{"content":[{"text":"Boolean algebras","bbox":{"left":0.427722,"top":0.162609,"width":0.14172400000000002,"height":0.015110000000000012}},{"text":"BIG FACT (Theorem 13.4.6): All ﬁnite Boolean algebras are isomorphic to \n (i.e., “look like”) the inclusion lattice of the power set of some ﬁnite set S.","bbox":{"left":0.181134,"top":0.197203,"width":0.634934,"height":0.03335000000000002}},{"text":"We say that the lattice [L; ∨, ∧] is isomorphic to the lattice [L′; ∨′, ∧′] if there \n is a bijection f from L to L′ such that for all x, y in L, f(x∨y) = f(x)∨′f(y) \n and f(x ∧ y) = f(x) ∧′ f(y). \n D30 (the set of divisors of 30 ordered by divisibility) is a Boolean algebra; \n 1 is the bottom element (the “0”), 30 is the top element (the “1”), meet \n corresponds to gcd, join corresponds to lcm, and the complement of a is \n 30/a.","bbox":{"left":0.181134,"top":0.246781,"width":0.634958,"height":0.141639}},{"text":"D30 is isomorphic to the power set of {2, 3, 5} ordered by inclusion. The \n isomorphism from D30 to P(2, 3, 5) sends an integer n to the set of its prime \n divisors. The reverse isomorphism from P(2, 3, 5) to D30 sends every subset \n of {2, 3, 5} to the product of the elements of the subset. Note that this \n requires that we treat the product of the elements of the empty set as 1; for \n more on this, see http://jamespropp.org/2190/factorial.pdf.","bbox":{"left":0.181134,"top":0.406267,"width":0.6349170000000001,"height":0.106686}},{"text":"More generally, Dn (ordered by divisibility) is a Boolean algebra whenever \n n is squarefree (i.e., has no repeated prime factors), and is isomorphic to \n the inclusion lattice of P({p, q, . . . , r}), where {p, q, . . . , r} is the set of prime \n divisors of n.","bbox":{"left":0.181133,"top":0.531178,"width":0.634917,"height":0.06982799999999989}},{"text":"What about nonsquarefree values of n? When n is a product of powers of \n two diﬀerent primes, say piqj, then Dn is isomorphic to (Ai × Aj, ⪯), in the \n notation of problem 13.1.6. Likewise, when n is piqjrk with p, q, r distinct \n primes, then Dn is isomorphic to Ai × Aj × Ak, equipped with the partial \n ordering in which (a, b, c) ⪯ (a′, b′, c′) ⇔ a ≤ a′ and b ≤ b′ and c ≤ c′.","bbox":{"left":0.181133,"top":0.619231,"width":0.634899,"height":0.09877200000000008}}],"images_bbox":[],"images_bbox_not_text_overlap":[]}