1. Lattice ideals

The concepts are the same, with $\vee$ for $+$ and $\wedge$ for $\cdot$, but it's easier to use these equivalent definitions:

(1) An ideal of $L$ is a nonempty downset closed under joins.

(2) An ideal $I$ is principal if $I = ( a ]$, i.e., $\{x \in L \vert x \leq a\}$, for some $a \in I$.

(3) An ideal $I$ is proper if $I < L$.

(4) An ideal $I$ is prime if $I$ is proper and $x \wedge y \in I$ implies $x \in I$ or $y \in I$.

(5) An ideal $I$ of $L$ is maximal if $I$ is proper and there is no ideal $J$ with $I < J < L$.

Figure: Ideals and a dual ideal