Question: Let R be a quasi-ordering and let S be the relation on the set of equivalence classes of R ∩ R-1 such that (C, D) belongs to S, where C and D are equivalence classes of R, if and only if there are elements c of C and d of D such that (c, d) belongs to R. Show that S is a partial ordering.
Let L be a lattice. Define the meet (∧) and join (∨) operations by x ∧ y = glb(x, y) and x ∨ y = lub(x, y).