We obta in here some ergodic resu l t s fo r p o s i t i v e operators ac t ing on a weakly s e q u e n t i a l l y complete Banach l a t t i c e E . The proofs are by a t runca t ion method in troduced in [ I ] . The f i r s t sect ion discusses general p rope r t i es of " t runcated l imi t s " . In the second sect ion we obta in necessary and s u f f i c i e n t cond i t ions fo r the ex is tence of i nva r i an t weak uni ts ( s t r i c t l y p o s i t i v e f i xed p o i n t s , in a d i f f e r e n t t e rm ino logy ) . One such cond i t ion is that t runcated l i m i t s of band p ro j ec t i ons of subsequences of averages do not vanish. In t h i r d sect ion we prove the strong convergence of (Anf) A @ , where A n are Cesaro averages of a p o s i t i v e c o n t r a c t i o n , f E E+ , and @ is an a r b i t r a r y i nva r i an t p o s i t i v e element. Without an add i t i ona l assumption on the Banach l a t t i c e , the t runcated l i m i t of Anf need not e x i s t .