Closure spaces and completions of posets

Abstract

Let ${{\mathcal {Z}}}$ be a subset selection. A ${{\mathcal {Z}}}$-completion of poset $P$ is a ${{\mathcal {Z}}}$-complete poset $E_{{{\mathcal {Z}}}}(P)$ together with a monotone mapping from $P$ into $E_{{{\mathcal {Z}}}}(P)$ that preserves existing suprema of ${{\mathcal {Z}}}$-sets and is universal among such mappings. First, for each subset selection ${{\mathcal {Z}}}$, we define two closure operators $\rho _{{{\mathcal {Z}}}}$ and $\hat{\rho }_{{{\mathcal {Z}}}}$ on each poset. We prove that if ${{\mathcal {Z}}}$ satisfies some natural conditions then: (i) for each poset the ${{\mathcal {Z}}}$-completion exists; (ii) each poset and its ${{\mathcal {Z}}}$-completion have isomorphic lattices of $\hat{\rho }_{{{\mathcal {Z}}}}$-closed sets; (iii) for any ${{\mathcal {Z}}}$-continuous poset the ${{\mathcal {Z}}}$-completion is ${{\mathcal {Z}}}$-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.