Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 120 / 3 / d / Solution

For a prime filter $P\in \hat{H}$, first suppose $a\Rightarrow b\in P$. If $Q\supseteq P$ and $a\in Q$, then $a\Rightarrow b\in Q$ andso $b\in Q$. Thus no prime filter above $P$ belongs to $a^{\ast }\setminus b^{\ast }$, and

Conversely, suppose $a\Rightarrow b\notin P$. The lattice filter generated by $P\cup \{a\}$ is disjoint from the principal lattice ideal $\downarrow b$. Indeed, an intersection would give some $p\in P$ with $p\wedge a\le b$, whence $p\le a\Rightarrow b$ and then $a\Rightarrow b\in P$, a contradiction. The Stone prime filter theorem therefore extends this filter to a prime filter $Q\supseteq P$ that omits $b$. Then $Q\in a^{\ast }\setminus b^{\ast }$, so $P\in \uparrow (a^{\ast }\setminus b^{\ast })$.

We have proved, for every $P$,which is the required identity