  We generalize the combinatorial algebraic approach first proposed by
Dhurandhar et al. to construct various classes of modified second-generation
time-delay interferometry (TDI) solutions. The main idea behind the algorithm
is to enumerate, in a given order, a specific type of commutator between two
monomials defined by the products of particular time-displacement operators. On
the one hand, the above commutators can be systematically rewritten as the
elements of a left ideal, defined by the l.h.s. of the relevant equation for
the TDI solution. On the other hand, these commutators are shown to vanish if
we only keep up the first-order contributions regarding the rate of change of
armlengths. In other words, each commutator furnishes a valid TDI solution
pertaining to the given type of modified second-generation combinations. In
this work, the original algorithm, which only involves time-delay operators, is
extended by introducing the time-advance ones and then utilized to seek
solutions of the Beacon, Relay, Monitor, Sagnac, and fully symmetric Sagnac
types. We discuss the relation between the present scheme's solutions and those
obtained by the geometric TDI approach, a well-known method of exhaustion of
virtual optical paths. In particular, we report the results on novel
Sagnac-inspired solutions that cannot be straightforwardly obtained using the
geometric TDI algorithm. The average response functions, floor noise power
spectral densities, and sensitivity functions are evaluated for the obtained
solutions.
