Commentary by cpressey on Abstract Algebra works

Interior Algebras and Varieties

So one big thing here is that while interior algebras are often defined not-entirely equationally:

I(0) = 0, I(1) = 1, I(a) ≤ a, I(I(a)) = I(a), and I(a)I(b) = I(ab) for all a, b ∈ R. These were considered by McKinsey and Tarski

This paper gives a purely equational definition, which is

1. I(a)I(b) = I(b)I(a),
2. aI(a) = I(a),
3. I(I(a)) = I(a), and
4. I(a)I(b) = I(ab)

The other big thing is that this paper points out that this interior (or closure) operator can be added to algebra other than boolean algebras; namely, to rings and to semigroups.

And these succumb to Birkhoff's theorem:

Thus the class of interior semigroups is a variety of algebras if one views I as an operation; similarly for interior rings.

Varieties of Interior Algebras

282 pages is about what I expected from a PhD thesis in terms of length, but I didn't expect it to be single-spaced.

Lots in here I'm sure I don't need. I would like to find Blok's equational formulation of the interior (or closure) operator axioms, to compare it to that other paper's formulation. Maybe someday, but honestly probably not.

The Convolution Algebra

start - MathStructures

A database of abstract algebras. Useful as well as interesting (this needs to be a seperate dimension of rating)

Mathematics and Computation | Alg

Definition:Congruence Relation - ProofWiki