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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Quasi-nilpotent sets in semigroups


Author: H. L. Chow
Journal: Proc. Amer. Math. Soc. 52 (1975), 393-397
MSC: Primary 43A05
DOI: https://doi.org/10.1090/S0002-9939-1975-0374809-2
MathSciNet review: 0374809
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Abstract: In a compact semigroup $S$ with zero $0$, a subset $A$ of $S$ is called quasi-nilpotent if the closed semigroup generated by $A$ contains $0$. A probability measure $\mu$ on $S$ is called nilpotent if the sequence $({\mu ^n})$ converges to the Dirac measure at $0$. It is shown that a probability measure is nilpotent if and only if its support is quasi-nilpotent. Consequently, the set of all nilpotent measures on $S$ is convex and everywhere dense in the set of all probability measures on $S$ and the union of their supports is $S$.


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Keywords: Quasi-nilpotent set, compact semigroup with zero, probability measure, support of a measure, nilpotent measure, nil semigroup
Article copyright: © Copyright 1975 American Mathematical Society