  The quotient bases for zero-dimensional ideals are often of interest in the
investigation of multivariate polynomial interpolation, algebraic coding
theory, and computational molecular biology, etc. In this paper, we discuss the
properties of zero-dimensional ideals with unique monomial quotient bases, and
verify that the vanishing ideals of Cartesian sets have unique monomial
quotient bases. Furthermore, we reveal the relation between Cartesian sets and
the point sets with unique associated monomial quotient bases.
