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On rings of sets

Published online by Cambridge University Press:  09 April 2009

T. P. Speed
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In the past a number of papers have appeared which give representations of abstract lattices as rings of sets of various kinds. We refer particularly to authors who have given necessary and sufficient conditions for an abstract lattice to be lattice isomorphic to a complete ring of sets, to the lattice of all closed sets of a topological space, or to the lattice of all open sets of a topological space. Most papers on these subjects give the conditions in terms of special elements of the lattice. We thus have completely join-irreducible elements — G. N. Raney [7]; join prime, completely join prime, and supercompact elements — V. K. Balachandran [1], [2]; N-sub-irreducible elements — J. R. Büchi [5]; and lattice bisectors — P. D. Finch [6]. Also meet-irreducible and completely meet-irreducible dual ideals play a part in some representations of G. Birkhoff & 0. Frink [4].

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 8 , Issue 4 , November 1968 , pp. 723 - 730
Copyright
Copyright © Australian Mathematical Society 1968

References

[1]Balachandran, V. K., ‘'A characterisation of ΣΔ-rings of subsets’, Fundamenta Mathematica 41 (1953), 38–41.CrossRefGoogle Scholar
[2]Balachandran, V. K., ‘On complete lattices and a problem of Birkhoff and Frink’, Proc. Amer. Math. Soc. 6 (1953), 548–553.CrossRefGoogle Scholar
[3]Birkhoff, G., Lattice Theory (Amer. Math. Soc. Colloq. New York 1948).Google Scholar
[4]Birkhoff, G. & Frink, O., ‘Representations of lattices by sets’, Trans. Amer. Math. Soc. 64 (1948), 299–316.CrossRefGoogle Scholar
[5]Büchi, J. R., ‘Representations of complete lattices by sets’, Portugaliae Mathematicae 11 (1952), 151–167.Google Scholar
[6]Finch, P. D., ‘On the lattice equivalence of topological spaces’, Journ. Aust. Math. Soc. 6 (1966), 495–511.CrossRefGoogle Scholar
[7]Raney, G. N., ‘Completely distributive complete lattices’, Proc. Amer. Math. Soc. 3 (1952), 677–680.CrossRefGoogle Scholar