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Arithmetic Properties Of Freely α-Generated Lattices

Published online by Cambridge University Press:  20 November 2018

Bjarni Jónsson
Bjarni Jónsson*
Affiliation:
University of Minnesota
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In § 1 we give a characterization of a lattice L that is freely α-generated by a given partially ordered set P. In § 2 we obtain a representation of an element of such a lattice as a sum (product) of additively (multiplicatively) irreducible elements which, although not unique, has some of the desirable features of the canonical representation, in Whitman (2), of an element of a free lattice. The usefulness of this representation is illustrated in § 3, where some further arithmetic properties of these lattices are derived.

We use + and . for the binary operations of lattice addition and multiplication, and Σ and II for the corresponding operations on arbitrary sets and sequences of lattice elements. In other respects the terminology will be the same as in Crawley and Dean (1).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 14 , 1962 , pp. 476 - 481
Copyright
Copyright © Canadian Mathematical Society 1962

References

1. Crawley, P. and Dean, R. A., Lattices with infinite operations. Trans. Amer. Math. Soc, 92 (1959), 35–47.Google Scholar
2. Whitman, P. M., Free lattices, Ann. Math. (2), 42 (1941), 325–330.Google Scholar