state-morphism mv-algebras and their generalizations · state-morphism mv-algebras and their...

State-morphism MV-algebras andTheir Generalizations

Anatolij DVURECENSKIJ

Mathematical Institute, Slovak Academy of Sciences,

Stefanikova 49, SK-814 73 Bratislava, Slovakia

E-mail: [email protected]

The talk given at 11th INTERNATIONAL CONFERENCE ON FUZZY SET THEORY

AND APPLICATIONS Liptovsk Jn, Slovak Republic, January 30 - February 3, 2012

ERDF OP R&D Projects CE meta-QUTE ITMS 26240120022.

State-morphism MV-algebras and Their Generalizations – p. 1

States on MV-algebras

• M - MV-algebra, we define a partial operation+, via a+ b is defined iff a ≤ b∗ iff a⊙ b = 0,then a+ b := a⊕ b.



•• M - MV-algebra, we define a partial operation+, via a+ b is defined iff a ≤ b∗ iff a⊙ b = 0,then a+ b := a⊕ b.

• + restriction of the ℓ-group addition




• + restriction of the ℓ-group addition• state- s : M → [0, 1], (i) s(a+ b) = s(a) + s(b),

(ii) s(1) = 1.





(ii) s(1) = 1.

• S(M) -set of states. S(M) 6= ∅.





(ii) s(1) = 1.

• S(M) -set of states. S(M) 6= ∅.

• extremal state s = λs1 + (1− λ)s2 forλ ∈ (0, 1) ⇒ s = s1 = s2.


•• {sα} → s iff limα sα(a) → s(a), a ∈ M.



• S(E) - Hausdorff compact topological space,∂eS(M)




• Krein-Mil’man S(M) = Cl(ConHul(∂eS(M))





• s is extremal iff s(a ∧ b) = min{s(a), s(b)} iff sis MV-homomorphism iff Ker(s) is a maximalideal.





• s is extremal iff s(a ∧ b) = min{s(a), s(b)} iff sis MV-homomorphism iff Ker(s) is a maximalideal.

• s ↔ Ker(s), 1-1 correspondence


•• every maximal ideal is a kernel of a uniquestate


••• every maximal ideal is a kernel of a uniquestate

• Kernel-hull topology = ∂eS(E) set of extremalstates




• Kroupa- Panti a 7→ a, a(s) := s(a),

s(a) =

∫

∂eS(M)

a(t)dµs(t)




• Kroupa- Panti a 7→ a, a(s) := s(a),

s(a) =

∫

∂eS(M)

a(t)dµs(t)

• µs - unique Borel regular σ-additive probabilitymeasure on B(S(M)) such thatµs(∂eS(M)) = 1. State-morphism MV-algebras and Their Generalizations – p. 4

State MV-algebras

••• MV-algebras with a state are not universalalgebras, and therefore, the do not provide analgebraizable logic for probability reasoningover many-valued events


State MV-algebras

•••• MV-algebras with a state are not universalalgebras, and therefore, the do not provide analgebraizable logic for probability reasoningover many-valued events

• Flaminio-Montagna - introduce analgebraizable logic whose equivalentalgebraic semantics is the variety of stateMV-algebras


State MV-algebras

•••• MV-algebras with a state are not universalalgebras, and therefore, the do not provide analgebraizable logic for probability reasoningover many-valued events

• Flaminio-Montagna - introduce analgebraizable logic whose equivalentalgebraic semantics is the variety of stateMV-algebras

• A state MV-algebra is a pair (M, τ), M -MV-algebra, τ unary operation on A s.t.


•••• τ(1) = 1


• τ(1) = 1

• τ(x⊕ y) = τ(x)⊕ τ(y ⊖ (x⊙ y))


• τ(1) = 1

• τ(x⊕ y) = τ(x)⊕ τ(y ⊖ (x⊙ y))

• τ(x∗) = τ(x)∗


• τ(1) = 1

• τ(x⊕ y) = τ(x)⊕ τ(y ⊖ (x⊙ y))

• τ(x∗) = τ(x)∗

• τ(τ(x)⊕ τ(y)) = τ(x)⊕ τ(y)


• τ(1) = 1

• τ(x⊕ y) = τ(x)⊕ τ(y ⊖ (x⊙ y))

• τ(x∗) = τ(x)∗

• τ(τ(x)⊕ τ(y)) = τ(x)⊕ τ(y)

• τ -internal operator, state operator


Properties

• τ 2 = τ


Properties

• τ 2 = τ

• τ(M) is an MV-algebra and τ on τ(M) -identity


Properties

• τ 2 = τ


• τ(x+ y) = τ(x) + τ(y)


Properties

• τ 2 = τ


• τ(x+ y) = τ(x) + τ(y)

• τ(x⊙ y) = τ(x)⊙ τ(y) if x⊙ y = 0.


Properties

• τ 2 = τ


• τ(x+ y) = τ(x) + τ(y)

• τ(x⊙ y) = τ(x)⊙ τ(y) if x⊙ y = 0.

• if (M, τ) is s.i., then τ(M) is a chain


Properties

• τ 2 = τ


• τ(x+ y) = τ(x) + τ(y)

• τ(x⊙ y) = τ(x)⊙ τ(y) if x⊙ y = 0.

• if (M, τ) is s.i., then τ(M) is a chain

• if (M, τ) is s.i., then M is not necessarily achain


• F -filter, τ -filter if τ(F ) ⊆ F.


•• F -filter, τ -filter if τ(F ) ⊆ F.

• 1-1 correspondence congruences andτ -filters




• M = [0, 1] × [0, 1], τ(x, y) = (x, x) s.i. - notchain





• state-morphism (M, τ), τ is an idempotentendomorphism





• state-morphism (M, τ), τ is an idempotentendomorphism

• s state on M , [0, 1]⊗M ,τs(α⊗ a) := α · s(a)⊗ 1


•• ([0, 1]⊗, τs) is an SMV-algebra.



• ([0, 1]⊗, τs) is an SMMV-algebra iff s is anextremal state




• if M is a chain, every SMV-algebra (M, τ) isan SMMV-algebra





• if τ(M) ∈ V(S1, . . . , Sn) for some n ≥ 1, then(M, τ) is an SMMV-algebra





• if τ(M) ∈ V(S1, . . . , Sn) for some n ≥ 1, then(M, τ) is an SMMV-algebra

• Iff τ((n+ 1)x) = τ(nx)


Other examples

•• Let (G, u) be a unital Riesz space and letA = Γ(G, u). Choose an endomorphisms : A → A such that s ◦ s = s and fix a realnumber λ ∈ [0, 1]. Define a mappingsλ : A2 → A by

sλ(x, y) = λs(x) + (1− λ)s(y), (x, y) ∈ A2,

and set τλ,s : A3 → A3 by

τλ,s(x, y, z) = (sλ(x, y), sλ(x, y), sλ(x, y)), (x, y, z) ∈ A3.


•••••• Then D(A)λ,s := (A3, τλ,s) is a stateMV-algebra such that it is a state morphismMV-algebra iff λ ∈ {0, 1}.

Ker(τλ,s) =

Ker(s)× Ker(s)× A if λ ∈ (0, 1),

A× Ker(s)× A if λ = 0,

Ker(s)× A× A if λ = 1.

Hence, if A is a subdirectly irreducibleMV-algebra and s is faithful, then neither(A, τ0,s) nor (A, τ1,s) is a subdirectly irreduciblestate morphism MV-algebra, State-morphism MV-algebras and Their Generalizations – p. 11

•••••• whilst if λ ∈ (0, 1), D(Aλ,s) is a subdirectlyirreducible state MV-algebra.


•••• whilst if λ ∈ (0, 1), D(Aλ,s) is a subdirectlyirreducible state MV-algebra.

• Let A = [0, 1] be the standard MV-algebra andlet s be the identity on A. If λ ∈ {0, 1}, thenD([0, 1]1,s) generates the variety of statemorphism MV-algebras


•••• whilst if λ ∈ (0, 1), D(Aλ,s) is a subdirectlyirreducible state MV-algebra.

• Let A = [0, 1] be the standard MV-algebra andlet s be the identity on A. If λ ∈ {0, 1}, thenD([0, 1]1,s) generates the variety of statemorphism MV-algebras

• Does D([0, 1]λ,s) for a fixed λ ∈ (0, 1) and theidentity s on [0, 1] generate the variety of stateMV-algebras ?


State BL-algebras

•••• M - BL-algebra. A map τ : M → M s.t.

(1)BL τ(0) = 0;

(2)BL τ(x → y) = τ(x) → τ(x ∧ y);

(3)BL τ(x⊙ y) = τ(x)⊙ τ(x → (x⊙ y));

(4)BL τ(τ(x)⊙ τ(y)) = τ(x)⊙ τ(y);

(5)BL τ(τ(x) → τ(y)) = τ(x) → τ(y)

state-operator on M, pair (M, τ) - state BL-algebra


State BL-algebras

••••• M - BL-algebra. A map τ : M → M s.t.

(1)BL τ(0) = 0;

(2)BL τ(x → y) = τ(x) → τ(x ∧ y);

(3)BL τ(x⊙ y) = τ(x)⊙ τ(x → (x⊙ y));

(4)BL τ(τ(x)⊙ τ(y)) = τ(x)⊙ τ(y);

(5)BL τ(τ(x) → τ(y)) = τ(x) → τ(y)

state-operator on M, pair (M, τ) - state BL-algebra

• If τ : M → M is a BL-endomorphism s.t.τ ◦ τ = τ, - state-morphism operator and thecouple (M, τ) - state-morphism BL-algebra.


••••• every state operator on a linear BL-algebra isa state-morphism


•••• every state operator on a linear BL-algebra isa state-morphism

• Example 0.2 Let M be a BL-algebra. On M ×Mwe define two operators, τ1 and τ2, as follows

τ1(a, b) = (a, a), τ2(a, b) = (b, b), (a, b) ∈ M×M.

(2.0)Then τ1 and τ2 are two state-morphism operators onM ×M.


•• every state operator on a linear BL-algebra isa state-morphism

• Example 0.3 Let M be a BL-algebra. On M ×Mwe define two operators, τ1 and τ2, as follows

τ1(a, b) = (a, a), τ2(a, b) = (b, b), (a, b) ∈ M×M.

(2.0)Then τ1 and τ2 are two state-morphism operators onM ×M.

• Ker(τ) = {a ∈ M : τ(a) = 1}.


•••• We say that two subhoops, A and B, of aBL-algebra M have the disjunction property iffor all x ∈ A and y ∈ B, if x ∨ y = 1, theneither x = 1 or y = 1.


••••• We say that two subhoops, A and B, of aBL-algebra M have the disjunction property iffor all x ∈ A and y ∈ B, if x ∨ y = 1, theneither x = 1 or y = 1.

• Lemma 0.5 Suppose that (M, τ) is a stateBL-algebra. Then:

(1) If τ is faithful, then (M, τ) is a subdirectlyirreducible state BL-algebra if and only ifτ(M) is a subdirectly irreducibleBL-algebra.

Now let (M, τ) be subdirectly irreducible.Then:


•• (2) Ker(τ) is (either trivial or) a subdirectlyirreducible hoop.

(3) Ker(τ) and τ(M) have the disjunctionproperty.


•• (2) Ker(τ) is (either trivial or) a subdirectlyirreducible hoop.

(3) Ker(τ) and τ(M) have the disjunctionproperty.

• Theorem 0.7 Let (M, τ) be a stateBL-algebra satisfying conditions (1), (2) and (3)in the last Lemma. Then (M, τ) is subdirectlyirreducible.


•• Theorem 0.8 A state-morphism BL-algebra(M, τ) is subdirectly irreducible irreducible ifand only if one of the following threepossibilities holds.



• (i) M is linear, τ = idM , and the BL-reduct Mis a subdirectly irreducible BL-algebra.




• (ii) The state-morphism operator τ is notfaithful, M has no nontrivial Booleanelements, and the BL-reduct M of (M, τ) is alocal BL-algebra, Ker(τ) is a subdirectlyirreducible irreducible hoop, and Ker(τ) andτ(M) have the disjunction property.


•• Moreover, M is linearly ordered if and only ifRad1(M) is linearly ordered, and in such acase, M is a subdirectly irreducibleBL-algebra such that if F is the smallestnontrivial state-filter for (M, τ), then F is thesmallest nontrivial BL-filter for M.


••••• Moreover, M is linearly ordered if and only ifRad1(M) is linearly ordered, and in such acase, M is a subdirectly irreducibleBL-algebra such that if F is the smallestnontrivial state-filter for (M, τ), then F is thesmallest nontrivial BL-filter for M.

• If Rad(M) = Ker(τ), then M is linearly ordered.


••••• (iii) The state-morphism operator τ is notfaithful, M has a nontrivial Boolean element.There are a linearly ordered BL-algebra A, asubdirectly irreducible BL-algebra B, and aninjective BL-homomorphism h : A → B suchthat (M, τ) is isomorphic as a state-morphismBL-algebra with the state-morphismBL-algebra (A× B, τh), whereτh(x, y) = (x, h(x)) for any (x, y) ∈ A× B.


Varieties of SMMV-algebras

•••••• Komori - countably many subvarieties ofMV-algebras



• Komori - countably many subvarieties ofMV-algebras

• V-variety of MV-algebras, Vτ -system ofSMMV-algebras (M, τ) s.t M ∈ V ∈ V .





• D(M) := (M ×M, τM )





• D(M) := (M ×M, τM )

• V(D) = V(M)τ





• D(M) := (M ×M, τM )

• V(D) = V(M)τ

• SMMV = V(D([0, 1]))





• D(M) := (M ×M, τM )

• V(D) = V(M)τ

• SMMV = V(D([0, 1]))

• Pτ = V(D(C)), P perfect MV-algebras, C-Chang


• Theorem: VI ⊆ VR ⊆ VL ⊆ Vτ . and allinclusions are proper of V is not finitelygenerated.


•• Theorem: VI ⊆ VR ⊆ VL ⊆ Vτ . and allinclusions are proper of V is not finitelygenerated.

• Theorem: Representable SMMV-algebras:

τ(x) ∨ (x → (τ(y) ↔ y)) = 1.




τ(x) ∨ (x → (τ(y) ↔ y)) = 1.

• also for BL-algebra




τ(x) ∨ (x → (τ(y) ↔ y)) = 1.

• also for BL-algebra• Theorem: VL - generated by those (M, τ), M

is local




τ(x) ∨ (x → (τ(y) ↔ y)) = 1.

• also for BL-algebra• Theorem: VL - generated by those (M, τ), M

is local•

(τ(x) ↔ x)∗ ≤ (τ(x) ↔ x).State-morphism MV-algebras and Their Generalizations – p. 21

Uncountable many subvarieties

•• [0, 1]∗ ultrapower, for positive infinitesimalǫ ∈ [0, 1]∗



• [0, 1]∗ ultrapower, for positive infinitesimalǫ ∈ [0, 1]∗

• X subset of prime numbers, A(X)MV-algebra generated by ǫ and n

ms.t





ms.t

• (1) either n = 0 or g.c.d(n,m) = 1





ms.t

• (1) either n = 0 or g.c.d(n,m) = 1

• ∀p ∈ X, p does not divide m





ms.t

• (1) either n = 0 or g.c.d(n,m) = 1


• τ(x)= standard part of x





ms.t

• (1) either n = 0 or g.c.d(n,m) = 1


• τ(x)= standard part of x

• (A(X), τ) is linearly ordered SMMV-algebra


• if X 6= Y , then V(A(X)) 6= V(A(Y ))


••••• if X 6= Y , then V(A(X)) 6= V(A(Y ))

• Theorem: Between MVI and MVR there isuncountably many varieties


Generators of SMBL-algebras

••••• t-norm- function t : [0, 1]× [0, 1] → [0, 1] suchthat (i) t is commutative, associative, (ii)t(x, 1) = x, x ∈ [0, 1], and (iii) t isnondecreasing.



•• t-norm- function t : [0, 1]× [0, 1] → [0, 1] suchthat (i) t is commutative, associative, (ii)t(x, 1) = x, x ∈ [0, 1], and (iii) t isnondecreasing.

• If t is continuous, we define x⊙t y = t(x, y)and x →t y = sup{z ∈ [0, 1] : t(z, x) ≤ y} forx, y ∈ [0, 1], thenIt := ([0, 1],min,max,⊙t,→t, 0, 1) is aBL-algebra.



•••• t-norm- function t : [0, 1]× [0, 1] → [0, 1] suchthat (i) t is commutative, associative, (ii)t(x, 1) = x, x ∈ [0, 1], and (iii) t isnondecreasing.

• If t is continuous, we define x⊙t y = t(x, y)and x →t y = sup{z ∈ [0, 1] : t(z, x) ≤ y} forx, y ∈ [0, 1], thenIt := ([0, 1],min,max,⊙t,→t, 0, 1) is aBL-algebra.

• The variety of all BL-algebras is generated byall It with a continuous t-norm t.


•••• T denotes the system of all BL-algebrasD(It), where t is a continuous t-norm on theinterval [0, 1],


•• T denotes the system of all BL-algebrasD(It), where t is a continuous t-norm on theinterval [0, 1],

• Theorem 0.15 The variety of allstate-morphism BL-algebras is generated bythe class T .


General Approach - State-Morphism Algebras

•• A an algebra of type F , τ an idempotentendomorphism of A, (A, τ) state-morphismalgebra




• θτ = {(x, y) ∈ A× A : τ(x) = τ(y)},




• θτ = {(x, y) ∈ A× A : τ(x) = τ(y)},

• φ ⊆ A2, Φ(φ), Φτ (φ) congruence generated byφ on A and (A, τ)




• θτ = {(x, y) ∈ A× A : τ(x) = τ(y)},

• φ ⊆ A2, Φ(φ), Φτ (φ) congruence generated byφ on A and (A, τ)

• Lemma: For any φ ∈ Con τ(A), we haveθφ ∈ Con (A, τ), and θφ ∩ τ(A)2 = φ. Inaddition, θτ ∈ Con (A, τ), φ ⊆ θφ, andΘτ(φ) ⊆ θφ.


•• Lemma: Let θ ∈ ConA be such that θ ⊆ θτ .Then θ ∈ Con (A, τ) holds.


•••• Lemma: Let θ ∈ ConA be such that θ ⊆ θτ .Then θ ∈ Con (A, τ) holds.

• Lemma: If x, y ∈ τ(A), thenΘ(x, y) = Θτ(x, y). Consequently,Θ(φ) = Θτ(φ) whenever φ ⊆ τ(A)2.


•••• Lemma: Let θ ∈ ConA be such that θ ⊆ θτ .Then θ ∈ Con (A, τ) holds.

• Lemma: If x, y ∈ τ(A), thenΘ(x, y) = Θτ(x, y). Consequently,Θ(φ) = Θτ(φ) whenever φ ⊆ τ(A)2.

• if (C, τ) → (B×B, τB), (C, τ) is said to be asubdiagonal state-morphism algebra


•••• Theorem 0.16 Let (A, τ) be a subdirectlyirreducible state-morphism algebra such thatA is subdirectly reducible. Then there is asubdirectly irreducible algebra B such that(A, τ) is B-subdiagonal.


•• Theorem 0.18 Let (A, τ) be a subdirectlyirreducible state-morphism algebra such thatA is subdirectly reducible. Then there is asubdirectly irreducible algebra B such that(A, τ) is B-subdiagonal.

• Theorem 0.19 For every subdirectlyirreducible state-morphism algebra (A, τ),there is a subdirectly irreducible algebra B

such that (A, τ) is B-subdiagonal.


•• Theorem 0.20 Let (A, τ) be a subdirectlyirreducible state-morphism algebra such thatA is subdirectly reducible. Then there is asubdirectly irreducible algebra B such that(A, τ) is B-subdiagonal.

• Theorem 0.21 For every subdirectlyirreducible state-morphism algebra (A, τ),there is a subdirectly irreducible algebra B

such that (A, τ) is B-subdiagonal.

• K of algebras of the same type, I(K), H(K),S(K) and P(K) D(K)


•••• Theorem 0.22 (1) For every class K ofalgebras of the same type F,V(D(K)) = V(K)τ .(2) Let K1 and K2 be two classes of same typealgebras. Then V(D(K1)) = V(D(K2)) if andonly if V(K1) = V(K2).


•• Theorem 0.24 (1) For every class K ofalgebras of the same type F,V(D(K)) = V(K)τ .(2) Let K1 and K2 be two classes of same typealgebras. Then V(D(K1)) = V(D(K2)) if andonly if V(K1) = V(K2).

• Theorem 0.25 If a system K of algebras ofthe same type F generates the whole varietyV(F ) of all algebras of type F, then the varietyV(F )τ of all state-morphism algebras (A, τ),where A ∈ V(F ), is generated by the class{D(A) : A ∈ K}.


•• Theorem 0.26 If A is a subdirectly irreduciblealgebra, then any state-morphism algebra(A, τ) is subdirectly irreducible.


•• Theorem 0.28 If A is a subdirectly irreduciblealgebra, then any state-morphism algebra(A, τ) is subdirectly irreducible.

• Theorem 0.29 A variety Vτ satisfy the CEP ifand only if V satisfies the CEP.


Applications

•• The variety of all state-morphismMV-algebras is generated by the diagonalstate-morphism MV-algebra D([0, 1]MV ).


Applications

••• The variety of all state-morphismMV-algebras is generated by the diagonalstate-morphism MV-algebra D([0, 1]MV ).

• The variety of all state-morphism BL-algebrasis generated by the class {D(It) : It ∈ T }.


Applications



• The variety of all state-morphismMTL-algebras is generated by the class{D(It) : It ∈ Tlc}.


Applications



• The variety of all state-morphismMTL-algebras is generated by the class{D(It) : It ∈ Tlc}.

• The variety of all state-morphismnaBL-algebras is generated by the class{D(Inat ) : It ∈ naT }.


••• If a unital ℓ-group (G, u) is double transitive,then D(Γ(G, u)) generates the variety ofstate-morphism pseudo MV-algebras.


References

•••••• A. Di Nola, A. Dvurecenskij, State-morphismMV-algebras, Ann. Pure Appl. Logic 161 (2009),161–173.


References

• A. Di Nola, A. Dvurecenskij, State-morphismMV-algebras, Ann. Pure Appl. Logic 161 (2009),161–173.

• A. Di Nola, A. Dvurecenskij, A. Lettieri, Erratum“State-morphism MV-algebras” [Ann. Pure Appl. Logic161 (2009) 161-173], Ann. Pure Appl. Logic 161(2010), 1605–1607.


References

• A. Di Nola, A. Dvurecenskij, State-morphismMV-algebras, Ann. Pure Appl. Logic 161 (2009),161–173.

• A. Di Nola, A. Dvurecenskij, A. Lettieri, Erratum“State-morphism MV-algebras” [Ann. Pure Appl. Logic161 (2009) 161-173], Ann. Pure Appl. Logic 161(2010), 1605–1607.

• A. Dvurecenskij, Subdirectly irreduciblestate-morphism BL-algebras, Archive Math. Logic50 (2011), 145–160.


• A. Dvurecenskij, T. Kowalski, F. Montagna,State morphism MV-algebras, Inter. J. Approx.Reasoning 52 (2011), 1215–1228.http://arxiv.org/abs/1102.1088



• M. Botur, A. Dvurecenskij, T. Kowalski, Onnormal-valued basic pseudo hoops, Soft ComputingDOI: 10.1007/s00500-011-0763-7




• A. Di Nola, A. Dvurecenskij, A. Lettieri, Onvarieties of MV-algebras with internal states, Inter. J.Approx. Reasoning 51 (2010), 680–694.




• A. Di Nola, A. Dvurecenskij, A. Lettieri, Onvarieties of MV-algebras with internal states, Inter. J.Approx. Reasoning 51 (2010), 680–694.

• L.C. Ciungu, A. Dvurecenskij, M. Hycko, StateBL-algebras, Soft Computing State-morphism MV-algebras and Their Generalizations – p. 34

• M. Botur, A. Dvurecenskij, State-morphismalgebras - general approach,http://arxiv.org/submit/230594


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