Tolerance relations and operator systems

We extend the scope of noncommutative geometry by generalizing construction algebra a quotient space to situations in which one is no longer dealing with an equivalence relation. For these so-called tolerance relations, passing associated relation looses crucial information as clear from examples such coarse graining physics or $$d(x, y) < \varepsilon$$ on metric space. Fortunately, thanks formalism operator systems extension possible and provides new invariants, C*-envelope propagation number. After thorough investigation structure (non-unital) we analyze corresponding state spaces. In particular, determine pure system for path measure