Closure spaces and completions of posets

Let $${{\mathcal {Z}}}$$Z be a subset selection. A $${{\mathcal {Z}}}$$Z-completion of poset $$P$$P is a $${{\mathcal {Z}}}$$Z-complete poset $$E_{{{\mathcal {Z}}}}(P)$$EZ(P) together with a monotone mapping from $$P$$P into $$E_{{{\mathcal {Z}}}}(P)$$EZ(P) that preserves existing suprema of $${{\mathcal {Z}}}$$Z-sets and is universal among such mappings. First, for each subset selection $${{\mathcal {Z}}}$$Z, we define two closure operators $$\rho _{{{\mathcal {Z}}}}$$ρZ and $$\hat{\rho }_{{{\mathcal {Z}}}}$$ρ^Z on each poset. We prove that if $${{\mathcal {Z}}}$$Z satisfies some natural conditions then: (i) for each poset the $${{\mathcal {Z}}}$$Z-completion exists; (ii) each poset and its $${{\mathcal {Z}}}$$Z-completion have isomorphic lattices of $$\hat{\rho }_{{{\mathcal {Z}}}}$$ρ^Z-closed sets; (iii) for any $${{\mathcal {Z}}}$$Z-continuous poset the $${{\mathcal {Z}}}$$Z-completion is $${{\mathcal {Z}}}$$Z-continuous. The results obtained here include the dcpo-completions and chain-completions of posets as special cases. From the general result, we also derive the sup-completions of posets.