• v∗Γ(ϕ #ψ) = (([ϕ # ψ]Γ,1),([¬(ϕ #ψ)]Γ,1),([◦(ϕ # ψ)]Γ,1)) and
v∗Γ(ϕ) # v∗Γ(ψ) = (([ϕ]Γ,1),([¬ϕ]Γ,1),([◦ϕ]Γ,1)) # (([ψ]Γ,1),([¬ψ]Γ,1),([◦ψ]Γ,1))
= {(c1, c2, c3) ∈ ((AΓ)∗)3 : c1 = ([ϕ]Γ,1) # ([ψ]Γ,1)}, for # ∈ {∧,∨,→}. Since ([ϕ]Γ,1) # ([ψ]Γ,1) def= ([ϕ]Γ # [ψ]Γ,1) and [ϕ #ψ]Γ def= [ϕ]Γ # [ψ]Γ , for # ∈
{∧,∨,→}, thenvΓ∗(ϕ #ψ)∈v∗Γ(ϕ) # vΓ∗(ψ).
• v∗Γ(¬ϕ) = (([¬ϕ]Γ,1),([¬¬ϕ]Γ,1),([◦¬ϕ]Γ,1)) and
¬(v∗Γ(ϕ)) = ¬(([ϕ]Γ,1),([¬ϕ]Γ,1),([◦ϕ]Γ,1)) = {(c1, c2, c3) ∈ ((AΓ)∗)3 : c1 = ([¬ϕ]Γ,1)}. So vΓ∗(¬ϕ)∈¬(vΓ∗(ϕ)).
• v∗Γ(◦ϕ) = (([◦ϕ]Γ,1),([¬◦ϕ]Γ,1),([◦ ◦ϕ]Γ,1)) and
◦(v∗Γ(ϕ)) = ◦(([ϕ]Γ,1),([¬ϕ]Γ,1),([◦ϕ]Γ,1)) = {(c1, c2, c3) ∈ ((AΓ)∗)3 : c1 = ([◦ϕ]Γ,1)}. So vΓ∗(◦ϕ)∈ ◦(vΓ∗(ϕ)).
Thus vΓ∗ is a valuation over (MCPLΓ +e)∗.
Now, suppose that vΓ∗(ϕ) = (([ϕ]Γ,1),([¬ϕ]Γ,1),([◦ϕ]Γ,1)) ∈ D
BCPL(AΓ)+e∗
. Then, ([ϕ]Γ,1) = (1Γ)∗ = (1Γ,1) and 1Γ = [p1∨¬p1]Γ. So, by a similar proof to that for the Fact
included in the proof of Theorem 4.1.8, we have Γ�CPL+e ϕ.
If we suppose thatΓ�CPL+e ϕ, again by a similar proof for the Factmentioned above, we have [ϕ]Γ = [p1∨¬p1]Γ = 1Γand therefore,vΓ∗(ϕ) = (([ϕ]Γ,1),([¬ϕ]Γ,1),([◦ϕ]Γ,1))
∈D
BCPL(AΓ)∗+e.
Hence, vΓ∗(γ) ∈ D
B(ACPLΓ)∗+e for every γ ∈ Γ, but vΓ∗(α) �∈ D
BCPL(AΓ)∗+e. From this, Γ�|=M at(KCPL+
e)α.
The full subcategory in MAlg(Σ) of swap structures forCPL+e will be denoted bySWCPL+e. That is, the class of objects ofSWCPL+e isKCPL+e, and the morphisms between two given swap structures are just the homomorphisms between them as multialgebras.
Let KmbC ={B∈KCPL+e : B is a swap structure for mbC} be the class of swap structures for mbC. The following is immediate:
Proposition 4.2.3. KmbC={B∈KCPL+e : |=M(B) (Ax10)∧(bc1)}.
Proof. Suppose that KmbC � {B ∈ KCPL+e : |=M(B) (Ax10)∧(bc1)} and let B be the domain of B. Then there exist a swap structure B ∈ KmbC and a valuation v over M(B) such that v((Ax10)∧(bc1)) �∈ DB, that is π1(v((Ax10)∧(bc1))) �= 1. But, by definition of BmbCA ={(c1, c2, c3)∈A3 : c1∨c2 = 1 and c1∧c2∧c3 = 0} we have that π1(v((Ax10)∧(bc1))) = 1, see below:
Since v((Ax10)∧(bc1)) ∈ v(Ax10)∧v(bc1), then π1(v((Ax10) ∧(bc1))) = π1(v(Ax10))∧π1(v(bc1)). But,π1(v(bc1)) =π1(v(◦α→(α →(¬α →β)))) =π1(v(◦α)) → (π1(v(α))→(π1(v(¬α))→π1(v(β)))) =∼(π1(v(◦α))∧(π1(v(α))∧(π1(v(¬α)))))∨π1(v(β)).
However, v(◦α)∈ ◦v(α) and v(¬α)∈¬v(α), thenπ1(v(◦α)) =π3(v(α)) and π1(v(¬α)) = π2(v(α)). So,∼(π1(v(◦α))∧(π1(v(α))∧(π1(v(¬α)))))∨π1(v(β)) =∼(π3(v(α))∧π1(v(α))∧ π2(v(α)))∨ π1(v(β)) = ∼0∨ π1(v(β)) = 1 ∨π1(v(β)) = 1. Similarly we show that π1(v(Ax10)) = 1.
Conversely, let B ∈ KCPL+e such that |=M(B) (Ax10)∧(bc1) and let p, q two different propositional variables. Let c = (c1, c2, c3) ∈ B and d ∈ π1[B]. Consider a valuation v over M(B) such that v(p) = cand π1(v(q)) =d. Then π1(v(p∨¬p)) = 1 and π1(v(◦p →(p→ (¬p→ q)))) = 1. That is, c1∨c2 = 1 and (c3 → (c1 →(c2 →d))) = 1.
From this, (c1∧c2∧c3)≤ d, for every d∈π1[B]. Since by Proposition 4.1.4, π1[B] is a classical implicative lattice included in A as subalgebra, then c1∧c2∧c3 = 0. From this B ∈KmbC.
Given a Boolean algebra A, let BAmbC be the unique swap structure for mbC with domainBmbCA such that, for every (a1, a2, a3) and (b1, b2, b3) in BmbCA :
(i) (a1, a2, a3)#(b1, b2, b3) = {(c1, c2, c3)∈BmbCA : c1 =a1#b1}, for #∈{∧,∨,→}; (ii) ¬(a1, a2, a3) ={(c1, c2, c3)∈BmbCA : c1 =a2};
(iii) ◦(a1, a2, a3) ={(c1, c2, c3)∈BmbCA : c1 =a3}.
The full subcategory in SWCPL+e of swap structures formbC will be denoted by SWmbC. Clearly,SWmbC is a full subcategory in MAlg(Σ). Thus, the class of objects of SWmbC is KmbC, and the morphisms between two given swap structures for mbC are the homomorphisms between them, seeing as multialgebras over Σ
Let BAlg be the category of Boolean algebras defined over signature ΣBA, with Boolean algebras homomorphisms as their morphisms.
Proposition 4.2.4. Let {Ai}i∈I be a family of Boolean algebras in BAlg such that, for every i∈I, Ai =�Ai,∧i,∨i,→i,0i,1i�. Let A=�i∈IAi be the standard construction of the cartesian product of the family of sets {Ai}i∈I with canonical projections πi :A→Ai
for every i∈I.3 Let A=�A,∧,∨,→,0,1� be an algebra such that its operations are given by:
(i) (a#b)(i) =a(i)#ib(i), for every a, b∈A and #∈{∧,∨,→}; (ii) 0A(i) = 0i;
(iii) 1A(i) = 1i. Then:
(a) A=�A,∧,∨,→,0,1� is a Boolean algebra;
(b) the canonical projections πi :A→Ai are homomorphisms of Boolean algebras;
(c) �A,{πi}i∈I� is the product of the family {Ai}i∈I in BAlg.
Proof. It is a well-known result that the family of Boolean algebras {Ai}i∈I has product in BAlg and that �A,{πi}i∈I� as described above is its product.
Notation 4.2.5. The Boolean algebra A will be denoted by �i∈IAi.
Consider again a family F = {Ai}i∈I of Boolean algebras such that I �= ∅, and let A = �i∈I Ai be its product in BAlg as described above. We want to show that the product B = �i∈IBmbCAi in MAlg(Σ) (recall Proposition 2.5.8) of the family of multialgebras {BmbCAi }i∈I is isomorphic inMAlg(Σ) (recall Proposition 2.5.3) to the multialgebra BAmbC.
To begin with, some notation is required: For i ∈ I and 1 ≤ j ≤ 3 let πij : (Ai)3 →Ai be the canonical projections. Observe that, if a∈|B|=�i∈IBmbCAi and i∈I
then a(i)∈BmbCAi ⊆(Ai)3. Thus, for every 1≤j ≤3 let zj ∈�i∈IAi such that, for every i∈I,zj(i) =πji(a(i)). Thenz = (z1, z2, z3) belongs to|A|3. Moreover, it can be proven that z belongs to BmbCA . Indeed, for every i∈I, z1(i)∨iz2(i) =πi1(a(i))∨iπ2i(a(i)) = 1i since a(i)∈BmbCAi . From this, z1∨z2 = 1A. Analogously it can be proven that z1∧z2∧z3 = 0A.
This allows to define a mappingfF :�i∈IBmbCAi →BmbC�
i∈IAi such that, for every a∈�i∈IBmbCAi ,fF(a) = z where z = (z1, z2, z3) is defined as above.
3 That is,A=� a∈� �
i∈IAi�I
: a(i)∈Ai for everyi∈I�, ifI�=∅; AndAis a singleton otherwise.
Proposition 4.2.6. Let F = {Ai}i∈I be a family of Boolean algebras such that I �= ∅. Then, the mapping fF :�i∈IBmbCAi →BmbC�
i∈IAi is an isomorphism in MAlg(Σ).
Proof. Clearly fF is a bijective mapping such that its inverse mapping is given by fF−1 : BmbC�
i∈IAi → �i∈I BmbCAi where fF−1(z1, z2, z3) =a, witha(i) = (z1(i), z2(i), z3(i)) for every i∈I. It is also clear that, for every a, b∈�i∈I BmbCAi and #∈{∧,∨,→}:
(i) fF[a#b] =fF(a)#fF(b);
(ii) fF[¬a] =¬fF(a); and (iii) fF[◦a] =◦fF(a)
Thus, the result follows from Proposition 2.5.3.
Proposition 4.2.7. The category SWmbC has arbitrary products.
Proof. LetF ={Bi}i∈I be a family of swap structures formbC, and assume thatI �=∅(the caseI =∅is trivial). By definition ofKmbC, for eachi∈Ithere is a Boolean algebraAi such thatBi ⊆BmbCAi . SinceSWmbC is a subcategory of MAlg(Σ) (whereΣ is the signature of mbC), and the latter has arbitrary products (cf. Proposition 2.5.8), there exists the product
�B,{πi}i∈I�ofF inMAlg(Σ). By the proof of Proposition 2.5.8, it is possible to defineBin such a way that B⊆�i∈I BAmbCi , where the multialgebra �i∈I BAmbCi is also constructed as in the proof of Proposition 2.5.8. Let h:B→�i∈IBmbCAi be the inclusion homomorphism.
Now, let G ={Ai}i∈I and let fG :�i∈I BAmbCi →BmbC�
i∈IAi be the isomorphism in MAlg(Σ) of Proposition 4.2.6. Then, the homomorphismfG◦h:B→BmbC�
i∈IAi is an injective function B� ��� h ��
fG◦h
��
�
i∈IBAmbCi
fG
��
B�mbC
i∈IAi
and so it induces an isomorphism fG◦h in MAlg(Σ) between B and the submultialgebra B� = (fG◦h)(B) of B�mbC
i∈IAi, by Proposition 2.5.9. This means that�B�,{πi◦(fG◦h)−1}i∈I� is another realization of the product of F in MAlg(Σ).
B
πi
��
� � fG◦h ��B�mbC
i∈IAi
Bi B�
(fG◦h)−1
�� ����
πi◦(fG◦h)−1
��
Given that SWmbC is a full subcategory of MAlg(Σ) and by observing that B� is an object ofSWmbC, it follows that�B�,{πi◦(fG◦h)−1}i∈I�is a construction for the product
The assignment A ∈ BAlg �→ BAmbC ∈ SWmbC is functorial, as it will be stated in Corollary 4.2.9 below.
Proposition 4.2.8. Let f :A→A� be a homomorphism between Boolean algebras. Then it induces a homomorphism f∗ : BmbCA →BAmbC� of multialgebras given by f∗(z1, z2, z3) = (f(z1), f(z2), f(z3)). Moreover, (f◦g)∗ = f∗◦g∗ and (idA)∗ = idBmbC
A , where idA :A→A and idBmbC
A :BmbCA →BAmbC are the corresponding identity homomorphisms.
Proof. Given a homomorphism f :A → A� between Boolean algebras, let f∗ :BmbCA → BmbCA� be the mapping such that f∗(z) = (f(z1), f(z2), f(z3)) for every z = (z1, z2, z3) ∈ BmbCA . If z = (z1, z2, z3), w = (w1, w2, w3) ∈ BmbCA and # ∈ {∧,∨,→} then, for every u = (u1, u2, u3) ∈(z#w), u1 = z1#w1 and so f(u1) =f(z1)#f(w1). That is, (f∗(u))1 = (f∗(z))1#(f∗(w))1. This means that f∗[z#w] = {f∗(u) : u ∈ (z#w)} ⊆ {u� ∈ BmbCA� :
u�1 = (f∗(z))1#(f∗(w))1}=f∗(z)#f∗(w).
On the other hand, if z = (z1, z2, z3)∈ BmbCA and u= (u1, u2, u3) ∈ ¬z then u1 = z2 whence (f∗(u))1 = f(u1) = f(z2) = (f∗(z))2. This means that f∗(u) ∈ {u� ∈ BmbCA� : u�1 = (f∗(z))2}=¬f∗(z) and so f∗[¬z]⊆¬f∗(z). Analogously it can be proven that f∗[◦z]⊆ ◦f∗(z). This shows that f∗ is indeed a homomorphism f∗ :BAmbC →BmbCA� in SWmbC. The rest of the proof is immediate, by the very definition of f∗.
Corollary 4.2.9. There is a functor F : BAlg → SWmbC given by F(A) = BAmbC for every Bolean algebra A, and F(f) =f∗ for every homomorphism f :A →A� in BAlg.
Proposition 4.2.10. The functor F :BAlg →SWmbC preserves arbitrary products.
Proof. It is an immediate consequence of Proposition 4.2.6 and the fact that SWmbC is a full subcategory of MAlg(Σ).
Proposition 4.2.11. The functor F : BAlg → SWmbC preserves subalgebras in the following sense: if A is a subalgebra of A� in BAlg then BAmbC ⊆ BAmbC� according to Definition 2.2.1.
Proof. LetAandA� be two Boolean algebras inBAlg such thatAis subalgebra ofA� and let f :A→A� be the inclusion morphism in BAlg given by f(a) =a for everya ∈|A|. By Proposition 4.2.8, f induces a homomorphism f∗ : BmbCA → BAmbC� of multialgebras given by f∗(z1, z2, z3) = (f(z1), f(z2), f(z3)) = (z1, z2, z3). By definition of |BmbCA | and
|BAmbC� | and since |A|⊆|A�|we have |BAmbC|⊆|BmbCA� |. Moreover,BAmbC ⊆BmbCA� .
Proposition 4.2.12. The functor F :BAlg →SWmbC preserves monomorphisms.
Proof. Let f : A → A� be a monomorphism between Boolean algebras, and let f∗ : BAmbC →BmbCA� be the induced homomorphism of multialgebras given by f∗(z1, z2, z3) = (f(z1), f(z2), f(z3)). It is well-known that every monomorphism in BAlg is an injective function, and then f is injective. From this it is immediate to see thatf∗ is also an injective function. As a consequence of Proposition 2.5.4, f∗ is a monomorphism in the category MAlg(Σ). Given that SWmbC is a full subcategory of MAlg(Σ), it follows that f∗ is a monomorphism in the category SWmbC.
As it was done in Definition 4.1.5, each B ∈ KmbC induces naturally a non-deterministic matrix M(B) = (B, DB). Moreover, in (CARNIELLI; CONIGLIO, 2016) it was proven that the classM at(KmbC) ={M(B) : B ∈KmbC}semantically characterizes mbC:
Theorem 4.2.13. (CARNIELLI; CONIGLIO, 2016, Theorem 6.4.8)LetΓ∪{α}⊆F or(Σ) be a set of formulas of mbC. Then: Γ�mbCα iff Γ|=M at(KmbC) α.
The non-deterministic matrix MmbC5 induced by the swap structure BmbCA2
defined over the two-element Boolean algebra A2 was originally introduced by A. Avron in (AVRON, 2005)4 in order to semantically characterize the logicmbC. The domain of the multialgebra BAmbC2 is the set BmbCA2 =�t, I, tI, f, fI
� such that t= (1,0,1), I = (1,1,0), tI = (1,0,0),f = (0,1,1), andfI = (0,1,0). Let D be the set of designated elements of the non-deterministic matrix MmbC5 =M�BAmbC2 �. Then, D = {t, I, tI}. Let ND =�f, fI
�
be the set of non-designated truth-values. The multioperations proposed by Avron over the set BmbCA2 (see the Example 2.1.6) corresponds exactly with that for BmbCA2 described after Proposition 4.2.3. It was proved in (AVRON, 2005) that mbCis adequate for MmbC5 : Theorem 4.2.14. (AVRON, 2005, Theorem 3.6) For every set of formulas Γ∪{α} ⊆ F or(Σ): Γ�mbCα iff Γ|=MmbC5 α.
As observed in (CARNIELLI; CONIGLIO, 2016, Chapter 6), Avron’s result means that the non-deterministic matrix induced by the swap structure BmbCA2 defined over the two-element Boolean algebra A2 is sufficient for characterizing the logic mbC, and so it represents, in a certain way, the whole class KmbC of swap structures for mbC.
One interesting question is to prove that the 5-element multialgebra BmbCA2 generates (in some sense to be determined) the class KmbC, in analogy to the fact that the 2-element Boolean algebra A2 generates the class of Boolean algebras.
4 In (AVRON, 2005) Avron denoted the systemmbCbyB, so the non-deterministic matrix formbCis
Recall that the power set ℘(I) of a given set I is a Boolean algebra, where the operations are the usual set-theoretic ones. A field of sets is any subalgebra of a power set Boolean algebra ℘(I). Birkhoff proves in 1935 (see (BIRKHOFF, 1935)) that every Boolean algebra A is isomorphic to afield of sets. Taking into account that ℘(I) is isomorphic, as a Boolean algebra, to the product �i∈IA2 of Boolean algebras, Birkhoff’s result is equivalent to the following: for every Boolean algebra A, there exists a set I and a monomorphism of Boolean algebras h:A→�i∈IA2.
From Birkhoff’s representation theorem for Boolean algebras, and taking into account the properties of the functor F :BAlg→SWmbC, a representation theorem for the class KmbC of swap structures for mbC can be obtained:
Theorem 4.2.15 (Representation Theorem for KmbC). Let B be a swap structure for mbC. Then, there exists a set I and a monomorphism of multialgebras ˆh:B→�i∈IBAmbC2 . Proof. Let B be a swap structure for mbC. Then, there is a Boolean algebra A such that B ⊆BAmbC. Letg :B→BAmbCbe the inclusion monomorphism inSWmbC. Using Birkhoff’s representation theorem for Boolean algebras5, there exists a set I and a monomorphism h:A→�i∈IA�iof Boolean algebras, whereA�i =A2, for everyi∈I. By Proposition 4.2.12, there is a monomorphism h∗ : BAmbC → B�mbC
i∈IA�i. Let fG : �i∈IBmbCA�i → B�mbC
i∈IA�i be the isomorphism in MAlg(Σ) of Proposition 4.2.6, where G ={A�i}i∈I. By definition ofA�i it follows that BAmbC�i = BAmbC2 , for every i∈I. Then ˆh:B→�i∈IBmbCA2 is a monomorphism in MAlg(Σ), where ˆh=fG−1◦h∗◦g.
From the previous result, it is natural to ask about the possibility of the class KmbC being a variety of multialgebras, that is, a class closed under products, submultialgebras and homomorphic images. We known that KmbC is closed under products (by Proposition 4.2.7) and submultialgebras (by the very definitions). Unfortunately, the
class is not closed under homomorphic images:
Proposition 4.2.16. The class KmbC of multialgebras is closed under submultialgebras and (direct) products, but it is not closed under homomorphic images.
Proof. Recall the notions of multicongruence (Definition 2.4.1), quotient multialgebra (Definition 2.4.5) and the canonical map p : A → A/Θ for every multicongruence Θ (Proposition 2.4.8). Now, let D = {z1, z2, z3} and ND = �z4, z5� be an enumeration of the elements of the domain BmbCA2 = D∪ND of the multialgebra BmbCA2 . Let Θ be the equivalence relation asociated to the partition {a, b}of BmbCA2 such that a={z1, z4} and b ={z2, z3, z5}. The relationΘhas the following property: for everyz ∈D there exists some w∈ND such that (z, w)∈Θ, and vice versa. From this, and by observing the definition of
5 In (DUNN; HARDEGREE 2001 Theorem 8.11.7, p. 307).
the multioperations in the multialgebra BmbCA2 , it follows that Θ is a multicongruence over BAmbC2 . It is easy to prove that the multioperations in the quotient multialgebra BmbCA2 /Θ are trivial, that is: for every x, y ∈{a, b} and # ∈{∧,∨,→}, (x#y) =¬x=◦x={a, b}.
Clearly BmbCA2 /Θ is not a swap structure for mbC: otherwise, it would generate a trivial non-deterministic matrix where the set of designated values is the whole domain. This would contradict (CARNIELLI; CONIGLIO, 2016, Proposition 6.4.5(ii)), where it was proven that no non-deterministic matrix in the class M at(KmbC) is trivial. This shows that BmbCA2 /Θ, the homomorphic image of the canonical map p:BmbCA2 →BAmbC2 /Θ, does not belong to the class KmbC, despite its domain BmbCA2 is in KmbC.
From this last result, a question that arises is: Why don’t we change/adapt the homomorphic images or the congruence definitions in order to show that KmbC is a variety of multialgebras? The problem is that, if we do this, we will lose some important results such as the method of completeness that we apply in the main theorems of this Thesis.