Tame congruence theory for infinite algebras?
- Hello all:For some time, I have thought it would be possible to do tame congruence theory for infinite algebras. There may not be minimal algebras as there are guaranteed to be with a finite algebra, but there are guaranteed to be minimal _filters_ of algebras in the lattice of filters. (Note, that to my way of thinking, filters should be ordered by reverse inclusion, so that an ultrafilter is minimal, in my opinion.) Is that all that's needed? If minimal filters of algebras aren't quite the thing, then there are minimal algebra objects in the category of filters. (There was a classic 1967, or something, paper with that title.)If we looked at these minimal objects, would there still always be 5 kinds of them, or would there be more? No idea here.I just wanted to throw this out there and see if anyone was interested in the idea. I have basically no free time to work on it.Bill Rowan
- Hello Bill,
finite filter sequences could be polynomials p(x) of a degree n. Take a
sequence of these polynomials with decreasing degree n and their
approximative power decreases. Now I can imagine a countably infinite
function space, but do not see a similar kind of order in that space.
Maybe the Galois theory can help. Imagine the sine being approximated by
an infinite number of triangles, built out of the functions sum, mod, abs
and sign. Next step imagine the sine being approximated by a finer grid of
triangles and so on. Thus you would have an increasing sequence of filters