  We give an axiomatic formulation of quantum structures like semilogics and
quasilogics which generalize the boolean semirings of events and fuzzy logics.
The notions of distributions, states, representations observables and
semiobservables are introduced and their Hilbert space realizations are found.
The closed and open structures in semilogics are introduced and the regular
distributions on the semilogics are studied.
