Generalized Ideals in Orthoalgebras

Since in 1936 Birkhoff and von Neumann regarded the lattice of all closed subspaces of a separable infinite-dimensional Hilbert space that is an orthomodular lattice as a proposition system for a quantum mechanical entity (Miklós, 1998), orthomodular lattices have been considered as a mathematical model for a calculus of quantum logic. With the development of the theory of quantum logics, orthoalgebras as a quantum structure that generalize orthomodular lattices, orthomodular posets, are also regarded as a mathematical model of quantum logic (Foulis et al., 1992). Because quantum structures are all algebraic structures, their algebraic properties play an important role in studying the theory of quantum logic (Miklós, 1998). We know that the notion of ideals (i.e., p-ideals) is a very powerful tool to study quantum logic ( Kalmbach, 1983). Hence, it is necessary to study ideals of orthoalgebras. From the point of logic, Foulis et al., studied local filters, local ideals, and obtained some properties of local filters (Foulis et al., 1992). In this note, we give the definitions of generalized ideals, generalized filters, prove the equivalence between generalized ideals and local ideals, and establish the relationship between generalized ideals and supports.