Wikipedia:Reference desk/Archives/Mathematics/2014 September 17

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September 17

A Boolean algebra

I'm taking a discrete math class. I'm confused by some of the terminology and I've already asked my teacher for a clarification/definition and I'm still confused. So, could someone explain what "a Boolean algebra" is? For instance, one of my homework questions says:

Let B be a Boolean algebra.

Define the relation ≤ on a Boolean algebra B by x≤y if xy=x.

Prove that if x is an atom of B, and y ∈ B, then either xy=0 or xy=x.

I'm not looking for help on the problem as much as I'm looking for an explanation of what an algebra or more specifically a Boolean algebra is. My teacher basically restated what's in our articles here about fields and rings and such. I think I'd be satisfied with a fairly simplistic explanation, if one can be given, similar to how electrons around an atom are often simply explained as a particle in an orbit when the full explanation is more complicated. Thanks, Dismas|^(talk) 05:14, 17 September 2014 (UTC)

Well, the long answer is that a Boolean algebra is a collection of elements and a pair of operations that satisfy a list of axioms. I assume that's the one your teacher gave. Another way to think of it is that a Boolean algebra is a collection of sets: You start with one big set, and take some of its subsets. 1 is the starting set. 0 is the empty set. xy means $x\cap y$, and x+y means $x\cup y$.--80.109.106.3 (talk) 10:40, 17 September 2014 (UTC)

Confusingly, a Boolean algebra is not an algebra. Most of the axioms for a Boolean algebra are the same as for a distributive lattice, but there is an additional operation, negation, and axioms associated with it. We have several basic articles on Boolean algebras but Boolean algebra (structure) gives an axiomatic treatment which is probably what you're looking for. Atom (order theory) defines atom in the sense used in the problem. --RDBury (talk) 14:38, 17 September 2014 (UTC)

In case it helps, our article is at Boolean algebra (structure). RDBury's (first) link goes to the wrong place, though to my mind it's a fairly natural mistake.
There are two basic notions called Boolean algebra, related but distinct, one that uses the term as a count noun (as you have used it) and one that uses it as a mass noun (roughly, the equations that are true in all Boolean algebras, and the methods of manipulating them).
There was a time that the straight search term Boolean algebra went to the article on the structure (the count-noun sense), but that resulted in too much confusion on the part of readers who were expecting the other notion. That did need correction. Unfortunately it was impossible to agree on exactly how to correct it, and it turned into a bloody war with a very unsatisfactory outcome. --Trovatore (talk) 16:51, 17 September 2014 (UTC)

Thanks, I missed that when I skimmed the articles. We also have Boolean algebras canonically defined which seems, and again I just skimmed it, to cover the subject from a constructive rather than axiomatic viewpoint. Perhaps analogous to the distinction between Group (mathematics) and Permutation group or between Synthetic geometry and Analytic geometry. --RDBury (talk) 20:49, 17 September 2014 (UTC)

The "canonically defined" article is a bit ... idiosyncratic. I really think something ought to be done about it but I don't have the belly for it anymore.

On the other hand, there certainly is an approach based on defining Boolean algebras in a particular representation rather than as models of certain axioms. This is the content of the Stone representation theorem, and yes, it's closely analogous to treating all groups as subgroups of some S_κ for sufficiently large κ, or all manifolds as submanifolds of Rⁿ for sufficently large n. It's not always the most natural representation — but then, neither are the other two examples. --Trovatore (talk) 21:44, 17 September 2014 (UTC)

Aside: Near the top of boolean algebra, it says "exists as a core data type in all modern programming languages generally abbreviated to as type bool". That "to as type bool" is awkward and needs changing, but I can't decide what to. Any suggestions? -- SGBailey (talk) 12:44, 18 September 2014 (UTC)

Good call, I fixed it. SemanticMantis (talk) 15:19, 18 September 2014 (UTC)