On $mathscr{H}$-complete topological semilattices

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.