User talk:Lord Farin/Backup/Definition:Boolean Algebra/Definition 2

I have applied the convention as in Robert Goldblatt's "Topoi: The categorial analysis of logic" to avoid the $\land,\lor$ and $\neg$ Awodey uses to signify the operations of a boolean algebra. This is done to make an easier distinction between the symbols used in logic, and their analogues in Boolean algebras. Symbology is as in Goldblatt. --Lord_Farin (talk) 09:11, 23 August 2012 (UTC)

Is $\left({S, \preceq}\right)$ supposed to be $\left({S, \le}\right)$, or the other way around in the rest of the page? --GFauxPas (talk) 15:53, 23 August 2012 (UTC)

Good catch. I literally copied the axioms from Awodey, and he uses $\le$, while $\preceq$ is preferred on PW. --Lord_Farin (talk) 16:47, 23 August 2012 (UTC)

Now I look at this, I notice that Awodey's object is not the same as Deskins's. The latter does not have an order imposed: that is, the pure object is just the underlying set and the operations. Awodey's object is Deskins's object, but with an ordering.

Checking further, I see that Deskins's object as we have defined it here as Definition:Boolean Algebra/Definition 1 is defined in 1965: Seth Warner: Modern Algebra as a Huntington Algebra: exercise 21.26, mistakenly referenced as Exercise 21.16 in the index. It then appears that Huntington's Theorem states that for every Huntington Algebra, there exists a unique lattice ordering $\preceq$ with the properties that make the resulting structure a Boolean algebra.

There are several exercises in this section which introduce the concepts of boolean ring (as defined in wikipedia, etc.), boolean algebra (the object created by adding that ordering to a Huntington algebra) and also Stone's Theorem is introduced which give a connection between a Boolean ring and a Boolean lattice (which seems at first glance the same as a Boolean algebra).

And there's more. I have just remembered another book I've got somewhere which goes into Huntington algebras in some detail (in the context of digital computer hardware, my major at University 30 years ago), but I can't for the life of me find it at the moment.

TL;DR: would it be appropriate to rename Definition:Boolean Algebra/Definition 1 to Definition:Huntington Algebra and make the (forthcoming) page "Equivalence of Definitions of Boolean Algebra" be "Huntington's Theorem" instead? We can take it onwards from there. --prime mover (talk) 19:39, 23 August 2012 (UTC)

That is the direction I was aiming for indeed (that def. 1 defines an ordering (by $a \preceq b \iff a * b = a$ (equivalently, $a \circ b = b$))). Only for the better it is that we can cover all this ground, see also Balg on MathWorld. Apparently we are about to call what MW calls Balg, a Huntington Algebra (and, further down the road, a Robbins algebra and a Wolfram algebra), but this is not bad. The algebraic descriptions of a boolean algebra are interesting in their own right. As we have a source, this sounds like an excellent idea. --Lord_Farin (talk) 21:38, 23 August 2012 (UTC)

Balg on MathWorld is confusing and inaccurate: it mentions the ordering but concretises it to subset inclusion (notwithstanding any poset is isomorphic to a subset of the power set of the underlying set), and then does not include the axioms defining that partial ordering. Therefore I would suggest it were not included as a source for the Boolean Algebra page. At this rate, it may well be that the only accurate, complete and consistent accounts of these structures will be found on various unassuming mathematics wikis. --prime mover (talk) 06:39, 24 August 2012 (UTC)

We now have the page Definition:Huntington Algebra. Fair-sized work package ahead to make everything consistent.--prime mover (talk) 22:11, 23 August 2012 (UTC)

I'll try to bend some brain cells and contribute tomorrow. --Lord_Farin (talk) 22:12, 23 August 2012 (UTC)

Think it's all done now. --prime mover (talk) 06:34, 24 August 2012 (UTC)