So, to every associative mean there corresponds a partial order << such that every interval contains a unique <<-minimal element, which the mean picks out. What does this tell us about *homogeneous* associative means, i.e., ones on R+ for which m(ta,tb) = t.m(a,b)? (Note 1. For a partial order, "minimal" is ambiguous. In this instance I mean that the element is actually <<-smaller than everything else in the interval. I think the term is usually used the other way.) (Note 2. We really want to ask the following more general question, but I don't know how to answer it. We have a total order (A,<) and a group G of its automorphisms; what associative means are there that are invariant under G?) We must have a << b => ta << tb. Another way of saying this: whether a << b depends only on b/a. Let S = { x : 1 << x } and T = { x : x << 1 }; then T = 1/S in the obvious sense by symmetry, 1 is in neither, and transitivity says that both are closed under multiplication. Every interval contains a <<-minimal element, which means that for any t [1,t] contains s such that everything other than 1 in [1/s,t/s] is >> 1. This is impossible unless one endpoint of the interval is 1; so for any t either everything in (1,t] is >> 1 or everything in [1/t,1) is >> 1. That is, either all of (1,oo) is >> 1 or all of (0,1) is >> 1. That is, the only homogeneous associative means on R+ are min and max. I suspect one can adapt this to the slightly less special case of the question in Note 2 where G=A and < is compatible with the group operation, but my brain's full of cotton wool right now so I shan't bother. -- g