Lattice theory by Garrett Birkhoff

By Garrett Birkhoff

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B:c), A (A. A. Monteiro, Rev. Union Mat. Argentine. 17 (1955), 148--60) 12*. Boolean Rings The analogy between the algebra of logic and ordinary algebra has been emphasized by many mathematicians, from Boole [lJ on. However, it was M. H. t A ring may be formed from the elements of any Boolean algebra by defining multiplication as meet and addition as the " symmetric difference ": ah = a /\ b and a (29) +b= (a A b') v (a' /\ b). It is obvious that both operations are commutative. tive law for addition: (a + b) + c = {[(a We next check the associa- /\ b') V (a' /\ b)] /\ c'} V {[(a /\ b') V (a' /\ b)]' /\ c}.

Y PROOF. BylA,x /I. u = x;byL3,Ll,x vu= xv (xv y) =(xv x) v y = x v y = u. The other cases are analogous, using symmetry in x and y and duality. (By IA, u v v = x v y v (x /I. ) COROLLARY. Let F 2 = 22 be the lattice of Figure 7, and kt a, b EL (L any FIGURE 7 lattice) be given. 9: F 2 -L. Then the mapping x - a, y - b can be e:dende,d to a morphism The preceding results are usually summarized in the statement that F 2 is the free lattice with generators x, y; it has just four elements, and is distributive (in fact, a Boolean lattice).

Also, if we define x' = 1 - x, then x /\ x' = x(I - x) = 0 and x v x' = x + (1 - x) + x(l - x) = l; hence our definition makes R into a complemented lattice. Finally, x /\ (y v z) = x(y + z yz) = xy + xz xyz = xy + xz - xyxz (x /I. y) v (x /I. z), and so R is a Boolean lattice. For any Boolean ring R(B) with unity obtained from a Boolean algebra B by the definitions (29), the Boolean lattice B(R(B)) obtained from the ring R(B) by (33) is isomorphic with B. We omit the proof. Stone [1] has defined a generalized Boolean algebra as a reWlively complemented distributive lattice with a 0 (but not necessarily a I), such as the following.

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