The Boolean Algebra is named after George Boole, a self-taught mathematician. 🙂 Let that be a lesson to you: Training trumps everything!
Over the next few weeks I’ll be posting up some reviews of some of the recent research in the intersection of complexity theory and cognitive science. However, these include references to the concepts of a Lattice, and more generally, to a mathematical structure called a Boolean Algebra. I’m posting this as a quicky reference guide.
You don’t need to know any of this to grok what I’ll be posting… but if you’re curious, this can be a jumping off point.
Boolean Algebra Definitions
- Boolean Algebra
- A Boolean Algebra is a kind of bounded distributive lattice where every element has a compliment .
Let’s unpack that a bit…
- A Lattice is a partially ordered set , such that for any , there exists a greatest lower bound , and least upper bound .
In logic we can define:
- as propositions
- as entailment
- as conjunction
- as disjunction
- Bounded Lattice
- We have a Bounded Lattice if there is a least element and a greatest element .
Which we can make more specific:
- Complimented Bounded Lattice
- We have a Complimented Bounded Lattice if for all , there exists its compliment such that the conjunction .
DeMorgan’s Laws For Our Lattice
Given that in a Boolean Algrebra, complements are unique, then . This means DeMorgan’s Laws apply because is order preserving and of period two. That is:
- if , then
Therefore, we get DeMorgan’s laws:
We also have:
- Distributive Lattice
- A lattice is Distributive if:
Orthocomplementation On A Bounded Lattice
In an arbitrary lattice, is an Orthocomplement if it is:
- period two
- order inverting
- is the compliment of
So, what we’ve built is an Orthocomplemented Lattice where was our original lattice, and is the orthocomplement.
Which is to say that a Boolean Algebra is a Complemented Distributive Lattice.
- A generalization of a Boolean Algebra that does not need to be distributive.
- See the paper Third Life Of Quantum Logic … , for a nice explanation. This is a take on that and others.
- Orthogonal Compliments & The Lattice Of Subspaces
Now go lift something heavy,