Talk:Boolean ring

Identity requirement

Sirs:

M.H.STONE defined a ring to be Boolean if every element is idempotent. This does not require the presence of an identity (neutral element for multiplication). When considering the equivalence of Boolean algebras with what you call Boolean rings, it is needed that the ring be with identity.

-- 193.205.5.2 — Preceding unsigned comment added by 193.205.5.2 (talk) 11:17, 20 August 2003 (UTC)

I don't really understand the comment. If you square x + x (instead of x + 1 as on the page), don't you find 2x = 4x anyway?

Charles Matthews 11:32, 20 August 2003 (UTC)

The point is that a Boolean algebra must have an identity (a top element in lattice-theoretic terms).

So a Boolean ring without identity would not yield a Boolean algebra, even though it would indeed satisfy all of the other requirements.

(Instead, it would yield a relatively complemented lattice.)

To some extent, 193.205.5.2 is still wrong, because every ring has an identity; otherwise it's just a rng.

But this is a relatively modern version of the definition of "ring", which some people still haven't come around to; in particular, identities weren't required in Stone's day.

You can still see the legacy of this in the term sigma-ring; the only difference between a sigma-ring and a sigma-algebra is that the former need not have an identity.

In measure theory, this can have some rather important consequences, especially to the development of the theory of non-sigma-finite measure spaces.

Similar (but less drastic) consequences arise in the study of Boolean rings, which are basically the same as sigma-algebras but without the countable unions that measure theory requires.

Thus Stone's definition of a Boolean ring would (I suppose) today be called a Boolean rng; and you can certainly find people today that haven't changed their definitions.

-- Toby Bartels 03:11, 24 August 2003 (UTC)

Change of notation

Boolean rings are in an important intermediate position between rings and lattices. The modern standard notation for lattices, ∧∨, minimises confusion and is used in this article. However, the traditional notation .+ is still used in many places, especially digital engineering. People who are used to this notation have a hard time reading articles in the modern notation. Here they have the additional problem that the notation with which they are familiar is used for the ring structure. I would guess that this makes the article essentially incomprehensible for them.

Since ring multiplication and Boolean algebra conjunction coincide here, they are not really such a big problem. Therefore I thought it best to write only addition in a new notation. I think this minimises the confusion for everybody. The particular choice of symbol for addition was easy, since ⊕, the obvious candidate, is already in use as a standard symbol for exclusive or; which is exactly what we have here.

The main disadvantage is that we also need the symbol $\ominus$, which does not yet display reliably in all browsers, so it will typically be served as a picture. --Hans Adler (talk) 17:21, 29 January 2008 (UTC)

This new use of ⊕ is a bad idea. There are already 3 different and incompatible systems of notation for Boolean rings/algebras, and inventing a 4th is not going to help the confusion. Wikipedia should record and explain the notations that are used and not invent its own new notation; see Wikipedia:Avoid neologisms. Moreover the symbol ⊕ is already used in commutative algebra for something quite different. R.e.b. (talk) 16:55, 19 February 2008 (UTC)

venn diagram

the diagram produces confusion, since it represents disjunction, not ring addition, which is the difference to a boolean algebra. i propose replacing it by a venn diagram for xy and x + y. a third diagram could be 1+x = ¬ x. or is it not usual to draw venn diagrams for boolean rings? --141.30.71.247 (talk) 15:40, 9 July 2009 (UTC)

i moved the existing diagram to the boolean algebras section, as a preliminary solution. --141.30.71.247 (talk) 13:40, 10 July 2009 (UTC)

boolean rings are not necessarily boolean algebras

The article currently says "A Boolean ring is essentially the same thing as a Boolean algebra..."

And this confusion also appears in this talk page, with some comment asserting that this is the case even though it obviously is not the case.

So let's go with a concrete example: if * is defined as "greatest common divisor" (or "least common multiple") and x is a non-negative integer, then x = x*x and we have a boolean ring. But we do not have any "not" operation here, so this is not a boolean algebra.

(I would be updating the main page, but I forgot my user name.) —Preceding unsigned comment added by 159.54.131.7 (talk) 14:42, 14 July 2010 (UTC)

Your counterexample isn't a counterexample -- you're not defining a ring at all, since you're giving only the ring multiplication, but not the corresponding addition operator. Using ordinary addition certainly doesn't work, as the ring would fail the distributive law since gcd(3,1+1)=1 does not equal gcd(3,1)+gcd(3,1)=2. –Henning Makholm (talk) 16:41, 9 January 2012 (UTC)

Use lcm as ring addition if you are using gcd as ring multiplication. --96.255.156.195 (talk) 16:13, 3 June 2012 (UTC)

Lcm does not have additive inverses, so it can’t make ring addition.—Emil J. 10:06, 19 June 2012 (UTC)

I am curious if the notion of Boolean algebra in logic is the same as algebra over [math] \mathbb F_2 [/math] in commutative algebra, because the typical finite algebras over [math]\mathbb F_2[/math] are well unital and associative, but usually not idempotent. To give a more concrete example [math] \mathbb F_4=\mathbb F_2(\omega)[/math] with [math]\omega^2 = \omega+1[/math], because [math] X^2+X+1[/math] is an irreducible polynomial, but [math] X^2+X [/math] is not.

MelchiorG (talk) 23:05, 23 March 2023 (UTC)

Unification decidible

According to , unification is decidible in boolean rings. Reportedly, an algorithm to solve every system of equations w.r.t. a boolean ring is given in . This should be noted somewhere in the article. Jochen Burghardt (talk) 07:54, 17 May 2013 (UTC)

References