  It is proved that the definition of an algebraic set of type ${\sf A}$ (a
notion related to the multidimensional Hamburger moment problem) does not
depend on the choice of a polynomial describing the algebraic set in question
and that an algebraic set of type ${\sf B}$ is always of type ${\sf A}$. This
answers in the affirmative two questions posed in 1992 by the second author. It
is also shown that an algebraic set is of type ${\sf A}$ if and only if it is
of type ${\sf C}$ (a notion linked to orthogonality of polynomials of several
variables). This, in turn, enables us to answer three questions posed in 2005
by Cicho\'n, Stochel, and Szafraniec.
