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On Birkhoff's Problem 73 For Monoids

Published online by Cambridge University Press:  20 November 2018

Barron Brainerd
Barron Brainerd*
Affiliation:
University of Toronto
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Birkhoff in [2] poses the following problem:

“Problem 73. Find necessary and sufficient conditions in order that the correspondence between the congruence relations and the (neutral) ideals of a lattice be one-one”.

This problem has been solved by Areškin [l] and Hashimoto [3]. Essentially the conditions reduce to the r e quirement that the lattice be a generalized Boolean algebra.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 3 , Issue 3 , 01 September 1960 , pp. 217 - 220
Copyright
Copyright © Canadian Mathematical Society 1960

References

1. Areškin, G. Ya., On congruence relations in distributive lattices with zero elements, Dokl. Akad, Nauk S.S.S.R. (NS), 90 (1953), 485-486.Google Scholar
2. Birkhoff, G., Lattice Theory, second edition, (New York, 1948.)Google Scholar
3. Hashimoto, J., Ideal theory for lattices, Math. Japon. 2 (1952), 149-186.Google Scholar
4. Preston, G. B., Inverse semi-groups, J. London Math. Soc. 29 (1954), 396-403.Google Scholar