Open projections do not form a right residuated lattice

for all x, y, z ∈ L. (So we do not require the dual arrow nor the associativity of .) In other words, the actions − y, y ( − : L→ L, for any fixed y ∈ L, provide a Galois connection on L. The Galois connection is also completely determined by the partial isomorphism between fixpoints of closure mapping y ( (− y) and coclosure mapping (y ( −) y. Thus we can restrict ,( to partial operations ·,→ where x · y is defined if x = y ( (x y) and y → z is defined if z = (y ( z) y. The partial isomorphism provides an equivalence