A uniform approach to inductive posets and inductive closure

The definition scheme, "A poset P is Z-inductive if it has a subposet B of Z-compact lements such that for every element of p of P there is a Z-set S in B such that p = {big square union}S, becomes meaningful when we replace the symbol of Z by such adjectives as "sirected", "chain", "pairwise compatible", "singleton", etc. Furthermore, several theorems have been proved that seem to differ only in their instantiations of Z. A simialr phenomena occurs when we comsider concepts such as Z-completeness of Z-comtinuity. This suggests that in all these different cases we are really talking about Z same thing. In this paper we show that this is indeed the case by abstracting out the essential common properties of the different instantiations of Z and proving common theorems within the resulting abstract framework. © 1978.