John Baez: I believe ‘special -ring’ is an old-fashioned term for what almost everyone now calls a lambda-ring, the ‘nonspecial’ ones having been found to be too general. This, at least, is what Hazewinkel says in his article cited on our page about lambda-rings. So I believe this page here should be folded in with lambda-ring.
At the very least, we should give both definitions of -ring — special and unspecial — over at lambda-ring. When I last checked, that page did not include a definition.
These form an abelian group under multiplication with the constant power-series as unit. What may be less familiar is that there is a commutative associative binary operation on this set that distributes over multiplication making into a commutative ring, for which the function taking to is a homomorphism. So what is usually called multiplication of power-series becomes addition in this ring, with as zero; very confusing. Of course, is a functor from rings to rings and is a natural transformation. In fact it is the counit of a comonad.
How is defined? We impose the condition
for and . It is now clear from distributivity that if then . But what happens if is not a product of linear factors? Newton’s theorem on symmetric polynomials comes to the rescue. For we note that the coefficient of in is a symmetric function in the and so can be expressed as a polynomial over the integers in the first coefficients of and of . In this way we have indicated that a universal formula for multiplication in exists, though we may not have written it down explicitly.
We use the same trick in defining the comultiplication
but now our old-fashioned notation and use of indeterminates starts to cause trouble. A power-series in is really just a sequence . We demand that
Again, to define on an arbitrary power-series, factorize it formally into linear factors, apply the rule and distributivity, and apply Newton’s theorem.
A special -ring is a coalgebra for the comonad above. If is the costructure map for such a coalgebra, we define unary operations on by the formula
The counit condition forces . It is also traditional to denote by . Note that and that .
The functor is representable by a commutative Hopf algebra , and so has a left adjoint. The underlying ring of is , the free special -ring on one generator ().
In the terminology of bimodels and its left adjoint is . So the theory of special -rings is a monadic extension of the theory of rings.