Semi-orthoposets

A semi-orthoposet (SOP) is a bounded posetP together with a unary operation :P P such thatab impliesba andaa for allaP. This structure generalizes all previously studied quantum logic frameworks and yet is rich enough so that nontrivial results can be proved. For various types of SOPs it is shown that a partially defined morphism has an extension to the full SOP and that there exists an order-determining set of morphisms with a specified range. These results are applied to obtain representations of SOPs in terms of SOPs of sets and SOPs of functions. Connections between SOPs and effect algebras as well as tensor products of SOPs are obtained.