  Properties of the lowest $0^{+}$ states of $^{12}\mathrm{C}$ are calculated
to study the role of three-body interactions in the $\alpha$-cluster model. An
additional short-range part of the local three-body potential is introduced to
incorporate the effects beyond the $\alpha$-cluster model. There is enough
freedom in this potential to reproduce the experimental values of the
ground-state and excited-state energies and the ground-state root-mean-square
radius. The calculations reveal two principal choices of the two-body and
three-body potentials. Firstly, one can adjust the potentials to obtain the
width of the excited $0_2^+$ state and the monopole $0_2^+ \to 0_1^+ $
transition matrix element in good agreement with the experimental data. In this
case, the three-body potential has strong short-range attraction supporting a
narrow resonance above the $0_2^+$ state, the excited-state wave function
contains a significant short-range component, and the excited-state
root-mean-square radius is comparable to that of the ground state. Next,
rejecting the solutions with an additional narrow resonance, one finds that the
excited-state width and the monopole transition matrix element are insensitive
to the choice of the potentials and both values exceed the experimental ones.
