Recall that for any distributive latticeL, the dual Priestley space ofL(XL, τ,≤) is defined as:
• XL is the space of all prime filters overL;
• τ is the topology generated by the basis β = {|a| − |b| ; a, b ∈ L}, where |a| = {p ∈
XL; a∈p} for anya∈L;
• ≤is the inclusion ordering onXL.
We first fix the following definitions.
Definition 3.3.1. LetLbe a distributive lattice, andA, B⊆Lsuch thatV
AandW
B exist in
L. V
Aisdistributive if for anyc∈L,V
A∨c=V
(A∨c), whereV
(A∨c) =V
{a∨c; a∈A}. Dually,W
Bis distributive if for anyc∈L,W
B∧c=W
(B∧c), whereW
(B∧c) =W
{b∧c; b∈B}. A filterpover L preserves V
A ifA ⊆pimpliesV
A ∈p. Dually, ppreserves W
B if W
B ∈p
impliesB∩p6=∅.
Goldblatt’s topological proof of the Rasiowa-Sikorski Lemma for distributive lattices relies on the following well-known fact:
Proposition 3.3.2. Let (X, τ) be a topological space. For any U ⊆X, −U ∩CU is nowhere dense, i.e.I(−U∩CU) =∅.
Proof.
I(−U∩CU) =I(−U)∩ICU=−CU∩ICU ⊆ −CU∩CU =∅.
The following lemma is of crucial relevance for the rest of the proof.
Lemma 3.3.3. LetLbe a distributive lattice,(XL, τ,≤)be its dual Priestley space, andA, B⊆L be such thatVAandWB exist and are distributive. Then:
1. |V
A|=I(T
a∈A|a|) 2. |WB|=C(S
b∈B|b|)
Proof. 1. Let A ⊆ L be such that VA exists and is distributive. Clearly, since filters are upward-closed and |VA| is open, we have
3.3. Goldblatt’s Proof of the Rasiowa-Sikorski Lemma for DL and HA
For the converse, assume there is some basic open set |c| − |d|such that |c| − |d| ⊆ \
a∈A
|a|.
Then for anya∈A,|c| − |d| ⊆ |a|, which entails that |c| ⊆ |a| ∪ |d|=|a∨d|.
But this means thatc≤a∨dfor anya∈A, and hence
c≤^(A∨d) =^A∨d
since, by assumption,V
Ais distributive. Hence
|c| ⊆ |^A∨d|=|^A| ∪ |d|,
which entails that|c| − |d| ⊆ |V
A|. Therefore, we have
I(\
a∈A
⊆ |^A|),
which completes the proof. 2. LetB⊆Lbe such thatW
B exists and is distributive. Since|W
B|is closed and filters are upward-closed, it follows that
C([
b∈B
|b|)⊆ |_B|.
For the converse, assume there is some basic closed set −|c| ∪ |d|such that [
b∈B
|b| ⊆ −|c| ∪ |d|.
Then for anyb∈B, we have
|b| ⊆ −|c| ∪ |d|,
which means that
|b∧c|=|b| ∩ |c| ⊆ |d|.
This implies that for any b∈B, we haveb∧c≤d, from which it follows that _
B∧c=_(B∧c)≤d
since by assumption WB is distributive. Hence
|_B∧c|=|_B| ∩ |c| ⊆ |d|,
which entails that
|_B| ⊆ −|c| ∪ |d|.
Therefore we have
|_B| ⊆C([
b∈B
|b|),
3.3. Goldblatt’s Proof of the Rasiowa-Sikorski Lemma for DL and HA
Combining two previous results, we obtain the following:
Lemma 3.3.4. LetLbe a distributive lattice, (XL, τ,≤)be its dual Priestley space, andA, B⊆
L such that VA and WB exist and are distributive. Then S
VA = S a∈A−|a| ∪ | VA| and SWB =S b∈B|b| ∪ −| W
B| are open dense sets inXL. Proof. Notice first that sinceS
a∈A−|a| and
S
b∈B|b|are both open, so are SVA andSW B. To
see that they are also dense, we show that−SVA and−SWB are both nowhere dense sets. In
the former case, notice that, by Lemma 3.3.3 1., we have −|^A|=−I(\ a∈A |a|) =C([ a∈A −|a|. Hence −SVA=−( [ a∈A −|a|)∩ C([ a∈A −|a|),
which by proposition 3.3.2 entails that−SVA is nowhere dense inXL. In the latter case, recall
that, by Lemma 3.3.3 2., we have
|_B|=C([ b∈B |b|) and therefore −SWB=−( [ b∈B |b|)∩C([ b∈B |b|).
By proposition 3.3.2 again, this implies that−SWB is nowhere dense inXL.
We can finally state and prove the Rasiowa-Sikorski Lemma for distributive lattices.
Lemma 3.3.5(Rasiowa-Sikorski Lemma for distributive lattices). LetLbe a distributive lattice, and letQM andQJbe two countable sets of existing distributive meets and joins inLrespectively. Then for anyab∈L, there exists a prime filterpoverLsuch thata∈p,b /∈p, andppreserves all meets inQM and all joins inQJ.
Proof. LetXL be the dual Priestely space, and let
SQ= \ VA∈QM SVA∩ \ WB∈QJ SWB.
By Lemma 3.3.4, SVA and SWB are both dense open sets in XL. Hence SQ is a countable
intersection of open and dense sets. Moreover, since XL is compact Hausdorff, by the Baire
Category Theorem, SQ is dense in XL. Now let a, b ∈ L such that a b. This means that
|a| − |b| 6=∅, and hence
SQ∩ |a| − |b| 6=∅
But clearly any prime filter inSQ preserves all meets inQM and all joins inQJ, and hence any
p∈SQ∩ |a| − |b|satisfies all requirements of the lemma.
Similarly to the case of Boolean algebras, the Rasiowa-Sikorski Lemma has the following important consequence:
3.3. Goldblatt’s Proof of the Rasiowa-Sikorski Lemma for DL and HA
Proof. Let (XQ,≤) be the set of all prime filters overLthat preserve all meets inQMand all joins
in QJ with the inclusion ordering, and let| · |:L→U p(XQ) such that|a|={p∈XQ; a∈p}.
Then the following hold:
1. |1|=XQ and|0|=∅ since anyp∈XQ is non-empty and proper.
2. |a∧b|=|a| ∩ |b|since filters are upward closed and downward-directed 3. |a∨b|=|a| ∪ |b|since anyp∈XQ is prime and upward-closed
4. |VA|=T
a∈A|a| for any
VA∈Q
M: the left-to-right direction is immediate since filters
are upward-closed. For the right-to-left direction, notice that for any p ∈ XQ, A ⊆ p
impliesV A∈psinceppreservesV A. 5. |W B|=S b∈B|b|for any W
B ∈ QJ: For the left-to-right direction, notice that WB ∈p
impliesB∩p6=∅for anyp∈XQ, sinceppreservesWB. The right-to-left direction follows
immediately from the fact that filters are upward-closed.
Hence|·|is a DL-homomorphism that preserves all meets inQMand all joins inQJ. Moreover,
by Lemma 3.3.5, for anya, b∈Lsuch thatab, there existsp∈XQ such thata∈pandb /∈p.
Hencea≤b iff|a| ⊆ |b|for anya, b∈L, which means that| · |is injective.