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Constructing locales from quantales

Published online by Cambridge University Press:  24 October 2008

Susan B. Niefield  and
Kimmo I. Rosenthal
Susan B. Niefield
Affiliation:
Department of Mathematics, Union College, Schenectady, New York 12308
Kimmo I. Rosenthal
Affiliation:
Department of Mathematics, Union College, Schenectady, New York 12308
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Extract

We recall that a locale is a complete lattice L satisfying a ∧ (∨bα) = ∨(a ∧ bα), for all a ∈ L, and {bα} ⊆ L. Examples of locales include the lattices (X) of open subsets of topological spaces X. Following Joyal and Tierney [7], a morphism f: L → M of locales is a ∨-, ∧-, and τ-preserving map. Such functions are sometimes called ‘frame homomorphisms’, in which case the right adjoint f*:M → L (which exists since f preserves ∨) is then called a ‘morphism of locales’.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 104 , Issue 2 , September 1988 , pp. 215 - 234
Copyright
Copyright © Cambridge Philosophical Society 1988

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References

REFERENCES

[1]Banaschewski, B. and Harting, R.. Lattice aspects of radical ideals and choice principles. Proc. London Math. Soc. (3) 50 (1985), 385–404.CrossRefGoogle Scholar
[2]Borceux, F. and van den Bossche, G.. Algebra in a Localic Topos with Applications to Ring Theory. Lecture Notes in Math. vol. 1038 (Springer-Verlag, 1983).CrossRefGoogle Scholar
[3]Borceux, F. and van den Bossche, G.. Quantales and their sheaves. Order 3 (1986), 61–87.CrossRefGoogle Scholar
[4]Dixmier, J.. C*-Algebras (North Holland, 1977).Google Scholar
[5]Dowker, C. H. and Strauss, D.. Quotient frames and subspaces. Proc. London Math. Soc. (3) 16 (1966), 275–96.CrossRefGoogle Scholar
[6]Johnstone, P. T.. Stone Spaces. Cambridge Studies in Advanced Math. no. 3 (Cambridge University Press, 1982).Google Scholar
[7]Joyal, A. and Tierney, M.. An Extension of the Galois Theory of Grothendieck. Mem. Amer. Math. Soc. no. 309 (American Mathematical Society, 1984).CrossRefGoogle Scholar
[8]Kirby, D.. Closure operations on ideals and submodules. J. London Math. Soc. 44 (1969), 283–291.CrossRefGoogle Scholar
[9]Mulvey, C. J.. &. Rend. Circ. Mat. Palermo (2) 12 (1986), 99–104.Google Scholar
[10]Nawaz, M.. Quantales: Quantal Sets. Ph.D. thesis, University of Sussex (1985).Google Scholar
[11]Niefield, S. B. and Rosenthal, K. I.. Spatial sublocales and essential primes. Topology Appl. 26 (1987), 263–269.CrossRefGoogle Scholar
[12]Niefield, S. B. and Rosenthal, K. I.. Lattice theoretic aspects of sheaves of ideals. (In preparation.)Google Scholar
[13]Niefield, S. B. and Rosenthal, K. I.. Componental nuclei. In Proceedings of Category Theory Meeting, Louvain-la-Neuve 1987, Lecture Notes in Math. (Springer-Verlag), to appear.Google Scholar
[14]Simmons, H.. A framework for topology. In Logic Colloquium 77, Studies in Logic and the Foundations of Math. no. 96. (North Holland, 1978), pp. 239–251.CrossRefGoogle Scholar
[15]Simmons, H.. Two-sided multiplicative lattices and ring radicals. (Preprint).Google Scholar