A question on Join homomorphism and Ideals On page 287 of the book Mathematical Methods in Linguistcs, by Barbara Partee, Alice Ter Meulen and Robert E. Wall (Dordrecht, Kluwer Academic Press, 1993), I find the following theorem, which they consider to establish, in their own words "a connection between join homomorphisms and ideals" (Ideals are referred here to lattice ideals; I am not sure whether they are equivalent to ring ideals. Could anyone clarify me that?):
THEOREM 11.4 $I$ is a proper ideal of the lattice $L$ if and only if there is a join homomorphism $G$ from $L$ onto the two element chain $C = \{0,1\}$ such that $I = G^{-1} (0)$, i.e. $I$ consists of those $x$ for which $G(x) = 0$
I am particulary interested in knowing:
1) Whether the connection stated in the theorem relates to some specific object of ring theory or to other results
2) How right and left ideals can be implemented within that theorem?
Thanks in advance (I hope the editing of the theorem comes up correctly)

The ideal in a lattice meant in the theorem is definitely the order theoretic notion of an ideal.  It has nothing to do with ring theory apart from some similarity in the definition.  Regarding (2): Left and right ideals are notions from (non-commutative) ring theory, so that they don't apply here.
In plain English the theorem tells you roughly the following.  Say you have some subset $I$ of a lattice $L$ (a partially ordered set with unique meets and joins) and you want to know if it forms an ideal in $L$, that is, it is closed under going down and forming joins.  The Theorem says that this happens exactly if you can find a lattice map (homomorphism) from $L$ to the two element lattice $C$ which respects joins and whose preimage of the lower element $0\in C$ is the set $I$.  The theorem is not hard to prove.  It is basically unwraping definitions.
