# How to define average operator on the Real (semi) ring

How does one define the average operator on groups/ring/filed of Real numbers with multiplication (and/or addition) operator $(\Re, ., +)$? Is this extendible to for example the semiring of $(\Re, \min,+)$?

• What properties do you want it to satisfy? As long as the semi ring contains inverses for the positive integers, I don't see why you wouldn't try the ordinary average. – rschwieb Jun 12 '14 at 1:05
• the semiring $(\Re \cup \infty, min, +)$ does have an inverse for the + operator (corresponding to division in $(\Re,+,.)$) but the division by 2 in sum-product ring, corresponds to -2 in this semi-ring which doesn't make sense or does it mean something and I'm missing it? Maybe geometric average is the way to go? – Siamak Jun 12 '14 at 2:40
• Hmm, in your specific case, the multiplication is idempotent and the characteristic is $2$, so the ordinary average isn't defined for an even number of elements, and it coincides with the sum of the numbers for odd number of elements. Seems like the normal average is not a good idea after all. I'd like to ask again: what properties do you want the average to satisfy? – rschwieb Jun 12 '14 at 13:03
• I'm looking for reasonable properties! maybe the question IS also what properties does average have? – Siamak Jun 12 '14 at 18:08
• Honestly I have no idea, but here are some conjectures. If there is a min operation, then this suggests there is a linear order on the semiring. Hopefully, there is also a Max counterpart. One property of the average that I would hope for is that for each set $S$ $min(S)\leq Avg(S)\leq max(S)$. In particular we'd also expect the average of any number of copies of the same element to return that element. – rschwieb Jun 13 '14 at 13:36