From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3416 Path: news.gmane.org!not-for-mail From: "David Ellerman" Newsgroups: gmane.science.mathematics.categories Subject: Re: Linear--structure or property? Date: Sun, 3 Sep 2006 11:32:20 -0700 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain;charset="us-ascii" Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241019291 8642 80.91.229.2 (29 Apr 2009 15:34:51 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:34:51 +0000 (UTC) To: Original-X-From: rrosebru@mta.ca Sun Sep 3 20:59:37 2006 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 03 Sep 2006 20:59:37 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1GK1s1-0007NW-43 for categories-list@mta.ca; Sun, 03 Sep 2006 20:59:21 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 6 Original-Lines: 121 Xref: news.gmane.org gmane.science.mathematics.categories:3416 Archived-At: 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" To: "Categories list" Subject: categories: Re: Linear--structure or property? > Dear Steve, > > It is true that constructing such examples with more than one object is ...