On Monday 02 May 2005 18:06, David P. Moulton wrote: [me:]
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.
[David:]
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).
Lovely! More generally: let << be a total order on R (or on whatever set our mean is supposed to be defined on) and let m(x,y) be the <<-smallest element of [x,y]. (Put whatever restrictions are needed on << to ensure that there is one.) Then m is an associative mean. Min and max arise when << is either < or >. My 0,1,2 example arises when << ranks 1 below 0 and 2. David's example is where << compares Conway simplicities. These are all the associative means. Let m be an associative mean. Say that x << y iff m(x,y)=x. If x << y << z then m(x,z) = m(m(x,y),z) = m(x,m(y,z)) = m(x,y) = x, so << is transitive. Clearly x << y << x implies x=y. Hence << is a partial order. Now suppose m(x,z) = y and x <= t <= z. Then I claim y << t. For m(y,t) <= m(y,z) = m(m(x,z),z) = m(x,m(z,z)) = m(x,z) = y and m(y,t) >= m(x,y) = m(x,m(x,z)) = m(m(x,x),z) = m(x,z) = y and hence m(y,t) = y. Thus m(x,z) is the <<-minimal element of [x,z]. Thus far, << isn't a total order, but we can break ties however we like (say, via <) and it won't change any of the above. -- g