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**A uniform approach to inductive posets and inductive closure.**
*(English)*
Zbl 0732.06001

Summary: The definition scheme: “A poset P is Z-inductive if it has a subposet B of Z-compact elements such that for every element p of P there is a Z-set S in B such that \(p=\sqcup S\),” becomes meaningful when we replace the symbol Z by such adjectives as “directed”, “chain”, “pairwise compatible”, “singleton”, etc. Furthermore, several theorems have been proved that seem to differ only in their instantiations of Z. A similar phenomenon occurs when we consider concepts such as Z-completeness or Z- continuity. This suggests that in all these different cases we are really talking about the same thing. 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.

### MSC:

06A06 | Partial orders, general |

68Q55 | Semantics in the theory of computing |

18A40 | Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.) |

06A15 | Galois correspondences, closure operators (in relation to ordered sets) |

### Keywords:

inductive posets; inductive closure; fixed-point semantics for programming languages; Z-set; Z-completeness; Z-continuity### Citations:

Zbl 0372.06002
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\textit{J. B. Wright} et al., Theor. Comput. Sci. 7, 57--77 (1978; Zbl 0732.06001)

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### References:

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This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.