Exercise 7, § 4

An element $a$ of a lattice $E$ is said to be irreducible if the relation $\sup(x,y)=a$ implies either $x=a$ or $y=a$.

(a) Show that in a finite lattice $E$ every element $a$ can be written as $\sup(e_1,\ldots,e_k)$, where the $e_i$ ($1\leq i\leq k$) are irreducible.

(b) Let $E$ be a finite lattice and let $J$ be the set of its irreducible elements. For each $x\in E$ let $S(x)$ be the set of all $y\in J$ which are $\leq x$. Show that the mapping $x\to S(x)$ is an isomorphism of $E$ onto a subset of $\mathfrak{P}(J)$, ordered by inclusion, and that $S(\inf(x,y))=S(x)\cap S(y)$.