  The satisfiability problem for multilevel syllogistic extended with the
Cartesian product operator (MLSC) is a long-standing open problem in computable
set theory. For long, it was not excluded that such a problem were undecidable,
due to its remarkable resemblance with the well-celebrated Hilbert's tenth
problem, as it was deemed reasonable that union of disjoint sets and Cartesian
product might somehow play the roles of integer addition and multiplication.
  To dispense with nonessential technical difficulties, we report here about a
positive solution to the satisfiability problem for a slight simplified variant
of MLSC, yet fully representative of the combinatorial complications due to the
presence of the Cartesian product, in which membership is not present and the
Cartesian product operator is replaced with its unordered variant.
  We are very confident that such decidability result can be generalized to
full MLSC, though at the cost of considerable technicalities.
