On Sat, 30 Apr 2005, Gareth McCaughan wrote:
On Friday 29 April 2005 19:48, Daniel Asimov wrote:
<< Must a Mean associate? -- Rich
...
Not all means are archimedean. For instance, min and max are nonarchimedean means; they are also associative. Here's another: let A = {0,1,2} and let m(0,0)=0, m(2,2)=2, m(x,y)=1 otherwise. Then m is associative.
Here's another example. Take m(x,y) to be the "simplest" number in the closed interval [x..y] (or [y..x] for x>y), with "simplest" as in Conway's surreal numbers. This means m(x,x) = x, and for x<y, m(x,y) is the unique integer in [x..y] of smallest absolute value, or, if there are no integers in the interval, then the unique diadic rational with the smallest denominator. This mean, however, fails to satisfy the identity m(ax,ay) = a m(x,y). David Moulton