LAUSR

In this paper we introduce the notion of $$Z_{\delta }$$Zδ-continuity as a generalization of precontinuity, complete continuity and $$s_{2}$$s2-continuity, where Z is a subset selection. And for each poset P, a closure space $$Z^{c}_{\delta }(P)$$Zδc(P) arises naturally. For any subset system Z, we define a new type of completion, called $$Z_{\delta }$$Zδ-completion, extending each poset P to a Z-complete poset. The main results are: (1) if a subset system Z is subset-hereditary, then $$cl_{Z}(\Psi (P))$$clZ(Ψ(P)), the Z-closure of all principal ideals $$\Psi (P)$$Ψ(P) of poset P in $$Z^{c}_{\delta }(P)$$Zδc(P), is a $$Z_{\delta }$$Zδ-completion of P and $$Z^{c}_{\delta }(P) \cong Z^{c}_{\delta }(cl_{Z}(\Psi (P)))$$Zδc(P)≅Zδc(clZ(Ψ(P))); (2) let Z be an HUL-system and P a $$Z_{\delta }$$Zδ-continuous poset, then the $$Z_{\delta }$$Zδ-completion of P is also $$Z_{\delta }$$Zδ-continuous, and a Z-complete poset L is a $$Z_{\delta }$$Zδ-completion of P iff P is an embedded $$Z_{\delta }$$Zδ-basis of L; (3) the Dedekind–MacNeille completion is a special case of the $$Z_{\delta }$$Zδ-completion.