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Constructive complete distributivity. I

Published online by Cambridge University Press:  24 October 2008

Barry Fawcett  and
R. J. Wood
Barry Fawcett
Affiliation:
Department of Mathematics, Statistics and Computing Science, Dalhousie University, Halifax, Canada
R. J. Wood
Affiliation:
Department of Mathematics, Statistics and Computing Science, Dalhousie University, Halifax, Canada
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Extract

The relationships, in many cases equivalences, between lattice distributivity, adjunction and continuity have been studied by many authors, for example [1, 3–8, 12, 13, 15, 17–20, 22, 23]. Very roughly, we refer to the following circle of ideas. Let L be an ordered set, and L a class of subsets of L, and suppose that L has a supremum for each element in L. We might say that L has -sups. The ‘distributivity’ we refer to is that of infs over -sups. The ‘adjunction’ is that given by a left adjoint to the map V: L→L. Now the latter has a left adjoint if and only if it preserves infs, and this means roughly that the -sup of an intersection is an inf of -sups. When one does succeed in identifying the -sup of an intersection as a -sup of infs, one has an instance of distributivity.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 107 , Issue 1 , January 1990 , pp. 81 - 89
Copyright
Copyright © Cambridge Philosophical Society 1990

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References

REFERENCES

[1]Bandelt, H.-J. and Erné, M.. The category of Z-continuous posets. J. Pure Appl. Algebra 30 (1983), 219–226.CrossRefGoogle Scholar
[2]Barr, M.. Toposes Triples and Theories (Springer-Verlag, 1985).CrossRefGoogle Scholar
[3]Bruns, G.. Distributivität und subdirekte Zerlegbarkeit vollständiger Verbände. Arch. Math. (Basel) 12 (1961), 61–66.CrossRefGoogle Scholar
[4]Bruns, G.. Darstellungen und Erweiterungen geordnete Mengen. I. J. Reine Angew. Math. 209 (1962), 167–200.CrossRefGoogle Scholar
[5]Bruns, G.. Darstellungen und Erweiterungen geordnete Mengen. II. J. Reine Angew. Math. 210 (1962), 1–23.CrossRefGoogle Scholar
[6]Büchi, J. R.. Die Boolesche Partialordnung und die Paarung von Gefügen. Portugal Math. 7 (1948), 119–190.Google Scholar
[7]Büchi, J. R.. Representation of complete lattices by sets. Portugal Math. 2 (1952), 149–186.Google Scholar
[8]Day, A.. Filter monads, continuous lattices and closure systems. Canad. J. Math. 27 (1975), 50–59.CrossRefGoogle Scholar
[9]Diaconescu, R.. Axiom of choice and complementation. Proc. Amer. Math. Soc. 51 (1975), 176–178.CrossRefGoogle Scholar
[10]Dugundji, J.. Topology (Allyn & Bacon Inc. Publishers, 1966).Google Scholar
[11]Fawcett, B. and Wood, R. J.. Constructive complete distributivity. II. (In preparation.)Google Scholar
[12]Hoffman, R.-E.. Continuous posets and adjoint sequences. Semigroup Forum 18 (1979), 173–188.CrossRefGoogle Scholar
[13]Johnstone, P. T.. Stone Spaces (Cambridge University Press, 1982).Google Scholar
[14]Kock, A. and Wraith, G. C.. Elementary toposes. Aarhus Universitet Lecture Series no. 30 (1971).Google Scholar
[15]Lawson, J. D.. The duality of continuous posets. Houston J. Math. 5 (1979), 357–386.Google Scholar
[16]Mikkelsen, C. J.. Lattice theoretic and logical aspects of elementary topoi. Aarhus Universitet Publ. Series no. 25 (1976).Google Scholar
[17]Novak, D.. Generalization of continuous posets. Trans. Amer. Math. Soc. 272 (1982), 645–667.CrossRefGoogle Scholar
[18]Raney, G. N.. Completely distributive complete lattices. Proc. Amer. Math. Soc. 3 (1952), 677–680.CrossRefGoogle Scholar
[19]Raney, G. N.. A subdirect-union representation for completely distributive lattices. Proc. Amer. Math. Soc. 4 (1953), 518–522.CrossRefGoogle Scholar
[20]Raney, G. N.. Tight Galois connections and complete distributivity. Trans. Amer. Math. Soc. 97 (1960), 418–426.CrossRefGoogle Scholar
[21]Rubin, H. and Rubin, J.. Equivalents of the Axiom of Choice (North Holland Publishing Company, 1963).Google Scholar
[22]Sikorski, R.. Representation and distributivity of Boolean algebras. Colloq. Math. 8 (1961), 1–13.CrossRefGoogle Scholar
[23]Sikorski, R.. Boolean Algebras (Academic Press, 1964).Google Scholar
[24]Street, R. and Walters, R. F. C.. Yoneda structures on 2-categories. J. Algebra 50 (1978), 350–379.CrossRefGoogle Scholar