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On semi-normal lattice rings

Published online by Cambridge University Press:  24 October 2008

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

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 ∧ (bc) = (ab) ∨ (ac), and a ∨ (bc) = (ab) ∧ (ac). For a non-empty subset M of a lattice group we define

Type
Research Article
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), 4169.Google Scholar