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Proceedings of the American Mathematical Society

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

 

 

Embedding partially ordered spaces in topological semilattices


Author: Lloyd D. Tucker
Journal: Proc. Amer. Math. Soc. 33 (1972), 203-206
MSC: Primary 06A10; Secondary 06A20
DOI: https://doi.org/10.1090/S0002-9939-1972-0292724-7
MathSciNet review: 0292724
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Abstract: A partial order $ \Gamma $ on a compact space S is called continuous if $ \Gamma $ is a closed subset of $ S \times S$. In this paper, we define and study an embedding $ \Phi $ of the arbitrary compact continuously partially ordered space $ (S,\Gamma )$ into a corresponding compact topological semilattice $ {S_\Gamma }$. We show that the structure of $ {S_\Gamma }$ entirely determines the structure of $ (S,\Gamma )$. We prove that the inverse images under $ \Phi $ of components in $ {S_\Gamma }$ are the order components of $ (S,\Gamma )$, where elements a and b of S are defined to be in the same order component of $ (S,\Gamma )$ if there exists no continuous monotonic map $ f:(S,\Gamma ) \to \{ 0,1\} $ which separates a and b. Finally, we show that $ {S_\Gamma }$ is connected if and only if $ (S,\Gamma )$ has only one order component.


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DOI: https://doi.org/10.1090/S0002-9939-1972-0292724-7
Keywords: Continuous orders, embedding, topological semilattice, order component, order connected
Article copyright: © Copyright 1972 American Mathematical Society