  We study the frustration properties of the Ising model on a decorated
triangular lattice with an arbitrary number of decorating spins on all lattice
bonds in the framework of an exact analytical approach based on the
Kramers--Wannier transfer matrix method. Expressions for the entropy, heat
capacity, and spontaneous magnetization of the lattice are obtained, including
the residual (zero-temperature) entropy and residual (zero-temperature)
spontaneous magnetization of the system. The existence of magnetic frustrations
in such a model and their influence on the behavior of the thermodynamic
functions of the system are shown. The new and most important result of our
study is related to the description of the possible coexistence of frustrations
and long-range magnetic order in partially ordered spin systems.
