# Lattices, Closure Operators, and Galois Connections

## Abstract

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.

## Keywords

Equivalence Relation Closure Operator Complete Lattice Great Element Neutral Element## References^{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 - 4.Garrett Birkhoff,
*Lattice Theory,*third edition, AMS Colloq. Publications, v. XXV, 1967. MR**37**#2638.Google Scholar - 32.
- 34.Serge Lang,
*Algebra,*Addison-Wesley, third edition, 1993. Reprinted as Springer GTM v. 211, 2002. MR**2003e**:00003.Google Scholar - 51.George M. Bergman,
*Constructing division rings as module-theoretic direct limits,*Trans. AMS.**354**(2002) 2079–2114. MR**2003b**:16017.Google Scholar - 55.George M. Bergman,
*Some notes on sets, logic and mathematical language,*supplementary course notes, 12 pp., accessible in several versions from http://math.berkeley.edu/~gbergman/ug.hndts/#sets_etc . - 75.A. Dundes,
*Interpreting Folklore,*Indiana University Press, 1980.Google Scholar - 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 - 106.Hans Kurzweil,
*Endliche Gruppen mit vielen Untergrupppen,*J. reine u. angewandte Math.**356**(1985) 140–160. MR**86f**:20024.Google Scholar - 108.Solomon Lefschetz,
*Algebraic Topology,*AMS Colloq. Pub. No. 27, 1942, reprinted 1963. MR**4**, 84f.Google Scholar - 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 - 143.Alan G. Waterman,
*The free lattice with 3 generators over N*_{5}*,*Portugal. Math.**26**(1967) 285–288. MR**42**#147.Google Scholar - 144.