On semi-normal lattice rings
S. J. Bernau · Mathematical Proceedings of the Cambridge Philosophical Society · 1965
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