H-closedness and absolute H-closedness

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.