3. Facts valid in all lattices $L$

(a) If $L$ is finite, every ideal is principal.

(b) The intersection of two ideals is an ideal, and the intersection of any family of ideals is either an ideal or the empty set. It follows that the ideals of $L$ under inclusion form a lattice Ideals$(L)$, and that if $L$ has a bottom element then Ideals$(L)$ is complete.

(c) For ideals $I$ and $J$ of $L$, $I \wedge J = I \cap J$ and $I \vee J = \{x \in L: x \leq i \vee j$    for some $i \in I, j \in J\}$. (Contrast with §(a) below.)

(d) An ideal $I$ is prime if and only if $L \! \setminus \! I$ is a dual ideal.

(e) A principal ideal is prime if and only if its top element is a meet-prime element.

(f) Prime ideals of $L$ are the same thing as meet-prime elements of Ideals$(L)$.