  The class of associative trialgebras, also known as triassociative algebras,
is characterized by three multiplications and eleven relations that generalize
associativity. In the current paper, we present a study of nilpotent
triassociative algebras. After some examples and basic results, we provide a
low-dimensional classification, a general monomial form, and an analogue of
Engel's Theorem. The main result shows that one of the three multiplication
operations behaves differently than the other two. In particular, if this
structure alone is nilpotent, then both other multiplications are nilpotent.
The converse is not true. Furthermore, the former is nilpotent if and only if
the entire algebra is nilpotent.
