Talk:Boolean algebras canonically defined

message: This is a tentative article to try out some ideas about Boolean algebras.  The existing article is already very good so this one may be useful only as a source of ideas and alternative perspectives.  --Vaughan Pratt 20:40, 7 August 2006 (UTC)
thread_title: Note moved from main page

message: :This rationale was moved here from the main article as being more appropriate to the talk page. ~~
The article on Boolean algebras defines Boolean algebras formally from the point of view of lattice theory.  Ring theory being no less a valid point of view, as explained below, the question arises as to whether Boolean algebras even have a "neutral" point of view from which they can be explained.  Category theory partially solves this with an account that is neutral between lattice theory and ring theory, but now from the point of view of category theory whose neutrality has yet to be generally accepted.  We solve this here by translating the category theoretic account into ordinary algebra; since Boolean algebra is a part of algebra one cannot get much more neutral than this.  
In more detail, Boolean algebras have traditionally been defined analogously to other structures such as groups, rings, vector spaces, and lattices.  These others all have characteristic operations and equational axioms, with no reason to suppose that there might be better operations or equations on which to base the definition.  In particular when we count the number of binary operations assumed by each definition, groups always employ one binary operation while rings and lattices always use two.  In contrast, although Boolean algebras are traditionally defined in terms of the two lattice operations of join (disjunction) and meet (conjunction), they can be defined equally rigorously in terms of a single binary operation, Sheffer stroke, NAND or ¬(x∧y).
One might acknowledge this alternative choice of operations while continuing to defend the essence of Boolean algebras as being lattices at heart.  In fact a Boolean algebra is exactly a lattice that is both distributive and complemented.  Marshall Stone [1936], in the course of pointing out the analogy between lattices and rings while writing his celebrated paper on the duality of Boolean algebras and Stone spaces, noticed that, for lattices that are Boolean algebras, this was more than just an analogy.  When the definition is based on exclusive-or instead of inclusive-or while retaining conjunction, a Boolean algebra is seen in fact to be a ring.  This leads to the definition of a Boolean algebra as simply any ring satisfying x2 = x, called a Boolean ring.
So while a Boolean algebra is fundamentally a lattice, it is at least as fundamentally a ring.  The question then arises as to whether the concept is one that has a multiplicity of equally good definitions, or whether there is some underlying common essence to all these definitions.
William Lawvere [1963] introduced the notion of (finitary) algebraic theory as a category with (finite) products, whose equations are represented as the commutative diagrams therein, and whose models are the product-preserving set-valued functors therefrom.  Since functors preserve commutativity of diagrams, these models satisfy those equations.  In this framework Boolean logic as the equational theory of Boolean algebras can be defined as the category of finite power sets and their functions.  A Boolean algebra is then any model of that category in Lawvere's sense.  An advantage of this definition is that it is unbiased: not only does it not express any preference for lattices or rings, it does not even mention either concept.
The present article amounts to a translation of Lawvere's categorical definition into the more widely used language of ordinary algebra.  In this translation a Boolean algebra becomes any model of the finitary equational theory of two values.  This definition requires operations to be finitary, but imposes no other restriction and hence implicitly equips the algebra of two values with all possible finitary operations.  The equational theory of Boolean algebras resides in the composition law of the category representing the Lawvere-style theory, which we translate into a more conventional equational axiom system, with a single axiom schema expressing the composition of that category in lieu of the more customary finite list of axioms.  This schema makes explicit the neutrality implicit in Lawvere's axiomatization: no Boolean operation is given preferential treatment because the axiom schema is instantiated with one axiom per Boolean operation. Vaughan Pratt 22:53, 8 July 2007 (UTC)
replies: text: Bravo! Clap.. clap clap! linas (talk) 13:32, 20 September 2012 (UTC)
thread_title: Rationale for this article

message: Jon, I appreciate that use of citation templates is a personal choice. Since I created the References section in the first place, I would suppose that my choice would take precedence over yours. But your reversion lost much more than that, because I had added ISSN links, page numbers, URL and format info, and so on. I have reverted back to my choice, but I will defer to Vaughan as to which we should use.
This gives me an opportunity to say two things. I'm enjoying watching the article evolve, and I extend thanks to Vaughan for writing it. Also, I had to guess that the references I cite are the ones intended; Vaughan, please drop a note here if I'm wrong. --KSmrqT 04:05, 15 August 2006 (UTC)
thread_title: References

message: There were numerous mistakes and missing data in the new references, which I have attempted to correct/add. Please check carefully for sanity. I have made the Boole reference a reprint with scholarly commentary; I hope that's OK. The Schröder volumes were reprinted by Chelsea (now owned by AMS), but I cite the original; in neither case could I find an ISBN; and it is not clear if a specific paper, or volume, or part, or year is intended. Unfortunately, the AMS Chelsea site does not list Schröder as currently available. In my younger days I was spoiled by access to great university libraries, but these days we rely on Internet resources, so I have included a link for the Lawvere dissertation (which includes much supplementary material). Also, much as I personally appreciate having classic references, I think it would be a kindness to our readers to provide some more accessible references as well, if possible. --KSmrqT 07:00, 17 August 2006 (UTC)
thread_title: Re: New references

message: JA: We need to be careful about the use of the word linear in this context.  I think it's best to save it for the algebraic sense of linear, as in the 2^n linear functions h : B^n -> B, B = {0, 1}. These are just all the different sums under the exclusive disjunction (+), with (n choose k) sums of rank k.  For example, the rank 0 sum is 0, the rank 1 sums are x_1, …, x_n, the rank 2 sums are all of the form x_i + x_j, and so on up to the rank n sum x_1 + x_2 + … + x_(n-1) + x_n.  If we save linear for that sense, then we need another word for the sense in a phrase like "An alternative to this linear disposition of the Boolean operations …".  If I read that right, anyway.  Jon Awbrey 01:32, 17 August 2006 (UTC)
thread_title: Linear Thinking

message: JA: This is a critter that goes back to Leibniz, I think.  In some dialects of graph theory it's called the link operator.  It's a discrete analogue of the point-deleted neighborhood in calculus.  I used to call it the boundary operator, because that's what it looks like in venn diagrams, and I think that there's an official name for it in co/homology theory, but I've forgotten what little I knew of that.
JA: If you are thinking of a n-cube (incubus) as your favorite graph, then each point of the cube can be represented as a conjunction of n posited or negated variables.  So the all 1's node is the product of all positives x_1 x_2 … x_[n-1] x_n and the all 0's node is the product of all negatives (x_1)(x_2) … (x_[n-1])(x_n), here using parentheses for negation a la Peirce.  These are called singular propositions (or singular boolean functions) because the fibre of truth under such a function is a single point of the cube.  To be continued … Jon Awbrey 17:08, 17 August 2006 (UTC)
thread_title: Minimal negation operator

message: Someone found a better way to work around the bug the WikiMedia software has with nested superscripts or subscripts. The trick is to use a  wrapper. Compare:
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Since the use of the wrapper preserves the intended semantics in the markup, I've replaced all instances in the article (I hope!). --KSmrqT 11:42, 17 September 2006 (UTC)
thread_title:  Bug workaround 

message: The article states "The free countably generated, countably distributive sigma-algebra has the same nice structure as free CABAs, being the power set algebra 22N". What is the proof of this? It would seem to me that its cardinality must instead only be 2N, along the lines of the proof that there are only 2N Borel sets of reals. -Chinju (talk) 08:56, 15 November 2008 (UTC)
replies: text: Sorry for not noticing this sooner (I need to check my watchlist more often).  Not sure what I was smoking when I wrote that, but it's clear that 22N elements will need more than countable joins and countable distributivity.  I don't see any interesting replacement for it so I'll just delete it.  Thanks.  --Vaughan Pratt (talk) 18:40, 25 November 2009 (UTC)
thread_title:  Free countably generated, countably distributive sigma-algebra 

message: I just wanted to say that I like this article. Very rich in information and well formulated. Thanks to the creators for this tremendous piece of work!
82.82.87.235 (talk) 01:11, 10 July 2009 (UTC)
replies: text: This article is essentially a gift from a single author: User:Vaughan Pratt. Contributions by others were mostly limited to copy editing and maintenance work. Hans Adler 15:52, 10 July 2009 (UTC)
thread_title:  great 

message: In the article is said: Such an algebra can be defined equivalently as a complete Boolean algebra that is atomic, meaning that every element is a sup of some set of atoms.
In Lattice_(order) article atomic and atomistic are defined as:
* Atomic if for every nonzero element x of L, there exists an atom a of L such that a ≤ x ;
* Atomistic if every element of L is a supremum of atoms. That is, for all a, b in L such that a\nleq b, there exists an atom x of L such that x\leq a and x\nleq b. 
So in our article it should be atomistic instead of atomic, shouldn't it? Is it an error?
VictorPorton (talk) 23:52, 26 November 2009 (UTC)
replies: text: I've never seen the word "atomistic". In any case, it makes no difference, the two notions are trivially equivalent for Boolean algebras. We should stick with "atomic", because that's what they are normally called in the literature. — Emil J. 11:03, 27 November 2009 (UTC)
thread_title:  Atomic vs. atomistic