Solution of Rota's problem on the order of series-parallel networks

Let N be a series-parallel network with n variable resistors. Letting the resistances independently take values r1,...,rn in the set [0, ∞], the total resistance of N will take a value R = R(r1,...,rn) in [0,∞]. When in particular ri ε{lunate} {0,∞} for each i, also R takes a value in {0, ∞}. Thus, the restriction R* of R to {0, ∞}n describes the open-short circuit performance of N. Given two networks N′ and N″, with their respective resistance functions R′ and R″, we say that N′ ≤ N″ iff R′(r1, ..., rn) ≤ R″(r1, ..., rn) for all r1, ..., rn ε{lunate} [0, ∞]. If we are only interested in comparing open-short circuit performances, we can write N′ ≤ *N″ iff R′* ≤ R″*. Rota asked if the two orders ≤ and ≤* coincide. We give a positive answer to the problem and discuss some applications.