  Exploiting symmetry in Groebner basis computations is difficult when the
symmetry takes the form of a group acting by automorphisms on monomials in
finitely many variables. This is largely due to the fact that the group
elements, being invertible, cannot preserve a term order. By contrast, inspired
by work of Aschenbrenner and Hillar, we introduce the concept of equivariant
Groebner basis in a setting where a_monoid_ acts by_homomorphisms_ on monomials
in potentially infinitely many variables. We require that the action be
compatible with a term order, and under some further assumptions derive a
Buchberger-type algorithm for computing equivariant Groebner bases. Using this
algorithm and the monoid of strictly increasing functions N -> N we prove that
the kernel of the ring homomorphism R[y_{ij} | i,j in N, i > j] -> R[s_i,t_i |
i in N], y_{ij} -> s_i s_j + t_i t_j is generated by two types of polynomials:
off-diagonal 3x3-minors and pentads. This confirms a conjecture by Drton,
Sturmfels, and Sullivant on the Gaussian two-factor model from algebraic
statistics.
