The largest cardinality among all the maximal independent sets is called the .
Of the many applications that arise, one in particular is in coding theory. We’ve discussed some basics of coding theory on this site as well.
In the converse direction, we have Stone’s representation theorem (see below), which asserts (among other things) that every abstract Boolean algebra is isomorphic to a concrete one (and even constructs this concrete representation of the abstract Boolean algebra canonically).
So, up to (abstract) isomorphism, there is really no difference between a concrete Boolean algebra and an abstract one.
An identity is a statement true for all possible values of its variable.
The first Boolean identity is that the sum of anything and zero is the same as the original.
As with Boolean algebras, one can now define an to be a set with the indicated objects, operations, and relations, which obeys axioms 1-5.
Again, every concrete -algebra is an abstract one; but is it still true that every abstract -algebra is representable as a concrete one?
Frank Harary should also be credited with his massive work in bringing applications of graph theory to the sciences and engineering with his famous textbook written in 1969.
My own research forced me to stumble into this area once my research partner, Jason Hathcock, suggested we explore the idea of viewing dependency relations in the sequences of variables we were studying as digraphs.