  Second-order transitive-closure logic, SO(TC), is an expressive declarative
language that captures the complexity class PSPACE. Already its monadic
fragment, MSO(TC), allows the expression of various NP-hard and even
PSPACE-hard problems in a natural and elegant manner. As SO(TC) offers an
attractive framework for expressing properties in terms of declaratively
specified computations, it is interesting to understand the expressivity of
different features of the language. This paper focuses on the fragment MSO(TC),
as well on the purely existential fragment SO(2TC)(E); in 2TC, the TC operator
binds only tuples of relation variables. We establish that, with respect to
expressive power, SO(2TC)(E) collapses to existential first-order logic. In
addition we study the relationship of MSO(TC) to an extension of MSO(TC) with
counting features (CMSO(TC)) as well as to order-invariant MSO. We show that
the expressive powers of CMSO(TC) and MSO(TC) coincide. Moreover we establish
that, over unary vocabularies, MSO(TC) strictly subsumes order-invariant MSO.
