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This came up on another thread (see particularly #36, #45, #47, and #60).
The usual notation for the group of nonzero elements of a field K is K×. An obvious generalisation is that the group of invertible elements of a monoid M should be M×. (The discussion above is really about whether this should be generalised from a monoid to a category.) I have never liked this.
With a field, at least ‘×’ indicates that you’re looking at multiplication rather than addition. With a monoid, multiplication is already the only operation, so the notation is counterintuitive. I decided once that a better symbol would be ‘÷’, so that M÷ is the group of invertible elements of the monoid M. (The notion of group can be nicely axiomatised using only the operation of division, so in a way, groups are about division while monoids are about multiplication.) As a special case (a case with extra structure, not merely extra property), R÷ is the group of invertible elements of a ring R. And of course, K÷ is the group of invertible elements of a field K.
However, to accomodate the classical notation K×, I also use R× for the monoid (not necessarily a group) of non-zero-divisors of R. Then K÷=K× when K is a field, although in general R÷⊂R×. There is no meaning of M× for an arbitrary monoid M, although it does make sense for a monoid with a (necessarily unique) absorbing element (an element z such that xz=z and zx=z for all x).
This is independent of the question of whether notation for a monoid should be extended to a category.
I have used this notation in zero-divisor.
A note: At least in commutative algebra, the standard notation for the group of units of a ring R is R*. The notation R× is meant to denote the entire multiplicative monoid of the ring. These are identical in the case of a field for the obvious reasons.
For this reason, I am opposed to using the R× notation the way you’ve used it.
The notation R× is meant to denote the entire multiplicative monoid of the ring.
I don’t undersand what you mean by this. This conflicts with the notation C× for the non-zero complex numbers that you championed on the other thread. And it is simply not true that
These are identical in the case of a field for the obvious reasons.
Instead, R* is R×∖{0} when R is a field (if R× is the entire multiplicative monoid of R).
It seems to me that you want R× to be the multiplicative monoid of non-zero elements of the ring R. Except that, unless R is an integral domain, this is not a monoid! However, it has a largest subset which is a monoid: the set of non-zero-divisors. And that is precisely what I want to denote by R×!
Yes, you’re right, my mistake. The notation for the group of units is still true at least.
So is R× used in my sense by commutative algebraists? Or do you just withdraw it entirely?
I have now introduced R* for the group of units (although it is just an aside) at zero-divisor.
Your use of R× generalizes the common usage and reduces correctly, which is why I withdrew my objection.
I’m not sure what I think about the notation M÷, but I observe that an element of a ring is a non-zero-divisor iff it is cancellable, i.e. xy=xz implies y=z and oppositely. And cancellability makes sense in any monoid.
@ Mike
Good point!
But then we get two different meanings of R× when R is an arbitrary rig. Probably the cancellability condition is actually the better one in a rig, and my definition at zero-divisor (or at least the claim that the definition makes sense in any rig) is wrong. So then, yes, M× makes sense for any monoid M.
For what it’s worth, what Harry was thinking of above was (R,×).
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