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

Abstract

In the paper we describe the structure of $\mathcal{AH}$-completions and $\mathcal{H}$-completions of the discrete semilattices $(\mathbb{N},\min )$ and $(\mathbb{N},\max )$. We give an example of an $\mathcal{H}$-complete topological semilattice which is not $\mathcal{AH}$-complete. Also we construct an $\mathcal{H}$-complete topological semilattice of cardinality $\lambda$ which has $2^{\lambda }$ many open-and-closed continuous homomorphic images which are not $\mathcal{H}$-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.