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On semi-normal lattice rings

Published online by Cambridge University Press:  24 October 2008

S. J. Bernau
S. J. Bernau
Affiliation:
University of Canterbury, New Zealand
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Extract

A lattice ring is a lattice group ((l), page 214) and a ring in which ab ≥ 0 whenever a ∧ b ≥ 0.

In any lattice group (commutative or not) we define a+ = a ∨ 0, a− = (−a) ∨ 0 and |a| = a+ + a−. Itisknown((1). pages 219,220) that a+ ∧ a− = 0, a = a+ − a−, |a| = a+ ∨ a−, and that a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c), and a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c). For a non-empty subset M of a lattice group we define

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 61 , Issue 3 , July 1965 , pp. 613 - 616
Copyright
Copyright © Cambridge Philosophical Society 1965

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References

REFERENCES

(1)Birkhoff, G.Lattice theory (Amer. Math. Soc; New York, 1948).Google Scholar
(2)Birkhoff, G. and Pierce, R. S.Lattice ordered rings. An. Acad. Brasil Ci. 28 (1956), 41–69.Google Scholar