Re: Linear--structure or property?

["David Ellerman" <david@ellerman.org>]

When a Boolean algebra B is treated as a Boolean ring in the usual manner,
the meet is the multiplication. In his little-known thesis, Herbrand noted
that a BA could also be construed as a ring with the join as the
multiplication (see Church's tome on logic). Gian-Carlo Rota noted that both
these Boolean rings were "opposite" quotients of what he called a "valuation
ring" V(B,Z_2) which carries *both* multiplications and one addition. What
is usually called "Boolean duality" (e.g., DeMorgan's law) is an
anti-isomorphism of the valuation ring that swaps the two multiplications
and leaves addition the same. The "trick" in constructing such rings was to
see that the bottom element z (representing the null set) should be a
separate element than the 0 of the ring. The usual Boolean ring constructed
from a BA is really the quotient of the valuation ring that identifies z and
0, i.e., V(B,Z_2)/(z). The Boolean ring noted by Herbrand is the quotient of
the valuation ring that identifies the element representing the top u with
0, i.e., V(B,Z_2)/(u). Remarkably, the valuation ring construction V(L,A)
works for any distributive lattice L and any commutative ring A, not just a
BA B and Z_2 and the anti-isomorphism works just as well. Thus we have
"Boolean duality" over arbitrary commutative rings A; it has nothing to do
with 0-1 nature of Z_2. This general theory of Boolean duality was developed
in a series of papers by Geissinger in Arch. Math. 1973. See Rota's book
"Finite Operator Calculus" for material on valuation rings.

For the opposite question of two additions and one multiplication in a
semi-ring, the natural setting is the algebraic treatment of series addition
a+b and parallel addition a:b = 1/((1/a)+(1/b)) (e.g., from electrical
circuit theory) in what might be called a "series-parallel algebra." Every
commutative group G (written multiplicatively) generates a series-parallel
division algebra SP(G) (think of all series-parallel circuits of resistors
that could be generated with the elements of G as the resistors). It is a
"division algebra" in the sense that the SP algebra is also a multiplicative
group where the inverse of any SP circuit is obtained by taking the
series-parallel conjugate circuit (see any circuit theory book) with the
atomic resistances from G replaced by their inverses in G. Then "taking
reciprocals" is the anti-isomorphism of the SP algebra that swaps the two
additions leaving multiplication the same, and it algebraically captures
series-parallel duality just as the anti-isomorphisms of the valuation rings
captured Boolean duality. The SP algebra SP({1}) of the trivial group is
just the positive rationals Q^+ (i.e., any rational resistance can be
obtained as a series-parallel circuit with unit resistances) and the
anti-isomorphism that swaps the two additions is just "taking the
reciprocal" r-->1/r. There is an 1892 paper by the great combinatorist Percy
MacMahon published in "The Electrician" that explains the notion of a
conjugate of a series-parallel circuit and shows that if each resistence is
1 (i.e., G = {1}) and the compound resistance of an SP circuit is R, then
the resistance of the conjugate SP circuit is 1/R (in case your library does
not carry "The Electrician" from 1892, see the Collected Papers of
MacMahon). Paying attention to the duality of series and parallel addition
on the positive rationals or reals gives some cute dualities. For instance,
instead of saying that the geometric series 1+x+x^2+... converges to 1/(1-x)
for any positive x<1, it is easier to say that 1+(1:x)+(1:x)^2+... converges
to 1+x for any positive x. And dually, the parallel sum infinte series
1:(1+x):(1+x)^2:... converges to 1:x for any positive x.

Both Rota's valuation rings and the series-parallel algebras are explained
in two chapters of my book: "Intellectual Trespassing as a Way of Life:
Essays in Philosophy, Economics, and Mathematics" (Rowman-Littlefield,
1995).

Cheers, David
__________________
David Ellerman

Visiting Scholar
University of California at Riverside

Email: david@ellerman.org
Webpage: www.ellerman.org

View my research on my SSRN Author page:
http://ssrn.com/author=294049


Now out in paperback: Helping People Help Themselves: From the World Bank to
an Alternative Philosophy of Development Assistance. University of Michigan
Press. 2006. For more information, see my website: www.ellerman.org . Book
available at better booksellers online.

-----Original Message-----
From: cat-dist@mta.ca [mailto:cat-dist@mta.ca] On Behalf Of Fred E.J. Linton
Sent: Sunday, September 03, 2006 2:27 AM
To: Categories list
Subject: categories: Re: Linear--structure or property?

George Janelidze and others answer affirmatively the question

> Could you have two (semi)ring structures on the same set with the same
> associative multiplication?

(attributed to Mike Barr) without ever noticing that the related question,
of having two (semi)ring structures on the same set, with the same addition,
also has answer YES.

For instance, take the additive group of 2x2 matrices with integer entries
(or entries from any semiring) and notice that, apart from the usual matrix
multiplication, there is also the sophomoric, or pointwise, multiplication
(so called since it is generally only sophomores in the first week of their
first linear algebra course who, following the analogous pointwise
definition of matrix addition, would wish to multiply two matrices by
multiplying their corresponding entries).

Not quite sure though how this impacts the situation with more than one
object.

-- Fred

------ Original Message ------
Received: Fri, 11 Aug 2006 01:06:56 PM EDT
From: "George Janelidze" <janelg@telkomsa.net>
To: "Categories list" <categories@mta.ca>
Subject: categories: Re: Linear--structure or property?

> Dear Steve,
>
> It is true that constructing such examples with more than one object is
...