Lattices, Closure Operators, and Galois Connections

  • George M. Bergman
Part of the Universitext book series (UTX)


Lattices and semilattices are developed, both as partially ordered sets where every pair of elements has a least upper bound and/or a greatest lower bound, and as algebraic structures, and various completeness conditions they can satisfy are examined.Such structures often arise from closure operators on sets, and this concept is developed.An insufficiently well known source of closure operators, which we develop, is the concept of a Galois connection between two sets. In the case for which the concept was named, the two sets are, respectively, the elements of a finite separable normal field extension, and the automorphisms of that extension. Another of the many examples noted relates the set of models of a formal language, and the set of propositions in that language.


Equivalence Relation Closure Operator Complete Lattice Great Element Neutral Element 
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  1. 3.
    George M. Bergman and Adam O. Hausknecht, Cogroups and Co-rings in Categories of Associative Rings, AMS Math. Surveys and Monographs, v.45, 1996. MR 97k :16001.Google Scholar
  2. 4.
    Garrett Birkhoff, Lattice Theory, third edition, AMS Colloq. Publications, v. XXV, 1967. MR 37 #2638.Google Scholar
  3. 32.
    Thomas W. Hungerford, Algebra, Springer GTM, v. 73, 1974. MR 50 #6693.Google Scholar
  4. 34.
    Serge Lang, Algebra, Addison-Wesley, third edition, 1993. Reprinted as Springer GTM v. 211, 2002. MR 2003e :00003.Google Scholar
  5. 51.
    George M. Bergman, Constructing division rings as module-theoretic direct limits, Trans. AMS. 354 (2002) 2079–2114. MR 2003b :16017.Google Scholar
  6. 55.
    George M. Bergman, Some notes on sets, logic and mathematical language, supplementary course notes, 12 pp., accessible in several versions from .
  7. 75.
    A. Dundes, Interpreting Folklore, Indiana University Press, 1980.Google Scholar
  8. 93.
    T. Ihringer, Congruence Lattices of Finite Algebras: the Characterization Problem and the Role of Binary Operations, Algebra Berichte v.53, Verlag Reinhard Fischer, München, 1986. MR 87c :08003.Google Scholar
  9. 106.
    Hans Kurzweil, Endliche Gruppen mit vielen Untergrupppen, J. reine u. angewandte Math. 356 (1985) 140–160. MR 86f :20024.Google Scholar
  10. 108.
    Solomon Lefschetz, Algebraic Topology, AMS Colloq. Pub. No. 27, 1942, reprinted 1963. MR 4, 84f.Google Scholar
  11. 126.
    Pavel Pudlák and Jiří T˚uma, Every finite lattice can be embedded in a finite partition lattice, Algebra Universalis 10 (1980) 74–95. MR 81e:06013.Google Scholar
  12. 143.
    Alan G. Waterman, The free lattice with 3 generators over N 5 , Portugal. Math. 26 (1967) 285–288. MR 42 #147.Google Scholar
  13. 144.
    D. J. A. Welsh, Matroid Theory, Academic Press, 1976. MR 55 #148.Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • George M. Bergman
    • 1
  1. 1.Department of MathematicsUniversity of California, BerkeleyBerkeleyUSA

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