  In this paper, we explore some algebraic properties of linear structural
equation modelsthat can be represented by an HTC-identifiable graph. In
particular, we prove that all mixedgraphs are HTC-identifiable if and only if
all the regression coefficients can be recovered fromthe covariance matrix
using straightforward linear algebra operations. We also find a set
ofpolynomials that generates the ideal that encompasses all the equality
constraints of the modelon the cone of positive definite matrices. We further
prove that this set of polynomials are theminimal generators of said ideal for
a subset of HTC-identifiable graphs.
