Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
There’s something just a bit odd though about the notion of idempotent monoid in Ab. When one says that a ring is a monoid in Ab, one means a monoid in the monoidal category (Ab,⊗) with the standard tensor product. However, this tensor product isn’t cartesian, and so expressing the notion of an idempotent monoid (where the axiom xx=x involves duplication of a variable) doesn’t go down so smoothly.
Added a bit about the monad for Boolean rings:
The free Boolean ring on a set X can be identified with 𝒫f𝒫fX, where 𝒫f:Set→Set assigns to any set the set of all its finite subsets. In fact 𝒫f:Set→Set can be made into a monad in two different ways: the monad for semilattices and the monad for vector spaces over the field with 2 elements. These two monads are related by a distributive law, making 𝒫f𝒫f into the monad for Boolean rings.
Also mentioned:
The category of Boolean algebras is discussed further in BoolAlg, but some of the results about this category are proved there by working with the equivalent category of Boolean rings.
J-B, the additive inverse in a Boolean ring does not interpret negation. (In fact, additive inversion is the identity!) Negation is given by x↦1−x, or equivalently by x↦1+x.
While not exactly wrong, I think it’s quite rare to refer to “characteristic 1”. Since “characteristic” usually refers to fields or perhaps algebras over fields, I’ve made an adjustment and hope no one minds.
Let boolean denote the type of booleans or decidable truth values. Since Booleans rings are 𝔽2-algebras, and 𝔽2 is equivalent to boolean, one could say that Boolean rings are boolean-algebras.
Ba-dum pah. Or: bool-yah!
It would be nice to describe explicitly the equivalence of categories between BoolAlg and BoolRing.
It is described in this entry how the two functors act on objects but not how they act on morphisms. Even less what are the natural isomorphisms F∘G≅Id and G∘F≅Id.
I’ll put something in.
Quick question. Are there any non-associative unital 𝔽2-algebras? Or are all unital 𝔽2-algebras associative?
By 𝔽2-algebra, I mean an associative one. But there are many examples of what you want, of unital bilinear magma structures that are nonassociative.
All you need to do is take any nonassociative unital magma M. (For example, take M={b,c,d,e}. and then create a multiplication table so that the first row and first column, each indexed by e, makes e the identity element, and then fill in the rest of the multiplication table by randomly choosing entries. Most of those tables will be nonassociative.) Then, take the free vector space 𝔽2[M] on M. The unique bilinear extension 𝔽2[M]×𝔽2[M]→𝔽2[M] of the multiplication table then gives a unital bilinear magma structure which is nonassociative because M is.
1 to 21 of 21