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

TL;DR

This paper investigates the structure of $\\mathscr{H}$-completions of certain discrete semilattices, providing new examples and counterexamples that clarify the distinctions between different types of topological semilattices.

Contribution

It characterizes $\\mathscr{H}$-completions of specific semilattices and constructs examples showing the non-equivalence of $\\mathscr{H}$-completeness and $\\mathscr{AH}$-completeness.

Findings

An example of an $\\mathscr{H}$-complete semilattice not $\\mathscr{AH}$-complete.

Construction of a large $\\mathscr{H}$-complete semilattice with many non-$\mathscr{H}$-complete images.

Negative answer to a question about the equivalence of $\\mathscr{H}$-completeness and $\\mathscr{AH}$-completeness.

Abstract

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 $\lambda$ 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 from the paper J. W. Stepp, {\it Algebraic maximal semilattices}. Pacific J. Math. {\bf 58}:1 (1975), 243-248.