A General Approach to State-Morphism MV-Algebras Anatolij DVURE - - PowerPoint PPT Presentation

▶
a general approach to state morphism mv algebras
SMART_READER_LITE
LIVE PREVIEW

A General Approach to State-Morphism MV-Algebras Anatolij DVURE - - PowerPoint PPT Presentation

Dec 16, 2023 203 likes •1.77k views

A General Approach to State-Morphism MV-Algebras Anatolij DVURE CENSKIJ Mathematical Institute, Slovak Academy of Sciences, Stef anikova 49, SK-814 73 Bratislava, Slovakia E-mail: dvurecen@mat.savba.sk The talk given at the Algebraic

slide-1
SLIDE 1

A General Approach to State-Morphism MV-Algebras

Anatolij DVURE ˇ CENSKIJ Mathematical Institute, Slovak Academy of Sciences, ˇ Stef´ anikova 49, SK-814 73 Bratislava, Slovakia E-mail: dvurecen@mat.savba.sk The talk given at the Algebraic Semantics for Uncertainty and Vagueness May 18–21, 2011, Palazzo Genovese, Salerno - Italy supported by Slovak-Italian project SK-IT 0016-08.

A General Approach to State-Morphism MV-Algebras – p. 1

slide-2
SLIDE 2

Quantum Mechanics

  • new physics, beginning 20th century

A General Approach to State-Morphism MV-Algebras – p. 2

slide-3
SLIDE 3

Quantum Mechanics

  • new physics, beginning 20th century
  • Newton mechanics fails in the micro world

A General Approach to State-Morphism MV-Algebras – p. 2

slide-4
SLIDE 4

Quantum Mechanics

  • new physics, beginning 20th century
  • Newton mechanics fails in the micro world
  • Heisenberg Uncertainty Principle

A General Approach to State-Morphism MV-Algebras – p. 2

slide-5
SLIDE 5

Quantum Mechanics

  • new physics, beginning 20th century
  • Newton mechanics fails in the micro world
  • Heisenberg Uncertainty Principle

σs(x)σs(y) ≥ > 0.

A General Approach to State-Morphism MV-Algebras – p. 2

slide-6
SLIDE 6

Quantum Mechanics

  • new physics, beginning 20th century
  • Newton mechanics fails in the micro world
  • Heisenberg Uncertainty Principle

σs(x)σs(y) ≥ > 0. x-momentum, y position of elementary particle, s state -probability measure

A General Approach to State-Morphism MV-Algebras – p. 2

slide-7
SLIDE 7

Quantum Mechanics

  • new physics, beginning 20th century
  • Newton mechanics fails in the micro world
  • Heisenberg Uncertainty Principle

σs(x)σs(y) ≥ > 0. x-momentum, y position of elementary particle, s state -probability measure

  • for classical mechanics

inf

s (σs(x)σs(y)) = 0.

A General Approach to State-Morphism MV-Algebras – p. 2

slide-8
SLIDE 8
  • Hilbert, 1900, 6th Problem:

A General Approach to State-Morphism MV-Algebras – p. 3

slide-9
SLIDE 9
  • Hilbert, 1900, 6th Problem:
  • To find a few physical axioms that, similar to the axioms
  • f geometry, can describe a theory for a class of

physical events that is as large as possible.

A General Approach to State-Morphism MV-Algebras – p. 3

slide-10
SLIDE 10
  • Hilbert, 1900, 6th Problem:
  • To find a few physical axioms that, similar to the axioms
  • f geometry, can describe a theory for a class of

physical events that is as large as possible.

  • Kolmogorov, probability theory, 1933,

A General Approach to State-Morphism MV-Algebras – p. 3

slide-11
SLIDE 11
  • Hilbert, 1900, 6th Problem:
  • To find a few physical axioms that, similar to the axioms
  • f geometry, can describe a theory for a class of

physical events that is as large as possible.

  • Kolmogorov, probability theory, 1933,
  • G. Birkhoff and J. von Neumann, 1936

quantum logic

A General Approach to State-Morphism MV-Algebras – p. 3

slide-12
SLIDE 12
  • Hilbert, 1900, 6th Problem:
  • To find a few physical axioms that, similar to the axioms
  • f geometry, can describe a theory for a class of

physical events that is as large as possible.

  • Kolmogorov, probability theory, 1933,
  • G. Birkhoff and J. von Neumann, 1936

quantum logic

  • C.C. Chang, 1958 MV-algebras

A General Approach to State-Morphism MV-Algebras – p. 3

slide-13
SLIDE 13
  • Hilbert, 1900, 6th Problem:
  • To find a few physical axioms that, similar to the axioms
  • f geometry, can describe a theory for a class of

physical events that is as large as possible.

  • Kolmogorov, probability theory, 1933,
  • G. Birkhoff and J. von Neumann, 1936

quantum logic

  • C.C. Chang, 1958 MV-algebras
  • J. Łukasiewicz, 1922 many-valued logic

A General Approach to State-Morphism MV-Algebras – p. 3

slide-14
SLIDE 14

Other Motivations

  • psychiatry

A General Approach to State-Morphism MV-Algebras – p. 4

slide-15
SLIDE 15

Other Motivations

  • psychiatry
  • compound systems of computers

A General Approach to State-Morphism MV-Algebras – p. 4

slide-16
SLIDE 16

Other Motivations

  • psychiatry
  • compound systems of computers
  • quantum chemistry

A General Approach to State-Morphism MV-Algebras – p. 4

slide-17
SLIDE 17

Other Motivations

  • psychiatry
  • compound systems of computers
  • quantum chemistry
  • quantum computing

A General Approach to State-Morphism MV-Algebras – p. 4

slide-18
SLIDE 18

Other Motivations

  • psychiatry
  • compound systems of computers
  • quantum chemistry
  • quantum computing
  • Bell inequalities

p(a) + p(b) − p(a ∧ b) ≤ 1,

A General Approach to State-Morphism MV-Algebras – p. 4

slide-19
SLIDE 19

Other Motivations

  • psychiatry
  • compound systems of computers
  • quantum chemistry
  • quantum computing
  • Bell inequalities

p(a) + p(b) − p(a ∧ b) ≤ 1,

  • (= p(a ∨ b)) test for a classical system

A General Approach to State-Morphism MV-Algebras – p. 4

slide-20
SLIDE 20

Quantum structures

  • Boolean algebras

A General Approach to State-Morphism MV-Algebras – p. 5

slide-21
SLIDE 21

Quantum structures

  • Boolean algebras
  • Orthomodular lattices

A General Approach to State-Morphism MV-Algebras – p. 5

slide-22
SLIDE 22

Quantum structures

  • Boolean algebras
  • Orthomodular lattices
  • Hilbert space H, L(H) the system of all

closed subspaces of H

A General Approach to State-Morphism MV-Algebras – p. 5

slide-23
SLIDE 23

Quantum structures

  • Boolean algebras
  • Orthomodular lattices
  • Hilbert space H, L(H) the system of all

closed subspaces of H

  • Orthomodular posets

A General Approach to State-Morphism MV-Algebras – p. 5

slide-24
SLIDE 24

Quantum structures

  • Boolean algebras
  • Orthomodular lattices
  • Hilbert space H, L(H) the system of all

closed subspaces of H

  • Orthomodular posets
  • D-posets -Kôpka and Chovanec 1992

A General Approach to State-Morphism MV-Algebras – p. 5

slide-25
SLIDE 25

Quantum structures

  • Boolean algebras
  • Orthomodular lattices
  • Hilbert space H, L(H) the system of all

closed subspaces of H

  • Orthomodular posets
  • D-posets -Kôpka and Chovanec 1992
  • effect algebras

A General Approach to State-Morphism MV-Algebras – p. 5

slide-26
SLIDE 26

Quantum structures

  • Boolean algebras
  • Orthomodular lattices
  • Hilbert space H, L(H) the system of all

closed subspaces of H

  • Orthomodular posets
  • D-posets -Kôpka and Chovanec 1992
  • effect algebras
  • MV-algebras - compatibility

A General Approach to State-Morphism MV-Algebras – p. 5

slide-27
SLIDE 27

States on Algebraic Structures

  • G. Boole: if M-alg. str. C = A + B, and P is a

probability, then P(A + B) = P(A) + P(B);

A General Approach to State-Morphism MV-Algebras – p. 6

slide-28
SLIDE 28

States on Algebraic Structures

  • G. Boole: if M-alg. str. C = A + B, and P is a

probability, then P(A + B) = P(A) + P(B);

  • the operation + is a partial one on M

A General Approach to State-Morphism MV-Algebras – p. 6

slide-29
SLIDE 29

States on Algebraic Structures

  • G. Boole: if M-alg. str. C = A + B, and P is a

probability, then P(A + B) = P(A) + P(B);

  • the operation + is a partial one on M
  • M is a BA, A + B := A ∪ B whenever

A ∩ B = ∅ ⇔ A ≤ B′

A General Approach to State-Morphism MV-Algebras – p. 6

slide-30
SLIDE 30

States on Algebraic Structures

  • G. Boole: if M-alg. str. C = A + B, and P is a

probability, then P(A + B) = P(A) + P(B);

  • the operation + is a partial one on M
  • M is a BA, A + B := A ∪ B whenever

A ∩ B = ∅ ⇔ A ≤ B′

  • A and B mutually excluding - summable -
  • rthogonal

A General Approach to State-Morphism MV-Algebras – p. 6

slide-31
SLIDE 31

States on Algebraic Structures

  • G. Boole: if M-alg. str. C = A + B, and P is a

probability, then P(A + B) = P(A) + P(B);

  • the operation + is a partial one on M
  • M is a BA, A + B := A ∪ B whenever

A ∩ B = ∅ ⇔ A ≤ B′

  • A and B mutually excluding - summable -
  • rthogonal
  • state or FAS on an algebraic structure

(M; +,′ , 0, 1), s : M → [0, 1] (i) s(1) = 1, (ii) s(a + b) = s(a) + s(b) if a + b ∈ M

A General Approach to State-Morphism MV-Algebras – p. 6

slide-32
SLIDE 32

States on L(H)

  • L(H), E(H) = {A ∈ B(H) : O ≤ A ≤ I}

A General Approach to State-Morphism MV-Algebras – p. 7

slide-33
SLIDE 33

States on L(H)

  • L(H), E(H) = {A ∈ B(H) : O ≤ A ≤ I}
  • sφ(M) = (PMφ, φ), M ∈ L(H), φ ∈ H, φ = 1

s(M) =

  • i

λisφi(M) = tr(TPM), M ∈ L(H). Gleason theorem, 1957, 3 ≤ dim H ≤ ℵ0,

A General Approach to State-Morphism MV-Algebras – p. 7

slide-34
SLIDE 34

States on L(H)

  • L(H), E(H) = {A ∈ B(H) : O ≤ A ≤ I}
  • sφ(M) = (PMφ, φ), M ∈ L(H), φ ∈ H, φ = 1

s(M) =

  • i

λisφi(M) = tr(TPM), M ∈ L(H). Gleason theorem, 1957, 3 ≤ dim H ≤ ℵ0,

  • If s is a FAS L(H), Aarnes

s = λs1 + (1 − λ)s2 s1 is a σ-additive, s2 a FAS vanishing on each finite-dimensional subspace of H.

A General Approach to State-Morphism MV-Algebras – p. 7

slide-35
SLIDE 35

Applications of Gleason’s Theorem

  • s(M) = tr(TPM), M ∈ L(H)

A General Approach to State-Morphism MV-Algebras – p. 8

slide-36
SLIDE 36

Applications of Gleason’s Theorem

  • s(M) = tr(TPM), M ∈ L(H)
  • dim H = 2 - Gleason’ Theorem not valid

A General Approach to State-Morphism MV-Algebras – p. 8

slide-37
SLIDE 37

Applications of Gleason’s Theorem

  • s(M) = tr(TPM), M ∈ L(H)
  • dim H = 2 - Gleason’ Theorem not valid
  • Gleason’s Theorem holds for nonseparable iff

dim H is a non-measurable cardinal

A General Approach to State-Morphism MV-Algebras – p. 8

slide-38
SLIDE 38

Applications of Gleason’s Theorem

  • s(M) = tr(TPM), M ∈ L(H)
  • dim H = 2 - Gleason’ Theorem not valid
  • Gleason’s Theorem holds for nonseparable iff

dim H is a non-measurable cardinal

  • Ulam, I- non-measurable cardinal if there

exists no probability measure on 2I vanishing

  • n each i ∈ I.

A General Approach to State-Morphism MV-Algebras – p. 8

slide-39
SLIDE 39

Applications of Gleason’s Theorem

  • s(M) = tr(TPM), M ∈ L(H)
  • dim H = 2 - Gleason’ Theorem not valid
  • Gleason’s Theorem holds for nonseparable iff

dim H is a non-measurable cardinal

  • Ulam, I- non-measurable cardinal if there

exists no probability measure on 2I vanishing

  • n each i ∈ I.
  • von Neumann algebra V - extension from

FAS from L(V ) to V .

A General Approach to State-Morphism MV-Algebras – p. 8

slide-40
SLIDE 40

States on prehilbert Q.L.

  • S-prehilbert - inner product space (·, ·)

A General Approach to State-Morphism MV-Algebras – p. 9

slide-41
SLIDE 41

States on prehilbert Q.L.

  • S-prehilbert - inner product space (·, ·)
  • E(S) = {M ⊆ S : M + M ⊥ = S} OMP

A General Approach to State-Morphism MV-Algebras – p. 9

slide-42
SLIDE 42

States on prehilbert Q.L.

  • S-prehilbert - inner product space (·, ·)
  • E(S) = {M ⊆ S : M + M ⊥ = S} OMP
  • F(S) = {M ⊆ S : M ⊥⊥ = M}

A General Approach to State-Morphism MV-Algebras – p. 9

slide-43
SLIDE 43

States on prehilbert Q.L.

  • S-prehilbert - inner product space (·, ·)
  • E(S) = {M ⊆ S : M + M ⊥ = S} OMP
  • F(S) = {M ⊆ S : M ⊥⊥ = M}
  • E(S) ⊆ F(S)

A General Approach to State-Morphism MV-Algebras – p. 9

slide-44
SLIDE 44

States on prehilbert Q.L.

  • S-prehilbert - inner product space (·, ·)
  • E(S) = {M ⊆ S : M + M ⊥ = S} OMP
  • F(S) = {M ⊆ S : M ⊥⊥ = M}
  • E(S) ⊆ F(S)
  • S complete iff F(S) OML

A General Approach to State-Morphism MV-Algebras – p. 9

slide-45
SLIDE 45

States on prehilbert Q.L.

  • S-prehilbert - inner product space (·, ·)
  • E(S) = {M ⊆ S : M + M ⊥ = S} OMP
  • F(S) = {M ⊆ S : M ⊥⊥ = M}
  • E(S) ⊆ F(S)
  • S complete iff F(S) OML
  • S complete iff F(S) σ-OMP

A General Approach to State-Morphism MV-Algebras – p. 9

slide-46
SLIDE 46

States on prehilbert Q.L.

  • S-prehilbert - inner product space (·, ·)
  • E(S) = {M ⊆ S : M + M ⊥ = S} OMP
  • F(S) = {M ⊆ S : M ⊥⊥ = M}
  • E(S) ⊆ F(S)
  • S complete iff F(S) OML
  • S complete iff F(S) σ-OMP
  • S complete iff E(S) = F(S)

A General Approach to State-Morphism MV-Algebras – p. 9

slide-47
SLIDE 47

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.

A General Approach to State-Morphism MV-Algebras – p. 10

slide-48
SLIDE 48

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.

  • + restriction of the ℓ-group addition

A General Approach to State-Morphism MV-Algebras – p. 10

slide-49
SLIDE 49

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.

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

(ii) s(1) = 1.

A General Approach to State-Morphism MV-Algebras – p. 10

slide-50
SLIDE 50

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.

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

(ii) s(1) = 1.

  • S(M) -set of states. S(M) = ∅.

A General Approach to State-Morphism MV-Algebras – p. 10

slide-51
SLIDE 51

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.

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

(ii) s(1) = 1.

  • S(M) -set of states. S(M) = ∅.
  • extremal state s = λs1 + (1 − λ)s2 for

λ ∈ (0, 1) ⇒ s = s1 = s2.

A General Approach to State-Morphism MV-Algebras – p. 10

slide-52
SLIDE 52
  • {sα} → s iff limα sα(a) → s(a), a ∈ M.

A General Approach to State-Morphism MV-Algebras – p. 11

slide-53
SLIDE 53
  • {sα} → s iff limα sα(a) → s(a), a ∈ M.
  • S(E) - Hausdorff compact topological space,

∂eS(M)

A General Approach to State-Morphism MV-Algebras – p. 11

slide-54
SLIDE 54
  • {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))

A General Approach to State-Morphism MV-Algebras – p. 11

slide-55
SLIDE 55
  • {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 s

is MV-homomorphism iff Ker(s) is a maximal ideal.

A General Approach to State-Morphism MV-Algebras – p. 11

slide-56
SLIDE 56
  • {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 s

is MV-homomorphism iff Ker(s) is a maximal ideal.

  • s ↔ Ker(s), 1-1 correspondence

A General Approach to State-Morphism MV-Algebras – p. 11

slide-57
SLIDE 57
  • every maximal ideal is a kernel of a unique

state

A General Approach to State-Morphism MV-Algebras – p. 12

slide-58
SLIDE 58
  • every maximal ideal is a kernel of a unique

state

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

states

A General Approach to State-Morphism MV-Algebras – p. 12

slide-59
SLIDE 59
  • every maximal ideal is a kernel of a unique

state

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

states

  • Kroupa- Panti a → ˆ

a, ˆ a(s) := s(a), s(a) =

  • ∂eS(M)

ˆ a(t)dµs(t)

A General Approach to State-Morphism MV-Algebras – p. 12

slide-60
SLIDE 60
  • every maximal ideal is a kernel of a unique

state

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

states

  • Kroupa- Panti a → ˆ

a, ˆ a(s) := s(a), s(a) =

  • ∂eS(M)

ˆ a(t)dµs(t)

  • µs - unique Borel σ-additive probability

measure on B(S(M)) such that µs(∂eS(M)) = 1.

A General Approach to State-Morphism MV-Algebras – p. 12

slide-61
SLIDE 61

State MV-algebras

  • MV-algebras with a state are not universal

algebras, and therefore, the do not provide an algebraizable logic for probability reasoning

  • ver many-valued events

A General Approach to State-Morphism MV-Algebras – p. 13

slide-62
SLIDE 62

State MV-algebras

  • MV-algebras with a state are not universal

algebras, and therefore, the do not provide an algebraizable logic for probability reasoning

  • ver many-valued events
  • Flaminio-Montagna - introduce an

algebraizable logic whose equivalent algebraic semantics is the variety of state MV-algebras

A General Approach to State-Morphism MV-Algebras – p. 13

slide-63
SLIDE 63

State MV-algebras

  • MV-algebras with a state are not universal

algebras, and therefore, the do not provide an algebraizable logic for probability reasoning

  • ver many-valued events
  • Flaminio-Montagna - introduce an

algebraizable logic whose equivalent algebraic semantics is the variety of state MV-algebras

  • A state MV-algebra is a pair (M, τ), M -

MV-algebra, τ unary operation on A s.t.

A General Approach to State-Morphism MV-Algebras – p. 13

slide-64
SLIDE 64
  • τ(1) = 1

A General Approach to State-Morphism MV-Algebras – p. 14

slide-65
SLIDE 65
  • τ(1) = 1
  • τ(x ⊕ y) = τ(x) ⊕ τ(t ⊖ (x ⊙ y))

A General Approach to State-Morphism MV-Algebras – p. 14

slide-66
SLIDE 66
  • τ(1) = 1
  • τ(x ⊕ y) = τ(x) ⊕ τ(t ⊖ (x ⊙ y))
  • τ(x∗) = τ(x)∗

A General Approach to State-Morphism MV-Algebras – p. 14

slide-67
SLIDE 67
  • τ(1) = 1
  • τ(x ⊕ y) = τ(x) ⊕ τ(t ⊖ (x ⊙ y))
  • τ(x∗) = τ(x)∗
  • τ(τ(x) ⊕ τ(y)) = τ(x) ⊕ τ(y)

A General Approach to State-Morphism MV-Algebras – p. 14

slide-68
SLIDE 68
  • τ(1) = 1
  • τ(x ⊕ y) = τ(x) ⊕ τ(t ⊖ (x ⊙ y))
  • τ(x∗) = τ(x)∗
  • τ(τ(x) ⊕ τ(y)) = τ(x) ⊕ τ(y)
  • τ -internal operator, state operator

A General Approach to State-Morphism MV-Algebras – p. 14

slide-69
SLIDE 69

Properties

  • τ 2 = τ

A General Approach to State-Morphism MV-Algebras – p. 15

slide-70
SLIDE 70

Properties

  • τ 2 = τ
  • τ(M) is an MV-algebra and τ on τ(M) -

identity

A General Approach to State-Morphism MV-Algebras – p. 15

slide-71
SLIDE 71

Properties

  • τ 2 = τ
  • τ(M) is an MV-algebra and τ on τ(M) -

identity

  • τ(x + y) = τ(x) + τ(y)

A General Approach to State-Morphism MV-Algebras – p. 15

slide-72
SLIDE 72

Properties

  • τ 2 = τ
  • τ(M) is an MV-algebra and τ on τ(M) -

identity

  • τ(x + y) = τ(x) + τ(y)
  • τ(x ⊙ y) = τ(x) ⊙ τ(y) if x ⊙ y = 0.

A General Approach to State-Morphism MV-Algebras – p. 15

slide-73
SLIDE 73

Properties

  • τ 2 = τ
  • τ(M) is an MV-algebra and τ on τ(M) -

identity

  • 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

A General Approach to State-Morphism MV-Algebras – p. 15

slide-74
SLIDE 74

Properties

  • τ 2 = τ
  • τ(M) is an MV-algebra and τ on τ(M) -

identity

  • 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 a

chain

A General Approach to State-Morphism MV-Algebras – p. 15

slide-75
SLIDE 75
  • F -filter, τ-filter if τ(F) ⊆ F.

A General Approach to State-Morphism MV-Algebras – p. 16

slide-76
SLIDE 76
  • F -filter, τ-filter if τ(F) ⊆ F.
  • 1-1 correspondence congruences and

τ-filters

A General Approach to State-Morphism MV-Algebras – p. 16

slide-77
SLIDE 77
  • F -filter, τ-filter if τ(F) ⊆ F.
  • 1-1 correspondence congruences and

τ-filters

  • M = [0, 1] × [0, 1], τ(x, y) = (x, x) s.i. - not

chain

A General Approach to State-Morphism MV-Algebras – p. 16

slide-78
SLIDE 78
  • F -filter, τ-filter if τ(F) ⊆ F.
  • 1-1 correspondence congruences and

τ-filters

  • M = [0, 1] × [0, 1], τ(x, y) = (x, x) s.i. - not

chain

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

endomorphism

A General Approach to State-Morphism MV-Algebras – p. 16

slide-79
SLIDE 79
  • F -filter, τ-filter if τ(F) ⊆ F.
  • 1-1 correspondence congruences and

τ-filters

  • M = [0, 1] × [0, 1], τ(x, y) = (x, x) s.i. - not

chain

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

endomorphism

  • s state on M, [0, 1] ⊗ M,

τs(α ⊗ a) := α · s(a) ⊗ 1

A General Approach to State-Morphism MV-Algebras – p. 16

slide-80
SLIDE 80
  • ([0, 1]⊗, τs) is an SMV-algebra.

A General Approach to State-Morphism MV-Algebras – p. 17

slide-81
SLIDE 81
  • ([0, 1]⊗, τs) is an SMV-algebra.
  • ([0, 1]⊗, τs) is an SMMV-algebra iff s is an

extremal state

A General Approach to State-Morphism MV-Algebras – p. 17

slide-82
SLIDE 82
  • ([0, 1]⊗, τs) is an SMV-algebra.
  • ([0, 1]⊗, τs) is an SMMV-algebra iff s is an

extremal state

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

an SMMV-algebra

A General Approach to State-Morphism MV-Algebras – p. 17

slide-83
SLIDE 83
  • ([0, 1]⊗, τs) is an SMV-algebra.
  • ([0, 1]⊗, τs) is an SMMV-algebra iff s is an

extremal state

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

an SMMV-algebra

  • if τ(M) ∈ V(S1, . . . , Sn) for some n ≥ 1, then

(M, τ) is an SMMV-algebra

A General Approach to State-Morphism MV-Algebras – p. 17

slide-84
SLIDE 84
  • ([0, 1]⊗, τs) is an SMV-algebra.
  • ([0, 1]⊗, τs) is an SMMV-algebra iff s is an

extremal state

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

A General Approach to State-Morphism MV-Algebras – p. 17

slide-85
SLIDE 85

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

A General Approach to State-Morphism MV-Algebras – p. 18

slide-86
SLIDE 86

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 the couple (M, τ) - state-morphism BL-algebra.

A General Approach to State-Morphism MV-Algebras – p. 18

slide-87
SLIDE 87
  • every state operator on a linear BL-algebra is

a state-morphism

A General Approach to State-Morphism MV-Algebras – p. 19

slide-88
SLIDE 88
  • every state operator on a linear BL-algebra is

a state-morphism

  • Example 0.2 Let M be a BL-algebra. On M × M

we 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 on

M × M.

A General Approach to State-Morphism MV-Algebras – p. 19

slide-89
SLIDE 89
  • every state operator on a linear BL-algebra is

a state-morphism

  • Example 0.3 Let M be a BL-algebra. On M × M

we 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 on

M × M.

  • Ker(τ) = {a ∈ M : τ(a) = 1}.

A General Approach to State-Morphism MV-Algebras – p. 19

slide-90
SLIDE 90
  • We say that two subhoops, A and B, of a

BL-algebra M have the disjunction property if for all x ∈ A and y ∈ B, if x ∨ y = 1, then either x = 1 or y = 1.

A General Approach to State-Morphism MV-Algebras – p. 20

slide-91
SLIDE 91
  • We say that two subhoops, A and B, of a

BL-algebra M have the disjunction property if for all x ∈ A and y ∈ B, if x ∨ y = 1, then either x = 1 or y = 1.

  • Lemma 0.5 Suppose that (M, τ) is a state

BL-algebra. Then:

(1) If τ is faithful, then (M, τ) is a subdirectly

irreducible state BL-algebra if and only if τ(M) is a subdirectly irreducible BL-algebra. Now let (M, τ) be subdirectly irreducible. Then:

A General Approach to State-Morphism MV-Algebras – p. 20

slide-92
SLIDE 92
  • (2) Ker(τ) is (either trivial or) a subdirectly

irreducible hoop.

(3) Ker(τ) and τ(M) have the disjunction

property.

A General Approach to State-Morphism MV-Algebras – p. 21

slide-93
SLIDE 93
  • (2) Ker(τ) is (either trivial or) a subdirectly

irreducible hoop.

(3) Ker(τ) and τ(M) have the disjunction

property.

  • Theorem 0.7 Let (M, τ) be a state

BL-algebra satisfying conditions (1), (2) and (3) in the last Lemma. Then (M, τ) is subdirectly irreducible.

A General Approach to State-Morphism MV-Algebras – p. 21

slide-94
SLIDE 94
  • Theorem 0.8 A state-morphism BL-algebra

(M, τ) is subdirectly irreducible irreducible if and only if one of the following three possibilities holds.

A General Approach to State-Morphism MV-Algebras – p. 22

slide-95
SLIDE 95
  • Theorem 0.9 A state-morphism BL-algebra

(M, τ) is subdirectly irreducible irreducible if and only if one of the following three possibilities holds.

  • (i) M is linear, τ = idM, and the BL-reduct M

is a subdirectly irreducible BL-algebra.

A General Approach to State-Morphism MV-Algebras – p. 22

slide-96
SLIDE 96
  • Theorem 0.10 A state-morphism BL-algebra

(M, τ) is subdirectly irreducible irreducible if and only if one of the following three possibilities holds.

  • (i) M is linear, τ = idM, and the BL-reduct M

is a subdirectly irreducible BL-algebra.

  • (ii) The state-morphism operator τ is not

faithful, M has no nontrivial Boolean elements, and the BL-reduct M of (M, τ) is a local BL-algebra, Ker(τ) is a subdirectly irreducible irreducible hoop, and Ker(τ) and τ(M) have the disjunction property.

A General Approach to State-Morphism MV-Algebras – p. 22

slide-97
SLIDE 97
  • Theorem 0.11 A state-morphism BL-algebra

(M, τ) is subdirectly irreducible irreducible if and only if one of the following three possibilities holds.

  • (i) M is linear, τ = idM, and the BL-reduct M

is a subdirectly irreducible BL-algebra.

  • (ii) The state-morphism operator τ is not

faithful, M has no nontrivial Boolean elements, and the BL-reduct M of (M, τ) is a local BL-algebra, Ker(τ) is a subdirectly irreducible irreducible hoop, and Ker(τ) and τ(M) have the disjunction property.

A General Approach to State-Morphism MV-Algebras – p. 22

slide-98
SLIDE 98
  • Theorem 0.12 A state-morphism BL-algebra

(M, τ) is subdirectly irreducible irreducible if and only if one of the following three possibilities holds.

  • (i) M is linear, τ = idM, and the BL-reduct M

is a subdirectly irreducible BL-algebra.

  • (ii) The state-morphism operator τ is not

faithful, M has no nontrivial Boolean elements, and the BL-reduct M of (M, τ) is a local BL-algebra, Ker(τ) is a subdirectly irreducible irreducible hoop, and Ker(τ) and τ(M) have the disjunction property.

A General Approach to State-Morphism MV-Algebras – p. 22

slide-99
SLIDE 99
  • Theorem 0.13 A state-morphism BL-algebra

(M, τ) is subdirectly irreducible irreducible if and only if one of the following three possibilities holds.

  • (i) M is linear, τ = idM, and the BL-reduct M

is a subdirectly irreducible BL-algebra.

  • (ii) The state-morphism operator τ is not

faithful, M has no nontrivial Boolean elements, and the BL-reduct M of (M, τ) is a local BL-algebra, Ker(τ) is a subdirectly irreducible irreducible hoop, and Ker(τ) and τ(M) have the disjunction property.

A General Approach to State-Morphism MV-Algebras – p. 22

slide-100
SLIDE 100
  • Moreover, M is linearly ordered if and only if

Rad1(M) is linearly ordered, and in such a

case, M is a subdirectly irreducible BL-algebra such that if F is the smallest nontrivial state-filter for (M, τ), then F is the smallest nontrivial BL-filter for M.

A General Approach to State-Morphism MV-Algebras – p. 23

slide-101
SLIDE 101
  • Moreover, M is linearly ordered if and only if

Rad1(M) is linearly ordered, and in such a

case, M is a subdirectly irreducible BL-algebra such that if F is the smallest nontrivial state-filter for (M, τ), then F is the smallest nontrivial BL-filter for M.

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

A General Approach to State-Morphism MV-Algebras – p. 23

slide-102
SLIDE 102
  • (iii) The state-morphism operator τ is not

faithful, M has a nontrivial Boolean element. There are a linearly ordered BL-algebra A, a subdirectly irreducible BL-algebra B, and an injective BL-homomorphism h : A → B such that (M, τ) is isomorphic as a state-morphism BL-algebra with the state-morphism BL-algebra (A × B, τh), where τh(x, y) = (x, h(x)) for any (x, y) ∈ A × B.

A General Approach to State-Morphism MV-Algebras – p. 24

slide-103
SLIDE 103

Varieties of SMMV-algebras

  • Komori - countably many subvarieties of

MV-algebras

A General Approach to State-Morphism MV-Algebras – p. 25

slide-104
SLIDE 104

Varieties of SMMV-algebras

  • Komori - countably many subvarieties of

MV-algebras

  • V-variety of MV-algebras, Vτ -system of

SMMV-algebras (M, τ) s.t M ∈ V ∈ V.

A General Approach to State-Morphism MV-Algebras – p. 25

slide-105
SLIDE 105

Varieties of SMMV-algebras

  • Komori - countably many subvarieties of

MV-algebras

  • V-variety of MV-algebras, Vτ -system of

SMMV-algebras (M, τ) s.t M ∈ V ∈ V.

  • D(M) := (M × M, τM)

A General Approach to State-Morphism MV-Algebras – p. 25

slide-106
SLIDE 106

Varieties of SMMV-algebras

  • Komori - countably many subvarieties of

MV-algebras

  • V-variety of MV-algebras, Vτ -system of

SMMV-algebras (M, τ) s.t M ∈ V ∈ V.

  • D(M) := (M × M, τM)
  • V(D) = V(M)τ

A General Approach to State-Morphism MV-Algebras – p. 25

slide-107
SLIDE 107

Varieties of SMMV-algebras

  • Komori - countably many subvarieties of

MV-algebras

  • V-variety of MV-algebras, Vτ -system of

SMMV-algebras (M, τ) s.t M ∈ V ∈ V.

  • D(M) := (M × M, τM)
  • V(D) = V(M)τ
  • SMMV = V(D([0, 1]))

A General Approach to State-Morphism MV-Algebras – p. 25

slide-108
SLIDE 108

Varieties of SMMV-algebras

  • Komori - countably many subvarieties of

MV-algebras

  • V-variety of MV-algebras, Vτ -system of

SMMV-algebras (M, τ) s.t M ∈ V ∈ V.

  • 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

A General Approach to State-Morphism MV-Algebras – p. 25

slide-109
SLIDE 109
  • Theorem: VI ⊆ VR ⊆ VL ⊆ Vτ. and all

inclusions are proper of V is not finitely generated.

A General Approach to State-Morphism MV-Algebras – p. 26

slide-110
SLIDE 110
  • Theorem: VI ⊆ VR ⊆ VL ⊆ Vτ. and all

inclusions are proper of V is not finitely generated.

  • Theorem: Representable SMMV-algebras:

τ(x) ∨ (x → (τ(y) ↔ y)) = 1.

A General Approach to State-Morphism MV-Algebras – p. 26

slide-111
SLIDE 111
  • Theorem: VI ⊆ VR ⊆ VL ⊆ Vτ. and all

inclusions are proper of V is not finitely generated.

  • Theorem: Representable SMMV-algebras:

τ(x) ∨ (x → (τ(y) ↔ y)) = 1.

  • also for BL-algebra

A General Approach to State-Morphism MV-Algebras – p. 26

slide-112
SLIDE 112
  • Theorem: VI ⊆ VR ⊆ VL ⊆ Vτ. and all

inclusions are proper of V is not finitely generated.

  • Theorem: Representable SMMV-algebras:

τ(x) ∨ (x → (τ(y) ↔ y)) = 1.

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

is local

A General Approach to State-Morphism MV-Algebras – p. 26

slide-113
SLIDE 113
  • Theorem: VI ⊆ VR ⊆ VL ⊆ Vτ. and all

inclusions are proper of V is not finitely generated.

  • Theorem: Representable SMMV-algebras:

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).

A General Approach to State-Morphism MV-Algebras – p. 26

slide-114
SLIDE 114

Uncountable many subvarieties

  • [0, 1]∗ ultrapower, fox positive infinitesimal

ǫ ∈ [0, 1]∗

A General Approach to State-Morphism MV-Algebras – p. 27

slide-115
SLIDE 115

Uncountable many subvarieties

  • [0, 1]∗ ultrapower, fox positive infinitesimal

ǫ ∈ [0, 1]∗

  • X subset of prime numbers, A(X)

MV-algebra generated by ǫ and n

m s.t

A General Approach to State-Morphism MV-Algebras – p. 27

slide-116
SLIDE 116

Uncountable many subvarieties

  • [0, 1]∗ ultrapower, fox positive infinitesimal

ǫ ∈ [0, 1]∗

  • X subset of prime numbers, A(X)

MV-algebra generated by ǫ and n

m s.t

  • (1) either n = 0 or g.c.d(n, m) = 1

A General Approach to State-Morphism MV-Algebras – p. 27

slide-117
SLIDE 117

Uncountable many subvarieties

  • [0, 1]∗ ultrapower, fox positive infinitesimal

ǫ ∈ [0, 1]∗

  • X subset of prime numbers, A(X)

MV-algebra generated by ǫ and n

m s.t

  • (1) either n = 0 or g.c.d(n, m) = 1
  • ∀p ∈ X, p does not divide m

A General Approach to State-Morphism MV-Algebras – p. 27

slide-118
SLIDE 118

Uncountable many subvarieties

  • [0, 1]∗ ultrapower, fox positive infinitesimal

ǫ ∈ [0, 1]∗

  • X subset of prime numbers, A(X)

MV-algebra generated by ǫ and n

m s.t

  • (1) either n = 0 or g.c.d(n, m) = 1
  • ∀p ∈ X, p does not divide m
  • τ(x)= standard part of x

A General Approach to State-Morphism MV-Algebras – p. 27

slide-119
SLIDE 119

Uncountable many subvarieties

  • [0, 1]∗ ultrapower, fox positive infinitesimal

ǫ ∈ [0, 1]∗

  • X subset of prime numbers, A(X)

MV-algebra generated by ǫ and n

m s.t

  • (1) either n = 0 or g.c.d(n, m) = 1
  • ∀p ∈ X, p does not divide m
  • τ(x)= standard part of x
  • (A(X), τ) is linearly ordered SMMV-algebra

A General Approach to State-Morphism MV-Algebras – p. 27

slide-120
SLIDE 120
  • if X = Y , then V(A(X)) = V(A(Y ))

A General Approach to State-Morphism MV-Algebras – p. 28

slide-121
SLIDE 121
  • if X = Y , then V(A(X)) = V(A(Y ))
  • Theorem: Between MVI and MVR there is

uncountably many varieties

A General Approach to State-Morphism MV-Algebras – p. 28

slide-122
SLIDE 122

Generators of SMBL-algebras

  • t-norm- function t : [0, 1] × [0, 1] → [0, 1] such

that (i) t is commutative, associative, (ii) t(x, 1) = x, x ∈ [0, 1], and (iii) t is nondecreasing in both components. Moreover, the variety of all BL-algebras is generated by all It with a continuous t-norm t.

A General Approach to State-Morphism MV-Algebras – p. 29

slide-123
SLIDE 123

Generators of SMBL-algebras

  • t-norm- function t : [0, 1] × [0, 1] → [0, 1] such

that (i) t is commutative, associative, (ii) t(x, 1) = x, x ∈ [0, 1], and (iii) t is nondecreasing in both components. Moreover, the variety of all BL-algebras is generated by all It with a continuous t-norm t.

  • If t is continuous, we define x ⊙t y = t(x, y)

and x →t y = sup{z ∈ [0, 1] : t(z, x) ≤ y} for x, y ∈ [0, 1], then It := ([0, 1], min, max, ⊙t, →t, 0, 1) is a BL-algebra.

A General Approach to State-Morphism MV-Algebras – p. 29

slide-124
SLIDE 124

Generators of SMBL-algebras

  • t-norm- function t : [0, 1] × [0, 1] → [0, 1] such

that (i) t is commutative, associative, (ii) t(x, 1) = x, x ∈ [0, 1], and (iii) t is nondecreasing in both components. Moreover, the variety of all BL-algebras is generated by all It with a continuous t-norm t.

  • If t is continuous, we define x ⊙t y = t(x, y)

and x →t y = sup{z ∈ [0, 1] : t(z, x) ≤ y} for x, y ∈ [0, 1], then It := ([0, 1], min, max, ⊙t, →t, 0, 1) is a BL-algebra.

  • Moreover, the variety of all BL-algebras is

A General Approach to State-Morphism MV-Algebras – p. 29

slide-125
SLIDE 125
  • T denotes the system of all BL-algebras It,

where t is a continuous t-norm on the interval [0, 1],

A General Approach to State-Morphism MV-Algebras – p. 30

slide-126
SLIDE 126
  • T denotes the system of all BL-algebras It,

where t is a continuous t-norm on the interval [0, 1],

  • Theorem 0.15 The variety of all

state-morphism BL-algebras is generated by the class T .

A General Approach to State-Morphism MV-Algebras – p. 30

slide-127
SLIDE 127

General Approach - State-Morphism Algeb

  • A an algebra of type F, τ an idempotent

endomorphism of A, (A, τ) state-morphism algebra

A General Approach to State-Morphism MV-Algebras – p. 31

slide-128
SLIDE 128

General Approach - State-Morphism Algeb

  • A an algebra of type F, τ an idempotent

endomorphism of A, (A, τ) state-morphism algebra

  • θτ = {(x, y) ∈ A × A : τ(x) = τ(y)},

A General Approach to State-Morphism MV-Algebras – p. 31

slide-129
SLIDE 129

General Approach - State-Morphism Algeb

  • A an algebra of type F, τ an idempotent

endomorphism of A, (A, τ) state-morphism algebra

  • θτ = {(x, y) ∈ A × A : τ(x) = τ(y)},
  • φ ⊆ A2, Φ(φ), Φτ(φ) congruence generated by

φ on A and (A, τ)

A General Approach to State-Morphism MV-Algebras – p. 31

slide-130
SLIDE 130

General Approach - State-Morphism Algeb

  • A an algebra of type F, τ an idempotent

endomorphism of A, (A, τ) state-morphism algebra

  • θτ = {(x, y) ∈ A × A : τ(x) = τ(y)},
  • φ ⊆ A2, Φ(φ), Φτ(φ) congruence generated by

φ on A and (A, τ)

  • Lemma: For any φ ∈ Con τ(A), we have

θφ ∈ Con (A, τ), and θφ ∩ τ(A)2 = φ. In addition, θτ ∈ Con (A, τ), φ ⊆ θφ, and Θτ(φ) ⊆ θφ.

A General Approach to State-Morphism MV-Algebras – p. 31

slide-131
SLIDE 131
  • Lemma: Let θ ∈ Con A be such that θ ⊆ θτ.

Then θ ∈ Con (A, τ) holds.

A General Approach to State-Morphism MV-Algebras – p. 32

slide-132
SLIDE 132
  • Lemma: Let θ ∈ Con A be such that θ ⊆ θτ.

Then θ ∈ Con (A, τ) holds.

  • Lemma: If x, y ∈ τ(A), then

Θ(x, y) = Θτ(x, y). Consequently, Θ(φ) = Θτ(φ) whenever φ ⊆ τ(A)2.

A General Approach to State-Morphism MV-Algebras – p. 32

slide-133
SLIDE 133
  • Lemma: Let θ ∈ Con A be such that θ ⊆ θτ.

Then θ ∈ Con (A, τ) holds.

  • Lemma: If x, y ∈ τ(A), then

Θ(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 a

subdiagonal state-morphism algebra

A General Approach to State-Morphism MV-Algebras – p. 32

slide-134
SLIDE 134
  • Theorem 0.16 Let (A, τ) be a subdirectly

irreducible state-morphism algebra such that A is subdirectly reducible. Then there is a subdirectly irreducible algebra B such that (A, τ) is B-subdiagonal.

A General Approach to State-Morphism MV-Algebras – p. 33

slide-135
SLIDE 135
  • Theorem 0.18 Let (A, τ) be a subdirectly

irreducible state-morphism algebra such that A is subdirectly reducible. Then there is a subdirectly irreducible algebra B such that (A, τ) is B-subdiagonal.

  • Theorem 0.19 For every subdirectly

irreducible state-morphism algebra (A, τ), there is a subdirectly irreducible algebra B such that (A, τ) is B-subdiagonal.

A General Approach to State-Morphism MV-Algebras – p. 33

slide-136
SLIDE 136
  • Theorem 0.20 Let (A, τ) be a subdirectly

irreducible state-morphism algebra such that A is subdirectly reducible. Then there is a subdirectly irreducible algebra B such that (A, τ) is B-subdiagonal.

  • Theorem 0.21 For every subdirectly

irreducible 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)

A General Approach to State-Morphism MV-Algebras – p. 33

slide-137
SLIDE 137
  • Theorem 0.22 (1) For every class K of

algebras of the same type F, V(D(K)) = V(K)τ.

(2) Let K1 and K2 be two classes of same type

  • algebras. Then V(D(K1)) = V(D(K2)) if and
  • nly if V(K1) = V(K2).

A General Approach to State-Morphism MV-Algebras – p. 34

slide-138
SLIDE 138
  • Theorem 0.24 (1) For every class K of

algebras of the same type F, V(D(K)) = V(K)τ.

(2) Let K1 and K2 be two classes of same type

  • algebras. Then V(D(K1)) = V(D(K2)) if and
  • nly if V(K1) = V(K2).
  • Theorem 0.25 If a system K of algebras of

the same type F generates the whole variety V(F) of all algebras of type F, then the variety V(F)τ of all state-morphism algebras (A, τ), where A ∈ V(F), is generated by the class {D(A) : A ∈ K}.

A General Approach to State-Morphism MV-Algebras – p. 34

slide-139
SLIDE 139
  • Theorem 0.26 If A is a subdirectly irreducible

algebra, then any state-morphism algebra (A, τ) is subdirectly irreducible.

A General Approach to State-Morphism MV-Algebras – p. 35

slide-140
SLIDE 140
  • Theorem 0.28 If A is a subdirectly irreducible

algebra, then any state-morphism algebra (A, τ) is subdirectly irreducible.

  • Theorem 0.29 A variety Vτ satisfy the CEP if

and only if V satisfies the CEP .

A General Approach to State-Morphism MV-Algebras – p. 35

slide-141
SLIDE 141

Applications

  • The variety of all state-morphism

MV-algebras is generated by the diagonal state-morphism MV-algebra D([0, 1]MV ).

A General Approach to State-Morphism MV-Algebras – p. 36

slide-142
SLIDE 142

Applications

  • The variety of all state-morphism

MV-algebras is generated by the diagonal state-morphism MV-algebra D([0, 1]MV ).

  • The variety of all state-morphism BL-algebras

is generated by the class {D(It) : It ∈ T }.

A General Approach to State-Morphism MV-Algebras – p. 36

slide-143
SLIDE 143

Applications

  • The variety of all state-morphism

MV-algebras is generated by the diagonal state-morphism MV-algebra D([0, 1]MV ).

  • The variety of all state-morphism BL-algebras

is generated by the class {D(It) : It ∈ T }.

  • The variety of all state-morphism

MTL-algebras is generated by the class {D(It) : It ∈ Tlc}.

A General Approach to State-Morphism MV-Algebras – p. 36

slide-144
SLIDE 144

Applications

  • The variety of all state-morphism

MV-algebras is generated by the diagonal state-morphism MV-algebra D([0, 1]MV ).

  • The variety of all state-morphism BL-algebras

is generated by the class {D(It) : It ∈ T }.

  • The variety of all state-morphism

MTL-algebras is generated by the class {D(It) : It ∈ Tlc}.

  • The variety of all state-morphism

naBL-algebras is generated by the class {D(Ina

t ) : It ∈ naT }.

A General Approach to State-Morphism MV-Algebras – p. 36

slide-145
SLIDE 145
  • If a unital ℓ-group (G, u) is double transitive,

then D(Γ(G, u)) generates the variety of state-morphism pseudo MV-algebras.

A General Approach to State-Morphism MV-Algebras – p. 37

slide-146
SLIDE 146

References

  • A. Di Nola, A. Dvureˇ

censkij, State-morphism

MV-algebras, Ann. Pure Appl. Logic 161 (2009),

161–173.

A General Approach to State-Morphism MV-Algebras – p. 38

slide-147
SLIDE 147

References

  • A. Di Nola, A. Dvureˇ

censkij, State-morphism

MV-algebras, Ann. Pure Appl. Logic 161 (2009),

161–173.

  • A. Di Nola, A. Dvureˇ

censkij, A. Lettieri, Erratum

“State-morphism MV-algebras” [Ann. Pure Appl. Logic 161 (2009) 161-173], Ann. Pure Appl. Logic 161

(2010), 1605–1607.

A General Approach to State-Morphism MV-Algebras – p. 38

slide-148
SLIDE 148

References

  • A. Di Nola, A. Dvureˇ

censkij, State-morphism

MV-algebras, Ann. Pure Appl. Logic 161 (2009),

161–173.

  • A. Di Nola, A. Dvureˇ

censkij, A. Lettieri, Erratum

“State-morphism MV-algebras” [Ann. Pure Appl. Logic 161 (2009) 161-173], Ann. Pure Appl. Logic 161

(2010), 1605–1607.

  • A. Dvureˇ

censkij, Subdirectly irreducible

state-morphism BL-algebras, Archive Math. Logic 50 (2011), 145–160.

A General Approach to State-Morphism MV-Algebras – p. 38

slide-149
SLIDE 149
  • A. Dvureˇ

censkij, T. Kowalski, F . Montagna,

State morphism MV-algebras, Inter. J. Approx.

Reasoning http://arxiv.org/abs/1102.1088

A General Approach to State-Morphism MV-Algebras – p. 39

slide-150
SLIDE 150
  • A. Dvureˇ

censkij, T. Kowalski, F . Montagna,

State morphism MV-algebras, Inter. J. Approx.

Reasoning http://arxiv.org/abs/1102.1088

  • M. Botur, A. Dvureˇ

censkij, T. Kowalski, On

normal-valued basic pseudo hoops,

A General Approach to State-Morphism MV-Algebras – p. 39

slide-151
SLIDE 151
  • A. Dvureˇ

censkij, T. Kowalski, F . Montagna,

State morphism MV-algebras, Inter. J. Approx.

Reasoning http://arxiv.org/abs/1102.1088

  • M. Botur, A. Dvureˇ

censkij, T. Kowalski, On

normal-valued basic pseudo hoops,

  • A. Di Nola, A. Dvureˇ

censkij, A. Lettieri, On

varieties of MV-algebras with internal states, Inter. J.

  • Approx. Reasoning 51 (2010), 680–694.

A General Approach to State-Morphism MV-Algebras – p. 39

slide-152
SLIDE 152
  • A. Dvureˇ

censkij, T. Kowalski, F . Montagna,

State morphism MV-algebras, Inter. J. Approx.

Reasoning http://arxiv.org/abs/1102.1088

  • M. Botur, A. Dvureˇ

censkij, T. Kowalski, On

normal-valued basic pseudo hoops,

  • A. Di Nola, A. Dvureˇ

censkij, A. Lettieri, On

varieties of MV-algebras with internal states, Inter. J.

  • Approx. Reasoning 51 (2010), 680–694.
  • L.C. Ciungu, A. Dvureˇ

censkij, M. Hyˇ cko, State

BL-algebras, Soft Computing

A General Approach to State-Morphism MV-Algebras – p. 39

slide-153
SLIDE 153
  • M. Botur, A. Dvureˇ

censkij, State-morphism

algebras - general approach,

http://arxiv.org/submit/230594

A General Approach to State-Morphism MV-Algebras – p. 40

slide-154
SLIDE 154

Thank you for your attention

A General Approach to State-Morphism MV-Algebras – p. 41