatorial number t... more We present a short ultrafilter proof of a result which has applications in combinatorial number theory and which has previously relied on the theory of compact semigroups.
Mathematical Proceedings of the Cambridge Philosophical Society, 1994
The Stone–Čech compactification βℕ of the discrete space ℕ of positive integer is a very large to... more The Stone–Čech compactification βℕ of the discrete space ℕ of positive integer is a very large topological space; for example, any countable discrete subspace of the growth ℕ* = βℕ/ℕ has a closure which is homeomorphic to βℕ itself ([23], §3·5] Now ℕ, while hardly inspiring as a discrete topological space, has a rich algebrai structure. That βℕ also has a semigroup structure which extends that of (ℕ, +) and in which multiplication is continuous in one variable has been apparent for about 30 years. (Civin and Yood [3] showed that βG was a semigroup for each discrete group G, and any mathematician could then have spotted that βℕ was a subsemigroup of βℕ.) The question which now appears natural was explicitly raised by van Douwen[6] in 1978 (in spite of the recent publication date of his paper), namely, does ℕ* contain subspaces simultaneously algebraically isomorphic and homeomorphic to βℕ? Progress on this question was slight until Strauss [22] solved it in a spectacular fashion: the...
Mathematical Proceedings of the Cambridge Philosophical Society, 1991
The maximal proper prime filters together with the ultrafilters of zero sets of any metrizable co... more 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 cons...
Page 1. Bull. London Math. Soc. Page 1 of 12 Cо2011 London Mathematical Society doi:10.1112/blms/... more Page 1. Bull. London Math. Soc. Page 1 of 12 Cо2011 London Mathematical Society doi:10.1112/blms/bdq116 Minimal determinants of topological centres for some algebras associated with locally compact groups Talin Budak, Nilgün Isık and John Pym ...
Le but de l'article est de proposer des modeles d'inspections pour des systemes sous proc... more Le but de l'article est de proposer des modeles d'inspections pour des systemes sous processus de deterioration de facon a evaluer diverses mesures de performance de ce systeme telles que le nombre attendu d'inspections avec differents resultats et la probabilite de detection
In this paper we determine the structure of the minimal ideal in the enveloping semigroup for the... more In this paper we determine the structure of the minimal ideal in the enveloping semigroup for the natural action of a connected semisimple Lie group on its maximal compact subgroup. In particular, if G = KAN is an Iwasawa decomposition of the group G, then the group in the minimal left ideal is isomorphic both algebraically and topologically with the normalizer M of AN in K. Complete descriptions are given for the enveloping semigroups in the cases G = SL(2, C) and G = SL(2, R).
For many semigroups (S, +) it is possible to prove the existence of extremal left translation inv... more For many semigroups (S, +) it is possible to prove the existence of extremal left translation invariant and extremal left translation invariant zero dimensional topologies. In this paper the origin of such topologies and their relation to idempotents in /3S, the Stone-tech compactification of S, for semigroups with identity is investigated.
ABSTRACT The paper begins by presenting a construction of the largest semigroup compactification ... more ABSTRACT The paper begins by presenting a construction of the largest semigroup compactification GLUC of a locally compact group as a quotient of the Stone-Cech compactification of the discrete group βGd. This presentation is used in a proof of the local structure theorem for GLUC, which gives a topological description of neighbourhoods of each point, and some new extensions of this result. These immediately imply Veech's Theorem. Finally a result is given which extends Veech's Theorem: for σ-compact groups the map g → gx is injective for all x ∈ GLUC on a set larger than G.
The equivalence between oids and semigroups which satisfy a condition involving finite sums is es... more The equivalence between oids and semigroups which satisfy a condition involving finite sums is established. Some of the already known results on the structure of Stone-Čech compactifications of discrete semigroups are obtained as immediate consequences. It is also shown that most commutative semigroups contain oids so that oid theory has applications to the Stone-Čech compactifications of many semigroups.
The existence of non-fixed, almost translation invariant ultrafilters on any infinite semigroupS ... more The existence of non-fixed, almost translation invariant ultrafilters on any infinite semigroupS satisfying some algebraic properties is established using an ultrafilter approach. The structure of the Stone-Čech compactification of any discrete semigroup is investigated using filters and closed subsets ofßS.
Proceedings of the American Mathematical Society, 1989
We present a short ultrafilter proof of a result which has applications in combinatorial number t... more We present a short ultrafilter proof of a result which has applications in combinatorial number theory and which has previously relied on the theory of compact semigroups.
Mathematical Proceedings of the Cambridge Philosophical Society, 1994
The Stone–Čech compactification βℕ of the discrete space ℕ of positive integer is a very large to... more The Stone–Čech compactification βℕ of the discrete space ℕ of positive integer is a very large topological space; for example, any countable discrete subspace of the growth ℕ* = βℕ/ℕ has a closure which is homeomorphic to βℕ itself ([23], §3·5] Now ℕ, while hardly inspiring as a discrete topological space, has a rich algebrai structure. That βℕ also has a semigroup structure which extends that of (ℕ, +) and in which multiplication is continuous in one variable has been apparent for about 30 years. (Civin and Yood [3] showed that βG was a semigroup for each discrete group G, and any mathematician could then have spotted that βℕ was a subsemigroup of βℕ.) The question which now appears natural was explicitly raised by van Douwen[6] in 1978 (in spite of the recent publication date of his paper), namely, does ℕ* contain subspaces simultaneously algebraically isomorphic and homeomorphic to βℕ? Progress on this question was slight until Strauss [22] solved it in a spectacular fashion: the...
Mathematical Proceedings of the Cambridge Philosophical Society, 1991
The maximal proper prime filters together with the ultrafilters of zero sets of any metrizable co... more 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 cons...
Page 1. Bull. London Math. Soc. Page 1 of 12 Cо2011 London Mathematical Society doi:10.1112/blms/... more Page 1. Bull. London Math. Soc. Page 1 of 12 Cо2011 London Mathematical Society doi:10.1112/blms/bdq116 Minimal determinants of topological centres for some algebras associated with locally compact groups Talin Budak, Nilgün Isık and John Pym ...
Le but de l'article est de proposer des modeles d'inspections pour des systemes sous proc... more Le but de l'article est de proposer des modeles d'inspections pour des systemes sous processus de deterioration de facon a evaluer diverses mesures de performance de ce systeme telles que le nombre attendu d'inspections avec differents resultats et la probabilite de detection
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