  In this work, we develop the basic formalism to study trions in semiconductor
layered materials using the Faddeev equations in momentum space for three
different particles lying in two dimensions. We solve the trion Faddeev coupled
integral equations for both short-range one-term separable Yamaguchi potential
and Rytova-Keldysh (RK) interaction applied to the MoS$_2$ layer. We devise two
distinct regularization methods to overcome the challenge posed by the
repulsive electron-electron RK potential in the numerical solution of the
Faddeev equations in momentum space. The first method regulates the repulsive
interaction in the infrared region, while the second regulates it in the
ultraviolet region. By extrapolating the trion energy to the situation without
screening, the two methods gave consistent results for the MoS$_2$ layer with a
trion binding energy of $-49.5(1)$~meV for the exciton energy of $-753.3$~meV.
We analyzed the trion structure for the RK and Yamaguchi potentials in detail,
showing their overall similarities and the dominant cluster structure, where
the strongly bound exciton is weakly bound to an electron. We found that this
property is manifested in the dominance of two of the Faddeev components over
the one where the hole is a spectator of the interacting electron pair.
