H-closedness and absolute H-closedness

Abstract

This paper focuses on the study of H-closedness and absolute H-closedness of the posets. First, we propose a counterexample to indicate that an absolutely H-closed topological semilattice may not be c-complete, which gives a negative answer to an open question proposed by Banakh and Bardyla. However, in the case of continuous semilattice with the Lawson topology, we prove that the absolutely H-closed topological semilattice implies c-completeness. Second, we obtain a characterization for quasicontinuous lattices utilizing the topological embedding mapping. Finally, enlightened by the definitions of H-closedness for Hausdorff spaces and absolute H-closedness for Hausdorff topological semilattices, we introduce the concepts of H-closedness and absolute H-closedness for posets with the Lawson topology.