  We study realizations of spirals and skyrmions in two-dimensional
antiferromagnets with a triangular lattice on an inversion-symmetry-breaking
substrate. As a possible material realization, we investigate the adsorption of
transition-metal atoms (Cr, Mn, Fe, or Co) on a monolayer of MoS$_2$, WS$_2$,
or WSe$_2$ and obtain the exchange, anisotropy, and Dzyaloshinskii-Moriya
interaction parameters using first-principles calculations. Using energy
minimization and parallel-tempering Monte-Carlo simulations, we determine the
magnetic phase diagrams for a wide range of interaction parameters. We find
that skyrmion lattices can appear even with weak Dzyaloshinskii-Moriya
interactions, but their stability is hindered by magnetic anisotropy. However,
a weak easy plane magnetic anisotropy can be beneficial for stabilizing the
skyrmion phase. Our results suggest that Cr$/$MoS$_2$, Fe$/$MoS$_2$, and
Fe$/$WSe$_2$ interfaces can host spin spirals formed from the 120$^{\circ}$
antiferromagnetic states. Our results further suggests that for other
interfaces, such as Fe$/$MoS$_2$, the Dzyaloshinskii-Moriya interaction is
strong enough to drive the system into a three-sublattice skyrmion lattice in
the presence of experimentally feasible external magnetic field.
