On Friday 29 April 2005 19:48, Daniel Asimov wrote:
<< Must a Mean associate? -- Rich
Well, from ((x+y)/2 + z)/2 = x/4 + y/4/ + z/2 we see that even the arithmetic mean fails to associate.
Let (A,<) be a totally ordered set. Define a "mean" on A to be a function m : AxA -> A such that m(x,x) = x for all x m(x,y) = m(y,x) for all x,y m(x,--) is monotone nondecreasing for all x. Suppose m is an associative mean. Then, in particular, m(x,m(x,y)) = m(m(x,x),y) = m(x,y). That is: if z = m(x,y) for some y then m(x,z) = z. Let's say that m is "archimedean" if, for any x and z, there exists y such that z = m(x,y). There is no associative archimedean mean on a set with more than one element; for if m is such a mean then the condition in the previous paragraph is vacuous and we have m(x,z)=z unconditionally, and likewise m(x,z)=x unconditionally, and so x=z unconditionally. 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. I suspect associativity is a very stringent condition; it also seems an unnatural one. (And not the one Gosper was actually pointing out the failure of, as Guy has noted.) -- g