States on orthoalgebras
An orthoalgebra is a quadruple \((L, 0, 1, \oplus)\) where \(L\) is a set, \(0,1\in L\) and \(\oplus\) is a partial binary operation on \(L\) such that (i) whenever one side of the commutativity or associativity law (for \(\oplus\)) is defined then so is the other and both sides are equal, (ii) for every \(a\in L\) there exists a unique \(b\in L\) with \(a\oplus b=1\), and (iii) 0 is the unique element \(a\) of \(L\) for which \(a\oplus a\) exists. Orthoalgebras can be considered as generalizations of orthomodular posets by interpreting \(\oplus\) as the join of orthogonal elements. It turned out that orthoalgebras are appropriate structures within the logico-algebraic foundation of quantum theories. States (i.e. finitely additive normed measures) on orthoalgebras are considered and a pasting technique for orthoalgebras is developed by means of which orthoalgebras are constructed whose state space has certain preassigned properties. The problem of extending states to larger orthoalgebras is also investigated.
scientific article; zbMATH DE number 844070
scientific article