  [Background] A complete set is a minimum set of observables which allows one
to determine the underlying reaction amplitudes unambiguously.
Pseudoscalar-meson photoproduction from the nucleon is characterized by four
such amplitudes and complete sets involve single- and double-polarization
observables.
  [Purpose] Identify complete sets of observables, and study how measurements
with finite error bars impact their potential to determine the reaction
amplitudes unambiguously.
  [Method] The authors provide arguments to employ the transversity
representation in order to determine the amplitudes in pseudoscalar-meson
photoproduction. It is studied whether the amplitudes in the transversity basis
for the $\gamma p \to K^+\Lambda$ reaction can be estimated without ambiguity.
To this end, data from the GRAAL collaboration and mock data from a realistic
model are analyzed.
  [Results] It is illustrated that the moduli of normalized transversity
amplitudes can be determined from precise single-polarization data. Starting
from mock data with achievable experimental resolution, it is quite likely to
obtain imaginary solutions for the relative phases of the amplitudes. Also the
real solutions face a discrete phase ambiguity which makes it impossible to
obtain a statistically significant solution for the relative phases at
realistic experimental conditions.
  [Conclusions] Single polarization observables are effective in determining
the moduli of the amplitudes in a transversity basis. Determining the relative
phases of the amplitudes from double-polarization observables is far less
evident. The availability of a complete set of observables does not allow one
to unambiguously determine the reaction amplitudes with statistical
significance.
