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)

For question 1, this link seems to provide an answer (that's rather extensive for planetmath). 
With regards to the second question, I'm not sure what you're aiming at. If you consider a lattice algebraically, with binary operations $\wedge$ and $\vee$, your structure will be commutative. What would it, for instance, mean if $a\wedge b\neq b\wedge a$? I therefore wouldn't expect any difference between the notions of ideal, left ideal, right ideal and two-sided ideal.
