On Stability of Regional Orthomodular Posets

The set of regions of a transition system, ordered by set inclusion, is an orthomodular poset, often referred to as quantum logic, here called regional logic. Regional logics, which result to be also regular and rich, are the main subject of investigation in this work. Given a regular, rich logic L, one can build a condition/event transition system A, such that L embeds into the regional logic of A. Call stable a logic if the embedding is an isomorphism. We give some necessary conditions for a logic to be stable, and show that under these, the embedding presents some stronger property. The full characterization of the class of stable logics is still an open problem. In particular, we show that any {0, 1}-pasting of n Boolean logics is stable, and that, whenever L contains n maximal Boolean sublogics sharing exactly one atom, L is stable