We now apply the same method as above to the case of modal Heyting and co-Heyting algebras. We begin with the following lemma:
Lemma 3.4.5.
1. Let (L,) be a modal Heyting algebra, (XL, τ,≤, R) its dual space, and p a prime filter overL. For any U ⊆XL, letUR[p]={q∈U ; pRq}. Then for anya, b∈L,(a→b)∈p
3.4. Generalization to co-Heyting Algebras and Modal Algebras
2. Let(M,♦)be a modal co-Heyting algebra,(XM, τ,≤, S)its dual space, andpa prime filter overL. For anyV ⊆XL, letVShpi={q∈V ; pSq}. Then for anya, b∈L,♦(a−< b)∈/p iff|a|Shpi⊆ |b|Shpi.
Proof.
1. Recall that for any p, q ∈XL, pRq iff p ⊆ q, wherep ={a ∈ L ; a∈ p}. We first
prove that for anya, b∈ L, if(a→b)∈p, then we have that |a|R[p] ⊆ |b|R[p]. Clearly,
if (a →b) ∈p, thena →b ∈q for any q such that pRq. But then, if a∈ q, it follows that b ∈qsince a∧(a→b)≤b. Conversely, assume (a→b)∈/ p. Then there exists q
such that pRq and a→b /∈ q. But this means that there exists q0 such that q ⊆q0 and
q∈ |a| − |b|. But then sincep⊆q⊆q0, it follows thatq0∈ |a|
R[p]− |b|R[p].
2. Recall that for anyp, q∈XM,pSq iffq⊆p♦, wherep♦={a∈M ; ♦a∈p}. Now assume
that ♦(a−< b)∈ pfor some a, b∈M. This means that a−< b ∈q for some q such that
pRq. But then, there exists q0 ⊆qsuch that q0 ∈ |a| − |b|. Moreover, since q0 ⊆q ⊆p♦, this means that q0 ∈ |a|Shpi− |b|Shpi. Hence if|a|Shpi ⊆ |b|Shpi, then
♦(a−< b)∈/ p. For the converse, assume there existsq∈ |a|Shpi− |b|Shpi. Then since a≤(a−< b)∨b andqis prime, it follows thata−< b∈q. But then♦(a−< b)∈p.
We now define the conditions on the setsQM andQJ that we will need for our generalization
of the Rasiowa-Sikorski Lemma:
Definition 3.4.6. Let (L,) be a modal Heyting algebra. A meetV
Aexisting inLisBarcan
ifV
A =V
{a; a ∈A}. Dually, if (M,♦) is a modal co-Heyting algebra, then a join W
B
existing inM isBarcan if♦W
B=W
{♦b; b∈B} Definition 3.4.7.
1. Let (L,) be a modal Heyting algebra, and letQM, QJ be two countable sets of Barcan
meets and joins in L. QM,QJ are (,→)-complete if for anyA, B⊆Landc, d∈L:
• ifVA
∈QM, then V((c→(A∨d)))∈QM;
• ifW
B∈QJ, thenV(((B∧c)→d))∈QM.
2. Dually, if (M,♦) is a modal Heyting algebra, andQM,QJ are two countable sets of Barcan
meets and joins inL, thenQM,QJ are (♦,−<)-complete if for any A, B⊆Landc, d∈L:
• ifV
A∈QM, then W(♦(c−<(A∨d)))∈QJ;
• ifWB∈Q
J, thenW(♦((B∧c)−< d))∈QJ.
Lemma 3.4.8.
1. Let(L,)be a modal Heyting algebra, and letQM,QJ be two(,→)- complete countable sets of Barcan meets and joins in L. Then for anyA, B∈L:
i) if V A∈QM, then|VA|R[p]= (I( T a∈A|a|))R[p] ii) if WB ∈QJ, then|WB|R[p]= (C(Sb∈B|b|))R[p]
2. Dually, if (M,♦) is a modal Heyting algebra, and QM, QJ are two (♦,−<)- complete countable sets of Barcan meets and joins in M, then for any A, B∈M:
3.4. Generalization to co-Heyting Algebras and Modal Algebras
iv) if WB∈Q
J, then|WB|Shpi= (C(Sb∈B|b|))Shpi Proof.
i) We only prove the non-immediate direction, i.e. the left-to-right direction. As usual, assume|c|R[p]− |d|R[p] ⊆ |a|R[p] for all a∈A. Then |c|R[p] ⊆ |a∨d|R[p], which means by
Lemma 3.4.5 (1.) that(c→(a∨d))∈pfor anya∈A. Hence sinceppreserves all meets in QM, it follows that
^
(c→(A∨d)) =^(c→(A∨d) =(c→(^A∨d)∈p.
By Lemma 3.4.5(1.) again, this implies that |c|R[p]⊆ |
^
A∨d|R[p]=|
^
A|R[p]∪ |d|R[p],
which in turn implies that
|c|R[p]− |d|R[p] ⊆ |
^
A|R[p].
ii) We only prove the right-to-left direction. Assume|b|R[p] ⊆(−|c|)R[p]∪ |d|R[p] for allb∈B.
Then
|b∧c|R[p]=|b|R[p]∩ |c|R[p]⊆ |d|R[p],
which by Lemma 3.4.5 (1.) implies that((b∧c)→d)∈pfor allb∈B. This implies that ^
((B∧c)→d) =^((B∧c)→d) =((_B∧c)→)∈p.
Therefore, by Lemma 3.4.5 (1.) again, we have that |_B|R[p]∩ |c|R[p]=|
_
B∩c|R[p]⊆ |d|R[p],
which implies that|W
b| ⊆(−|c|)R[p]∪ |d|R[p].
iii) The proof of the left-to-right direction is similar to the proof in i) above. Assume that |c|Shpi− |d|Shpi⊆ |a|Shpi
for alla∈A. This implies that|c|Shpi⊆ |a∨d|Shpi, which means by Lemma 3.4.5 (2.) that
♦(c−<(a∨d)∈/ p) for anya∈A. Hence sinceppreserves all joins inQJ, we have that
_
(c−<(A∨d)) =_(c−<(A∨d)) =(c−<(_A∨d))∈/ p.
By Lemma 3.4.5(2.) again, this implies that |c|Shpi⊆ |^
A∨d|Shpi=|^
A|Shpi∪ |d|Shpi,
which in turn implies that
|c|Shpi− |d|Shpi⊆ |^
3.4. Generalization to co-Heyting Algebras and Modal Algebras
iv) Once again, the proof of the left-to-right direction is similar to the proof in ii) above. Assume that
|b|Shpi⊆(−|c|)Shpi∪ |d|Shpi for allb∈B. Then
|b∧c|Shpi=|b|Shpi∩ |c|Shpi⊆ |d|Shpi,
which by Lemma 3.4.5 (2.) implies that♦((b∧c)−< d)∈/pfor allb∈B. Hence _
♦((B∧c)−< d) =♦_((B∧c)−< d) =♦((_B∧c)−<)∈p.
Hence by Lemma 3.4.5 (2.) again, we have that |_B|Shpi∩ |c|Shpi=|_
B∩c|Shpi⊆ |d|Shpi,
which yields that
|_B| ⊆(−|c|)Shpi∪ |d|Shpi.
As in the case for Heyting and co-Heyting algebras, the last step required before proving the Rasiowa-Sikorski lemma is the following:
Lemma 3.4.9. 1. Let (L,) be a modal Heyting algebra, (XL, τ,≤, R) be the dual modal Priestley space of L, and letQM andQJ be two countable(,→)-complete sets of Barcan meets and joins respectively. Then for any p∈XL which preserves all meets inQM and all joins in QJ and any A, B⊆L:
i) R[p] is closed in(XL, τ) ii) if VA
∈ QM, then (SVA)R[p] is dense in (R[p], τR[p]), and if WB ∈ QJ, then
(SWB)R[p] is dense in (R[p], τR[p]).
2. Dually, if(M,♦)is a modal co-Heyting algebra, (XM, σ,≤, S)is the dual modal Priestley space ofM, andQM andQJ are two countable(♦,−<)-complete sets of Barcan meets and joins respectively, then for any p∈XM which preserves all meets in QM and all joins in
QJ and anyA, B ⊆M: iii) Shpiis closed in(XL, τ)
iv) if VA ∈ Q
M, then (SVA)Shpi is dense in (Shpi, τShpi), and if WB ∈ QJ, then
(SWB)Shpi is dense in (Shpi, τShpi).
Proof.
i) This is a consequence of the fact thatR[p] =T
a∈p|a|
ii) This is an immediate consequence of Lemma 3.3.3 and Lemma 3.4.8 i) and ii). iii) This follows from the fact thatShpi=T
♦b /∈p−|b|
3.4. Generalization to co-Heyting Algebras and Modal Algebras
Lemma 3.4.10.
1. Let (L,) be a modal Heyting algebra, and let QM and QJ be two countable (,→)- complete sets of Barcan meets and joins respectively. Then for any prime filter pover L
that preserves all meets in QM and all joins inQJ, and any a∈L, if a /∈p, then there exists a prime filter q such that p ⊆q, a /∈ q, and q preserves all meets in QM and all joins inQJ.
2. Dually, if(M,♦)is a modal co-Heyting algebra, andQM andQJ are two countable(♦,−<)- complete sets of Barcan meets and joins respectively, then for any prime filter p over M
which preserves all meets in QM and all joins inQJ and any b∈M, if♦b∈p, then there exists a prime filter q such that q⊆ p♦, b ∈q, andq preserves all meets in QM and all joins inQJ.
Proof.
1. Let (XL, τ,≤, R) be the dual modal Priestley space of L, and let p ∈ XL such that p
preserves all meets inQM and all joins inQJ. Note that by Lemma 3.4.9 i), (R[p], τR[p]) is
compact Hausdorff, and therefore satisfies the conditions of the Baire Category Theorem. Moreover, by Lemma 3.4.9 ii), (SQ)R[p] is a countable intersection of open dense sets in
R[p], and is therefore dense by the Baire Category Theorem. Now assumea /∈pfor some
a∈L. This means that|a|R[p] is non-empty, and hence |a|R[p]∩(SQ)R[p]6=∅. But any q
in this set satisfies all the requirements of the lemma.
2. Dually, let (XM, τ,≤, S) be the dual modal space of M. A similar argument shows that
for any p preserving all meets in QM and all joins in QJ, (SQ)Shpi is dense in Shpi by
Lemma 3.4.9 iii) and iv). Now for any b∈M such that♦b∈p, |b|Shpi is non-empty. But this means that|b|Shpi∩(S
Q)Shpi6=∅. Since anyqin this set satisfies all the requirements,
the lemma is proved.