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Free Decompositions of a Lattice

Published online by Cambridge University Press:  20 November 2018

G. Grätzer  and
J. Sichler
G. Grätzer
Affiliation:
University of Manitoba, Winnipeg, Manitoba
J. Sichler
Affiliation:
University of Manitoba, Winnipeg, Manitoba
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Two basic questions have been raised for free products of lattices:

1. Do any two free products have a common refinement?

2. Can every lattice be decomposed into a free product of freely indecomposable lattices?

Both questions have been around for some time and attempts at solving them were made especially after the Structure Theorem for Free Products was discovered (see G. Grâtzer, H. Lasker, and C. R. Piatt [3]). Partial answer to question one was supplied in A. Kostinsky [7].

In this paper we answer both questions. Our basic observation is that the proper framework for these results is the theory of free K-products, that is, free products in an arbitrary equational class K of lattices.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 27 , Issue 2 , 01 April 1975 , pp. 276 - 285
Copyright
Copyright © Canadian Mathematical Society 1975

References

1. Gràtzer, G., Lattice theory. First concepts and distributive lattices (W. H. Freeman and Co., San Francisco, 1971).Google Scholar
2. Grâtzer, G., B. Jônsson, and Lakser, H., The amalgamation property in equational classes of modular lattices, Pacific J. Math. 45 (1973), 507–524.Google Scholar
3. Grâtzer, G., Lakser, H., and Piatt, C. R., Free products of lattices, Fund. Math. 69 (1970), 233–240.Google Scholar
4. Jônsson, B., Modular lattices and Desargues Theorem, Math. Scand. 2 (1954), 295–314.Google Scholar
5. Jônsson, B., Sublattices of a free lattice, Can. J. Math. 13 (1961), 256–264.Google Scholar
6. Jônsson, B., Relatively free products of lattices, Algebra Universalis 1 (1972), 362–373.Google Scholar
7. Kostinsky, A., Some problems for rings and lattices within the domain of general algebra, Ph.D. Thesis, Berkeley, 1971.Google Scholar