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Some applications of double-negation sheafification

Published online by Cambridge University Press:  20 January 2009

D. S. Macnab
D. S. Macnab
Affiliation:
College of Education, Aberdeen
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In this note we point out that certain algebraic-topological constructions are particular cases of one construction, namely double-negation sheafification. The principal cases we have in mind are concerned with booleanpowers, completions of boolean algebras, and maximal rings of quotients.We conjecture that several other constructions—particularly completion-type constructions—will turn out also to be examples of double-negation sheafification.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 20 , Issue 4 , September 1977 , pp. 279 - 285
Copyright
Copyright © Edinburgh Mathematical Society 1977

References

REFERENCES

(1) Banaschewski, B., Maximal Rings of Quotients of Semi-simple Commutative Rings, Archiv. der Math., 16 (1965), 414–420,CrossRefGoogle Scholar
(2) Daigneault, A., Boolean Powers in Algebraic Logic, Z. Math. Grundlagen Math., 17 (1971), 411–420.CrossRefGoogle Scholar
(3) Davey, B. A., Sheaf Spaces and Sheaves of Universal Algebras, Math. Z. 134 (1973), 275–290.CrossRefGoogle Scholar
(4) Ellerman, D. P., Sheaves of Structures and Generalised Ultraproducts,Ann. Math. Logic, 7 (1974), 163–195.CrossRefGoogle Scholar
(5) Fine, N. J., Gillman, L. and Lambek, J., Rings of Quotients of Rings of Functions (McGill University Press, Montreal, 1965).Google Scholar
(6) Foster, A. L., Generalised “Boolean” Theory of Universal Algebras, Math. Z. 58 (1953), 306–338 and 59 (1953), 191-199.CrossRefGoogle Scholar
(7) Freyd, P. J., Aspects of Topoi, Bull. Austral. Math. Soc. 7 (1972), 1–76.Google Scholar
(8) Godement, R., Theorie des Faisceaux (Hermann, Paris, 1964).Google Scholar
(9) Gratzer, G., Universal Algebra (Van Nostrand, Princeton, 1963).Google Scholar
(10) Grothendieck, A., and Verdier, J. L., Theorie des Topos et Cohomologie Etale des Schemas (Springer-Verlag, Lecture Notes 269, Berlin, 1972).CrossRefGoogle Scholar
(11) Hofmann, K. H., Representation of Algebras by Continuous Sections, Bull. Amer. Math. Soc, 78 (1972), 291–373.CrossRefGoogle Scholar
(12) Lambek, J., Lectures on Rings and Modules (Blaisdell, Mass., 1966).Google Scholar
(13) Lawvere, F. W., Quantifiers and Sheaves, Actes Congres Intern. Math. (1970), Tome 1, 329–334.Google Scholar
(14) Lawvere, F. W., Toposes, Algebraic Geometry and Logic (Springer-Verlag, Lecture Notes 274, Berlin, 1972).Google Scholar
(15) MacDonald, I. G., Algebraic Geometry (Benjamin, New York, 1968).Google Scholar
(16) Macnab, D. S., The Structure of Polyadic Algebras (M.Sc. Thesis, Aberdeen,1973).Google Scholar
(17) Mansfield, R., Theory of Boolean Ultrapowers, Ann. Math. Logic, 2 (1971), 297–325.CrossRefGoogle Scholar
(18) Mitchell, B., Theory of Categories (Academic Press, New York, 1965).Google Scholar
(19) Mulvey, C. J., On Ringed Spaces (Ph.D. Thesis, Sussex, 1970).Google Scholar
(20) Mulvey, C. J., Intuitionistic Algebra and the Representation of Rings, in Recent Advances in the Representation Theory of Rings and C*-algebras, Mem. Amer. Math. Soc, 148 (1974).Google Scholar
(21) Ribenboim, P., Boolean Powers, Fund. Math. 65 (1969), 243–268.CrossRefGoogle Scholar
(22) Roos, J.-E., Locally Distributive Spectral Categories and Strongly Regular Rings, Reports of the Midwest Category Seminar I (Springer-Verlag, Lecture Notes 47, Berlin, 1967).Google Scholar
(23) Stenstrom, B., Rings and Modules of Quotients (Springer-Verlag, Lecture Notes 237, Berlin, 1971).CrossRefGoogle Scholar
(24) Swan, R., Theory of Sheaves (University of Chicago Press, 1964).Google Scholar
(25) Tierney, M., Sheaf Theory and the Continuum Hypothesis, Toposes, Algebraic Geometry and Logic (Springer-Verlag, Lecture Notes 274, Berlin, 1972).Google Scholar
(26) Wraith, G. C., Lectures on Elementary Topoi, Model Theory and Topoi (Springer-Verlag, Lecture Notes 445, Berlin, 1975).Google Scholar