The PCF Trichotomy Theorem does not hold for short sequences

Abstract. The PCF Trichotomy Theorem deals with sequences of ordinal functions on an infinite $\kappa$ modulo some ideal I. If a $<_I$-increasing sequence of ordinal functions has regular length which is larger than $\kappa^+$, then by the Trichotomy Theorem the sequence satisfies one of three structural conditions. It was of some interest to find out if the Trichotomy Theorem could hold also for sequences of length $\kappa^+$. It is shown that this is not the case.