Abstract
In the paper we present various characterizations of chain-compact and chain-finite topological semilattices. A topological semilattice X is called chain-compact (resp. chain-finite) if each closed chain in X is compact (finite). In particular, we prove that a (Hausdorff) \(T_1\)-topological semilattice X is chain-finite (chain-compact) if and only if for any closed subsemilattice \(Z\subset X\) and any continuous homomorphism \(h:Z\rightarrow Y\) to a (Hausdorff) \(T_1\)-topological semilattice Y the image h(Z) is closed in Y.
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Communicated by Jimmie D. Lawson.
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Banakh, T., Bardyla, S. Characterizing chain-compact and chain-finite topological semilattices. Semigroup Forum 98, 234–250 (2019). https://doi.org/10.1007/s00233-018-9921-x
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DOI: https://doi.org/10.1007/s00233-018-9921-x