On the additivity of lattice completeness
Israel Halperin, María J. Wonenburger · Pacific Journal of Mathematics · 1962
Neumann-geometry, respectively a von Neumann geometry.In any relatively complemented modular lattice, if a ^ b then [a -b] will denote an arbitrary (but fixed) element such that [ab] (j b = a (the dot indicates that the summands in the union are independent).We write a ~ b to denote: a is perspective to 6, and a < b to denote: a ~ 6 X for some b x ^ b.Elements α, b are called completely disjoint, (notation: (α, b)P) if: a x ~ b u a x ^ α, 6 X <^ 6 together imply α x = 0.3* The additivity of completeness theorem* In this section α, 6, c, a?*, will denote elements in a given relatively complemented modular lattice L.If [0, a U c] is upper ^-complete W e shall write u(a, c, ^) to mean:(3.1) Whenever x a ^ a U c for all ael (with / ^ K a Π (U(%l/3eF)) = 0 /or αZi j^mίe F cz I, then a Π (Ufel αe ^)) = 0.It is important to note: if u(a, c, ^) holds then u(a r , c', ^) holds for all a' ^ a, c'