HTC-identifiable graphs are a large family of graphs known to be generically
identifiable. We explore some algebraic properties of linear structural equation
models that can be represented by an HTC-identifiable graph. In particular, we prove
that all mixed graphs are HTC-identifiable if and only if all the regression coefficients
can be recovered from the covariance matrix using straightforward linear algebra
operations on specified equations. Furthermore, given an HTC-identifiable graph,
we provide a set of polynomials that generates the ideal that encompasses
all the equality constraints of the corresponding graphical model on the
cone of positive definite matrices. We further prove that, for a subset of
HTC-identifiable graphs, this set of polynomials are the minimal generators of said
ideal.