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

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- Solution c

Solution

The Stone prime filter theorem says that if a lattice filter

F

and a lattice ideal

I

of a distributive lattice are disjoint, then there is a prime filter of a distributive lattice

P

such that

F \subseteq P and P \cap I = \emptyset .

Equivalently, whenever

a ≰ b

, there is a prime filter containing

a

and omitting

b

.

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