a question about equational classes

a question about equational classes

[James J. Madden]

There are many examples of functors between equational classes F:V -> W
that are defined by letting F(A) be the free W-algebra on the elements of
A modulo an equivalence generated by certain expressions of the form:

w(v(a)) = w'(v'(a)).

Here, w, w' are tuples of W-words, v(a) and v'(a) are tuples of V-words
that we fill in in all possible ways with elements of A.

A nice example is Joyal's definition of the "spectrum" functor from
commutative rings to distributive lattices.  Here, F(A) is the free
distributive lattice on the underlying set of A, modulo the relations:

1 = top,
0 = bottom,
a*b = a inf b,
a+b sup a sup b = a sup b.

(Under strong enough set-theoretic assumptions, F(A) is the distributive
lattice of compact opens in the spectrum of A, and if we let 'a' stand for
the element of F(A) corresponding to an element a from A, then 'a' may be
identified the "cozero" set of a.)

Several similar examples of spectrum functors defined by generators and
relations appear in Johnstone's book on Stone spaces.  There are many
other examples, e.g., free group over a monoid, group rings,
booleanization of a lattice.  There are also examples from K-theory (the
Steinberg group, I believe---but now I'm recalling very old stuff so don't
pin me down).  Less well-known, there a kind of universal valuation that
I've been interested in, which is a functor from commutative rings to
lattice-ordered monoids defined by letting F(A) be the free commutative,
totally distributive lattice-ordered-monoid with infinity on A modulo:

1_A = 0_F(A),
0_A = infinity,
a *_A b = a +_F(A) b,
a +_A b geq a sup_F(A) b.

(These, of course, are equational versions of the usual conditions for a valuation.)

My question is, has there ever been an attempt to make a general theory
about functors of this kind? Any relevant references that anyone knows
about?


-- 
James J. Madden
Department of Mathematics
Louisiana State University
Baton Rouge LA 70803-4918