# FOM: Boolean algebra vs Boolean ring

Stephen G Simpson simpson at math.psu.edu
Mon Mar 16 13:30:03 EST 1998

Vaughan Pratt has been playing a dishonest game of refusing to admit
that there is any distinction whatsoever between Boolean algebras and
Boolean rings.  (In particular, Pratt has refused to admit that
Boolean algebras and Boolean rings have different signatures.)  I
think this game was for Pratt's own amusement, or maybe it was his
intention to disrupt the FOM list.  In any case, the task of cleaning
up the mess created by Pratt has fallen to me.

In support of his frivolous claims, Pratt 10 Mar 1998 20:39:59 quotes
Sikorski:

> 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....

but Pratt carefully omits the surrounding context.  This context makes
Sikorski's statement mathematically correct and supports the generally
accepted view of the matter, which is the opposite of Pratt's.

The full statement of Sikorski's theorem 17.1 reads as follows:

17.1.  Every Boolean algebra is a Boolean ring with the following

(6)      A+B = (A-B)v(B-A)

(7)      A.B = A^B

Conversely, every Boolean ring is a Boolean algebra with the
following definitions of join, meet and complement:

(8)      AvB = A+B+(A.B)

(9)      A^B = A.B

(10)      -A = 1+A

In both cases the algebraic zero and unit coincide with the Boolean
zero and unit, respectively.

-- Steve