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An Unusual Compact Monothetic Semigroup

An Unusual Compact Monothetic Semigroup G. BROWN AND W. MORAN 1. Introduction In a previous paper, [1], we gave examples of compact monothetic semigroups (with separately continuous multiplication) having respectively (i) uncountably many idempotents, (ii) countably many idempotents, (iii) an arbitrarily prescribed finite non-zero number of idempotents. For all our examples in cases (ii) and (iii) the idempotents are totally ordered by the usual relation, e^f ifandonlyif ef = e. We present here a particulary simple example of a compact monothetic semigroup having exactly four idempotents e e , e , e , which satisfy u 2 3 4 exej = e j = 1,2,3,4; e±e - e j = 1, 2 , 3 , 4 ; ee=e\ u } p 2z x in other words, we have the lattice diagram 2. Mise-en-scene As with all the examples in [1], the object of our concern is a monothetic sub- semigroup of a homomorphic image of the carrier space of the algebra of measures on the circle. In fact let T denote the circle group (regarded as the additive reals factored by the integers) and let M(T) denote the convolution algebra of bounded regular Borel measures on T. We recall that a generalized character of http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

An Unusual Compact Monothetic Semigroup

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/3.3.291
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Abstract

G. BROWN AND W. MORAN 1. Introduction In a previous paper, [1], we gave examples of compact monothetic semigroups (with separately continuous multiplication) having respectively (i) uncountably many idempotents, (ii) countably many idempotents, (iii) an arbitrarily prescribed finite non-zero number of idempotents. For all our examples in cases (ii) and (iii) the idempotents are totally ordered by the usual relation, e^f ifandonlyif ef = e. We present here a particulary simple example of a compact monothetic semigroup having exactly four idempotents e e , e , e , which satisfy u 2 3 4 exej = e j = 1,2,3,4; e±e - e j = 1, 2 , 3 , 4 ; ee=e\ u } p 2z x in other words, we have the lattice diagram 2. Mise-en-scene As with all the examples in [1], the object of our concern is a monothetic sub- semigroup of a homomorphic image of the carrier space of the algebra of measures on the circle. In fact let T denote the circle group (regarded as the additive reals factored by the integers) and let M(T) denote the convolution algebra of bounded regular Borel measures on T. We recall that a generalized character of

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Nov 1, 1971

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