CONTRAEXAMPLES IN DIFFERENCE POSETS AND ORTHOALGEBRAS

We show that every orthoalgebra (difference orthoposet) uniquely determines a difference orthoalgebraic structure. We give examples of posets on which there exist mom than one difference operation. In spite of that, every finite chain is a uniquely determined difference poset. On a difference poset there need not exist any orthoalgebraic operation, but the category of difference orthoposets is isomorphic with the category of orthoalgebras. But a difference poset which is also an orthoposet need not be a difference orthoposet. Moreover, there exist complete lattices on which there does not exist any difference operation. Finally, we show that difference operations and orthoalgebraic operations need not be extendable on a MacNeille completion of the base poset.