Wright, J. B.; Wagner, E. G.; Thatcher, J. W. A uniform approach to inductive posets and inductive closure. (English) Zbl 0732.06001 Theor. Comput. Sci. 7, 57-77 (1978). 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. Cited in 4 ReviewsCited in 47 Documents 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 PDFBibTeX XMLCite \textit{J. B. Wright} et al., Theor. Comput. Sci. 7, 57--77 (1978; Zbl 0732.06001) Full Text: DOI References: [1] Goguen, J. A.; Thatcher, J. W.; Wagner, E. G.; Wright, J. B., A junction between computer science and category theory: I, Basic definitions and examples, Part 1, IEM Research Report RC 4526 (1973) [2] Goguen, J. A.; Thatcher, J. W.; Wagner, E. G.; Wright, J. B., Initial algebra semantics and continuous algebras, J. ACM., 24, 68-95 (1977), IBM Research Report RC 5701 (3 November 1975) · Zbl 0359.68018 [3] Goguen, J. A.; Tatcher, J. W.; Wagner, E. G.; Wright, J. B., A junction between computer science and category theory: I, Basic definitions and examples, Part 2, IBM Research Report RC 5908 (1976) [4] Birkhoff, G., Lattice Theory, Vol. 25 (1948), Amer. Math. Soc. Colloq. Pub: Amer. Math. Soc. Colloq. Pub New York · Zbl 0126.03801 [5] Bloom, S. L., Varieties of ordered algebras, J. Comp. Sys. Sci., 13, 200-212 (1976) · Zbl 0337.06008 [6] Courcelle, B.; Nivat, M., Algebraic families of interpretations, Proc. 17th Ann. IEEE Symp. on Foundation of Computing, 137-146 (1976), Houston, Texas [7] Egli, H.; Constable, R. L., Computability concepts for programming language semantics, Theoret. Comput. Sci., 2, 133-145 (1976) · Zbl 0352.68042 [8] Mac Lane, S., Category Theory for the Working Mathematician (1971), Springer-Verlag: Springer-Verlag Berlin · Zbl 0232.18001 [9] Markowsky, G., Categories of chain-complete posets, IBM Research Report RC 5100 (1974) [10] Markowsky, G.; Rosen, B. K., Bases for chain complete posets, IBM J. Res. Dev., 20, 138-147 (1976) · Zbl 0329.06001 [11] Scott, D., Outline of a mathematical theory of computation, Proc. 4th Ann. Princeton Conf. on Information Sciences and Systems, 169-176 (1970) [12] Scott, D., Continuous lattices, (Oxford University Computing Laboratory Technical Monograph PRG 7. Oxford University Computing Laboratory Technical Monograph PRG 7, Lecture Notes in Mathematics 274 (1971), Springer-Verlag: Springer-Verlag Berlin), 97-136 [13] Scott, D., Data types as lattices (1972), unpublished notes, Amsterdam [14] Tiuryn, J., Regualr algebras (extended abstract) manuscript (1976), Warsaw University 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.