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