# FOM: Boolean algebra vs Boolean ring

Vaughan Pratt pratt at cs.Stanford.EDU
Tue Mar 10 23:39:59 EST 1998

```From: Stephen Simpson 3/10/98 6:04PM
>Seriously, my statement about Boolean algebras vs Boolean rings is
>true (for everybody!), and the distinction is useful for all
>mathematicians.

From: Sikorski, R., "Boolean Algebras", Springer-Verlag, 1960:
>17.1  Every Boolean algebra is a Boolean ring...
>Conversely, every Boolean ring is a Boolean algebra....

Simpson:
>For a start, could you please state your definitions of Boolean ring
>and Boolean algebra?  After you have stated them, I'll compare them
>with definitions that are usual in textbooks.  And we'll try to come to
>some agreement.  After that, we'll examine whether they are isomorphic.

Sikorski (op.cit.):
>A Boolean algebra is a non-empty set with three operations AUB, A^B, -A
>satisfying axioms (A1)-(A5) [omitted here]
>A ring is said to be a Boolean ring provided it contains a unit 1 and
>A.A = A for every A.

From: Colin McLarty 3/10/98 7:48PM
>Now Vaughan, like many other logicians and mathematicians, understands an
>algebraic theory without distinguishing "basic" operations from "derived"
>ones--that is primitive from defined operations. There is nothing arcane