So, here's something that had been bothering me for a while. Groups are isomorphic to their reverse groups, by inverses. But general rings[0] shouldn't be isomorphic to their reverse rings, right? But what's a counterexample? What noncommutative rings do I know? Matrix rings? No, they've got transpose. Quaternions? Conjugation. Group rings? Reverse (invert) the group.
Two summers ago I spent a bunch of time trying to fine a counterexample. I was trying finite rings. I think I may well have searched every possible ring of order 8 - I don't quite remember - but I do remember that all the ones I did try had some reverse-automorphism, even if not immediately obvious. Anyway, today I was thinking about it again and managed to find one.
Take F a field of order q, then take the ring R of matrices of the following form:
( a b c )
( 0 d 0 )
( 0 0 e )
for a,b,c,d,e∈F. Then R has 2q²+4q+12 left ideals, but q²+6q+13 right ideals. (And always 13 two-sided, for what it's worth.) These are never equal for q an integer, so R is never isomorphic to its reverse for F finite. (Probably true for infinite fields as well, but no way am I going to try to prove that.)
So in particular I've got one of order 32. I'm tempted to wonder if there's a smaller one (it would have to be order 8, 16, 24, or 27, and 24 is possible iff 8 is), but I really have no intention of trying that.
OK, thought, now that I've wasted all this time on this: it's really easy to find a monoid not isomorphic to its reverse. Maybe if I had just taken a commutative ring and adjoined such a monoid instead of a group, that would have been easier? Well, screw it, I'm not going to try that either. Though, actually, recalling my massive investigation two summers ago, I think it safe to say that won't work in general; I'm quite certain many of the ones I tried could be expressed in that form. So, I'm definitely not going to try that. Indeed, I'm going to put this down now, because it's pretty unimportant. Maybe someday I'll get back to the question of if it's the smallest, but not now.
Incidentally, I was down in TANSTAAFL and some Thompson guys were boasting about the Smash tournament. They couldn't actually go, so they said they'd just find whoever won and beat him later. Ha. Probably not going to be so easy, guys.
-Harry
[0]Meaning, with identity. Without identity, there's an easy counterexample of 4 elements (and very definitely smallest possible), but who cares about rings without identity? (And yes, taking that example and adjoining an identity completely screws it up. I don't remember how, and I don't feel like checking, but it very definitely does.)
