  The existence of incompatibility is one of the most fundamental features of
quantum theory, and can be found at the core of many of the theory's
distinguishing features, such as Bell inequality violations and the
no-broadcasting theorem. A scheme for obtaining new observables from existing
ones via classical operations, the so-called simulation of observables, has led
to an extension of the notion of compatibility for measurements. We consider
the simulation of observables within the operational framework of general
probabilistic theories and introduce the concept of simulation irreducibility.
While a simulation irreducible observable can only be simulated by itself, we
show that any observable can be simulated by simulation irreducible
observables, which in the quantum case correspond to extreme rank-1
positive-operator-valued measures. We also consider cases where the set of
simulators is restricted in one of two ways: in terms of either the number of
simulating observables or their number of outcomes. The former is seen to be
closely connected to compatibility and $k$-compatibility, whereas the latter
leads to a partial characterization for dichotomic observables. In addition to
the quantum case, we further demonstrate these concepts in state spaces
described by regular polygons.
