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Sublattices of a Free Lattice

Published online by Cambridge University Press:  20 November 2018

Bjarni Jónsson
Bjarni Jónsson*
Affiliation:
University of Minnesota
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Professor R. A. Dean has proved (1, Theorem 3) that a completely free lattice generated by a countable partially ordered set is isomorphic to a sublattice of a free lattice. In particular, it follows that a free product of countably many countable chains can be isomorphically embedded in a free lattice. Generalizing this we show (2.1) that the class of all lattices that can be isomorphically embedded in free lattices is closed under the operation of forming free lattice-products with arbitrarily many factors. We also prove (2.4) that this class is closed under the operation of forming simply ordered sums with denumerably many summands. Finally we show (2.7) that every finite dimensional sublattice of a free lattice is finite.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 13 , 1961 , pp. 256 - 264
Copyright
Copyright © Canadian Mathematical Society 1961

References

1. Dean, R. A., Sublattices of free lattices. Google Scholar
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4. Whitman, P. M., Free lattices I, Ann. Math. (2), 42 (1941), 325–330.Google Scholar
5. Whitman, P. M., Free lattices II, Ann. Math. (2), 43 (1942), 104–115.Google Scholar