  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.
