O. Gutik | Searcharxiv

arXiv2013

On pseudocompact topological Brandt $λ^0$-extensions of semitopological monoids

In the paper we investigate topological properties of a topological Brandt $λ^0$-extension $B^0_λ(S)$ of a semitopological monoid $S$ with zero. In particular we prove that for every Tychonoff pseudocompact (resp., Hausdorff countably compact, Hausdorff compact) semitopological monoid $S$ with zero there exists a unique semiregular pseudocompact (resp., Hausdorff countably compact, Hausdorff compact) extension $B^0_λ(S)$ of $S$ and establish theirs Stone-Čech and Bohr compactifications. We also describe a category whose objects are ingredients in the constructions of pseudocompact (resp., countably compact, sequentially compact, compact) topological Brandt $λ^0$-extensions of pseudocompact (resp., countably compact, sequentially compact, compact) semitopological monoids with zeros.

math.GR↗

arXiv2013

On $\mathscr{H}$-complete topological semilattices

In the paper we describe the structure of $\mathscr{AH}$-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 $\mathscr{AH}$-complete. Also we construct an $\mathscr{H}$-complete topological semilattice of cardinality $λ$ which has $2^λ$ 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 from the paper J. W. Stepp, {\it Algebraic maximal semilattices}. Pacific J. Math. {\bf 58}:1 (1975), 243-248.

math.GN↗