  In this paper we study the problem of a possibility to use quantum
observables to describe a possible combination of the order effect with
sequential reproducibility for quantum measurements. By the order effect we
mean a dependence of probability distributions (of measurement results) on the
order of measurements. We consider two types of the sequential reproducibility:
adjacent reproducibility ($A-A$) and separated reproducibility($A-B-A$). The
first one is reproducibility with probability 1 of a result of measurement of
some observable $A$ measured twice, one $A$ measurement after the other. The
second one, $A-B-A$, is reproducibility with probability 1 of a result of $A$
measurement when another quantum observable $B$ is measured between two $A$'s.
Heuristically, it is clear that the second type of reproducibility is
complementary to the order effect. We show that, surprisingly, for an important
class of quantum observables given by positive operator valued measures
(POVMs), this may not be the case. The order effect can coexist with a
separated reproducibility as well as adjacent reproducibility for both
observables $A$ and $B.$ However, the additional constraint in the form of
separated reproducibility of the $B-A-B$ type makes this coexistence
impossible. Mathematically, this paper is about the operator algebra for
effects composing POVMs. The problem under consideration was motivated by
attempts to apply the quantum formalism outside of physics, especially, in
cognitive psychology and psychophysics. However, it is also important for
foundations of quantum physics as a part of the problem about the structure of
sequential quantum measurements.
