if A and B are two equivalence classes in P(ω)/Fin, we define to be the equivalence class of , where a and b are some (any) elements of A and B respectively.Now let a0, a1, … be pairwise disjoint infinite sets of naturals, and let A0, A1, … be their corresponding equivalence classes in P(ω)/Fin. Then given any upper bound X of A0, A1, … in P(ω)/Fin, we can find a lesser upper bound, by removing from a representative for X one element of each an. Therefore the An have no supremum. Properties of complete Boolean algebras Every subset of a complete Boolean algebra has a supremum, by definition; it follows that every subset also has an infimum (greatest lower bound). For a complete boolean algebra, both infinite distributive laws hold if and only if it is isomorphic to the powerset of some set. For a complete boolean algebra infinite de-Morgan's laws hold. A Boolean algebra is complete if and only if its Stone space of prime ideals is extremally disconnected. Sikorski's extension theorem states that if A is a subalgebra of a Boolean algebra B, then any homomorphism from A to a complete Boolean algebra C can be extended to a morphism from B to C. The completion of a Boolean algebra The completion of a Boolean algebra can be defined in several equivalent ways: The completion of A is (up to isomorphism) the unique complete Boolean algebra B containing A such that A is dense in B; this means that for every nonzero element of B there is a smaller non-zero element of A. The completion of A is (up to isomorphism) the unique complete Boolean algebra B containing A such that every element of B is the supremum of some subset of A. The completion of a Boolean algebra A can be constructed in several ways: The completion is the Boolean algebra of regular open sets in the Stone space of prime ideals of A. Each element x of A corresponds to the open set of prime ideals not containing x (which is open and closed, and therefore regular). The completion is the Boolean algebra of regular cuts of A. Here a cut is a subset U of A+ (the non-zero elements of A) such that if q is in U and p ≤ q then p is in U, and is called regular if whenever p