Let K1+K2 be the coproduct of two Kripke polynomial functors. It is well
know thatXY+Z ∼=XY ×XZ. Therefore a predicate lifting
λ: 2(−)−→2(K1+K2)(−)∼= 2K1(−)×2K2(−),
is in fact a pair of predicate liftings
λ1: 2(−)−→2K1(−) λ2: 2(−)−→2K2(−).
Assume λ = (λ1, λ2), where λ1 and λ2 are predicate liftings for K1 andK2
respectively. It is easy to see that for a setX⊆A,
λA(X) =λ1,A(X)∪λ2,A(X).
The same holds for polyadic predicate liftings.
Now we characterize singleton liftings for the coproduct. An element p ∈
K1(2η) +K2(2η) belongs to exactly one of the factors. Assumep∈K1(2η). The
predicate lifting for the coproduct associated with{p} is (λp, λ⊥), where λ⊥ is the predicate lifting defined on page 43 and λp is the predicate lifting for K1
associated with{p}. This predicate lifting acts as follows. (λp, λ⊥)A(−X→) =λp(−X→),
where −X→= (Xj)j∈η Symmetrically if p∈K2(2η) the predicate lifting for the
coproduct associated with{p} is (λ⊥, λp).
A coalgebra for the coproduct is a function α: A −→ K1A+K2A, i.e for
each states∈Aeither α(s)∈K1Aor eitherα(s)∈K2A. The semantics for a
singleton lifting associated withp∈K1(2η) is
s|=α[λp, λ⊥](ϕj)j∈η iffα(s)∈λp([ϕj])j∈η.
Notice that this previous formula impliesα(s)∈K1A. The semantics forp∈
4.4.6
Product of functors
Let a pair of functorsK1, K2:Set−→Setbe given. In this section we will in-
vestigate the Coalgebraic modal language for the product functorK1×K2:Set
−→Set.
We first studied singleton liftings to understand predicate liftings for the product. The following paragraphs will show that singleton liftings enormously simplify the presentation of predicate liftings for the product of functors.
Let (p1, p2) ∈ K1(2η)×K2(2η). According to equation 4.3, the predicate
lifting for the product associated with{(p1, p2)}acts as follows: Given a family
of subsets (Xj)j∈η ofA
λ(p1,p2),A(X) ={(t1, t2)∈K1A×K2A|(K1×K2)(< χXj >j∈η)(t1, t2) = (p1, p2))}.
In other words this can be stated as follows
λ(p1,p2),A(Xj)j∈η=λp1(Xj)j∈η×λp2(Xj)j∈η,
whereλp1 andλp2 are the singleton predicate liftings associated withp1 forK1
and associated withp2forK2.
The semantics for such singleton predicate lifting is:
s|=α[λ(p1,p2)](ϕj)j∈η iffπ1α(s)∈λp1([ϕj])j∈η andπ2α(s)∈λp2([ϕ])j∈η.
Remark 4.4.1. Notice that the predicate liftings for the product are associated with sets C ⊆ K12×K2. If the set C is the product of two sets we can give
a nice characterization as we did in the case of singleton liftings. If not a characterization using the predicate liftings of the factors is much more obscure. Singleton liftings solve this problem because now the predicate lifting associated withC can be represented as the union of the predicate liftings of its elements.
4.4.7
Hom(D, K))
functors
Now we will compose a Kripke polynomial functor, say K, with a homomor- phism functor, sayHom(D,−). A coalgebra for this functor is a functionα:A
−→Hom(D, KA).
A singleton predicate lifting will be associated with a function P : D −→
K(2η). The predicate lifting associated with {P} acts as follows: Given a family of subsets (Xj)j∈η ofA
λP,A(Xj)j∈η={h:D−→T A|K([Xj]j∈η)h=P}. The semantic for such predicate lifting is:
s|=α[λP](ϕj)j∈η iff (∀d∈D)(α(s)(d)∈λP(d)(Xj)j∈η), whereλP(d)is the singleton predicate lifting associated withP(d) forK.
4.4.8
PK
functors
A coalgebra for this functor can be seen as a Kripke frame of type K, i.e is a function α : A −→ PKA. A singleton lifting will be associated with a set
P ⊆K(2). Equation 4.2 tell us that the function ofλ(P,A): 2A−→2PKA acts
as follows: A setX⊆Ais mapped to
λ(P,A)(X) ={U ⊆KA|K(χX)[U] =P},
where K(χX)[U] is the direct image of U under the functionK(χX). Notice that K(χX) : KA−→K2, therefore its direct image goes fromPKAto PK2. Using this we can see that the elements inλ(P,A)(X) are the setsU ⊆KAsuch
that
(∀u∈U)(∃p∈P)(K(χX)(u) =p)∧ ∀(p∈P)(∃u∈U)(K(χX)(u) =p).
Now we will provide a characterization that will be used to translate the predicate liftings for a functor of the form PK into the Moss’ language for the same functor.
We will use capital P to denote subsets of K2 and we will use small pto denote elements ofK2. Consider the predicate lifting associated withP ={p}, where p ∈ K2. The A-component of this predicate lifting goes from 2A to 2PKA. Rewriting the previous characterization the elements ofλ
{p}(X) are the setsU ⊆KAsuch that
(∀u∈U)(K(χX)(u) =p)∧(∃u∈U)(K(χX)(u) =p).
Now consider the predicate lifting associated with p∈ K2 for the functor K. We already know that theA-componentλp: 2A−→2KA acts as follows
λp(X) ={u∈KA|K(χX)(u) =p}.
Using this we can see that the formula
(∀u∈U)(K(χX)(u) =p) is equivalent to the formula
U ⊆λp(X).
The formula
(∃u∈U)(K(χX)(u) =p)
is expressing the fact thatU is not empty. We conclude that the singleton lifting
λ{p}: 2A−→2PKAcan be characterized in terms ofλp as follows:
λ({p},A)(X) ={U ⊆T A|U 6=∅ ∧U ⊆λp(X)}.
More general, we can see that forP ⊆K2 the formula (∀u∈U)(∃p∈P)(K(χX)(u) =p) is equivalent to the formula
U ⊆ [
p∈P
We can also see that the formula
∀(p∈P)(∃u∈U)(K(χX)(u) =p) is equivalent to the formula
(∀p∈P)(U∩λp(X)6=∅).
Using that we can characterizeλ(P,A) : 2A−→2PKA in terms of the predicate
liftings forK associated with the elements ofP as follows
λ(P,A)(X) ={U ⊆KA|U ⊆
[ p∈P
λp(X) ∧ (∀p∈P)(U∩λp(X)6=∅)},
where aλp: 2A−→2KAis the singleton lifting associated withpfor the functor
K.