If $L$ is a lattice then the ideal $I$ of $L$ is a nonempty lower segment closed under join. I need to show that the set of ideals $I(L)$ of $L$ forms a lattice under $\subseteq$
I know the construction of join in case the ideals are of the ring. ($i_{1} + i_{2}$ from $I_1$ and $I_2$). But I cant figure out a similar definition of join for Ideals of Lattices.
On
If a lattice is finite, then ideals are principal, which means that they are generated by a single element, i.e., they have the following form: $$\downarrow a:=\{x\in L:x\leqslant a\}.$$
Consequently, join and meet are given as follows:
$$(\downarrow a)\otimes(\downarrow b)=\,\downarrow(a\wedge b),$$
$$(\downarrow a)\oplus(\downarrow b)=\,\downarrow (a\vee b).$$
Obviously, then $L\cong I(L)$.
For infinite lattices there are some results,... to be continued (if you want).
Observe $I(L)$ is closed under intersection. This yields the meet. If $A,B\in I(L)$ then consider the intersection of all ideals containing $A\cup B$ (which is nonempty since $A\cup B\subseteq L\in I(L)$).
More generally, any semilattice which is complete with respect to one lattice operation can have the other lattice operation defined in this way, making it a (not just semi) lattice.