On H-complete topological semilattices

Author S. Bardyla, O. Gutik
sbardyla@yahoo.com, o_gutik@franko.lviv.ua, ovgutik@yahoo.com
Ivan Franko National University of Lviv

Abstract In the paper we describe the structure of $\mathscr{A\!H}$-completions and $\mathscr{H}$-completions of the discrete semilattices $(\mathbb{N},\min)$ and $(\mathbb{N},\max)$. We give an example of an $\mathscr{H}$-complete topological semilattice which is not $\mathscr{A\!H}$-complete. Also for an arbitrary infinite cardinal $\lambda$ we construct an $\mathscr{H}$-complete topological semilattice of cardinality $\lambda$ which has $2^\lambda$ many open-and-closed continuous homomorphic images which are not $\mathscr{H}$-complete topological semilattices. The constructed examples give a negative answer to Question 17 in the paper J. W. Stepp, Algebraic maximal semilattices, Pacific J. Math., 58 (1975), no.1, 243--248.
Keywords topological semilattice; free filter; complete semigroup; chain
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Pages 118-123
Volume 38
Issue 2
Year 2012
Journal Matematychni Studii
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