Algebra and probability in Lukasiewicz...

Algebra and probability in Lukasiewicz logic

Ioana Leustean

Faculty of Mathematics and Computer ScienceUniversity of Bucharest

Probability, Uncertainty and RationalityCertosa di Pontignano (Siena), 1-3 November, 2009

Ioana Leustean (UB) Algebra and Probability in Lukasiewicz logic PURa 09 1 / 32

LAP

Logic

Algebra

Probability


LAP

Logic

Algebra

Probability

lap = movement once arround a course


LAP

Logic

Algebra

Probability


Classical logic

Boolean algebras

Classical probability theory: the set of events is a Boolean algebra


LAP

Logic

Algebra

Probability


Classical logic

Boolean algebras

Classical probability theory: the set of events is a Boolean algebra

LAP interaction in Lukasiewicz logic


Lukasiewicz logic

3-valued (2-valued) Lukasiewicz logic classical logic

l l0, 1

2 , 1 0, 1

‖ ‖

L3 L2

J. Lukasiewicz, 1920.


Lukasiewicz logic

n-valued (2-valued) Lukasiewicz logic classical logic

l l0, 1

n−1 ,2

n−1 , . . . ,n−2n−1 , 1 0, 1

‖ ‖

Ln L2




Lukasiewicz logic

∞-valued n-valued (2-valued) Lukasiewicz logic Lukasiewicz logic classical logic

l l l[0, 1] 0, 1

n−1 ,2

n−1 , . . . ,n−2n−1 , 1 0, 1

‖ ‖

Ln L2



J. Lukasiewicz, A. Tarski, 1930.


(∞-valued) Lukasiewicz logic:

Connectives

¬ and −→

• Lukasiewicz considered also Mp for ”p is possible”• Tarski defined Mp = ¬p −→ p



Connectives

¬ and −→


”truth-tables”

¬p := 1− p, p −→ q := min(1− p + q, 1) (p, q ∈ [0, 1] )



Connectives

¬ and −→


”truth-tables”

¬p := 1− p, p −→ q := min(1− p + q, 1) (p, q ∈ [0, 1] )

Lukasiewicz logic is truth-functional


(∞-valued) Lukasiewicz logic L:

Axioms

(L1) ϕ −→ (ψ −→ ϕ);(L2) (ϕ −→ ψ) −→ ((ψ −→ χ) −→ (ϕ −→ χ));(L3) ((ϕ −→ ψ) −→ ψ) −→ ((ψ −→ ϕ) −→ ϕ);(L4) (¬ψ −→ ¬ϕ) −→ (ϕ −→ ψ).

the deduction rule is modus ponens:ϕ,ϕ −→ ψ ⊢ ψ



Axioms

(L1) ϕ −→ (ψ −→ ϕ);(L2) (ϕ −→ ψ) −→ ((ψ −→ χ) −→ (ϕ −→ χ));(L3) ((ϕ −→ ψ) −→ ψ) −→ ((ψ −→ ϕ) −→ ϕ);(L4) (¬ψ −→ ¬ϕ) −→ (ϕ −→ ψ).

the deduction rule is modus ponens:ϕ,ϕ −→ ψ ⊢ ψ

L + ((ϕ −→ ¬ϕ) −→ ¬ϕ) ⇒ classical logic

L + An + Ak | k ∈ 2, . . . , (n − 2), k 6| (n − 1) ⇒ n-valued logic



McNaughton Theorem, 1951 (formulas as functions)

If f : [0, 1]n → [0, 1], TFAE:(a) f = ϕ[0,1] for some formula ϕ of Lukasiewicz logic,(b) f is continuous such that

∃q1, . . ., qk : Rn → R ∀(a1, . . . , an) ∈ [0, 1]n ∃i ∈ 1, kf (a1, . . . , an) = qi (a1, . . . , an),

where qi (a1, . . . , an) = m0i + m1ia1 + · · ·+ mnian, mji ∈ Z ∀i , j .







Normal form representation theorem

A. Di Nola, A.Lettieri, 2004







Normal form representation theorem

A. Di Nola, A.Lettieri, 2004

Rose and Rosser (1958)

A formula ϕ is a [0, 1]-tautology of L iff it can be derived from the axiomsusing modus ponens and substitution.


Algebra: historical overview

Gr.C. Moisil: 3-valued and 4-valued Lukasiewicz algebras (1940)n-valued Lukasiewicz algebras (1941)




A. Rose (1965): for n ≥ 5 the n-valued Lukasiewicz algebraKn = 0, 1

n−1 ,n−2n−1 , 1 is not closed to Lukasiewicz implication






Gr.C. Moisil: ϑ-valued Lukasiewicz algebras (1968)







Argentininan school: A. Monteiro, R.Cignoli, L. Monteiro, M.Abad, etc.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

C.C. Chang: MV-algebras (1958)








. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


R. Grigolia: MVn-algebras (1977)








. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



R. Cignoli: proper n-valued Lukasiewicz algebras (1982)








. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .




J.M. Font, A.J. Rodriguez, A. Torrens: Wajsberg algebras (1984)








. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .




J.M. Font, A.J. Rodriguez, A. Torrens: Wajsberg algebras (1984)

MV-algebras and Wajsberg algebrasMVn-algebras and proper n-valued Lukasiewicz algebras


the n-valued case

Boolean logicւ ց

Lukasiewiczn-valued logic

Postn-valued logic

ւ ց ↓ L-proper LMn-algebras ≃ MVn-algebras ⊃ Postn-algebras(Cignoli,1982) (Grigolia,1977) (Rosenbloom,1942)

∩LMn-algebras

(Moisil,1941)


n-valued Lukasiewicz-Moisil algebras

LMn-algebra

(L,∨,∧,∗ , ϕ1, . . . , ϕn−1, 1)(LM0)(L,∨,∧,∗ ) De Morgan algebra (LM1) ϕi (ϕj(x)) = ϕj(x)(LM2) ϕi (x ∨ y) = ϕi (x) ∨ ϕi (y) (LM3) ϕi (x) ∨ (ϕi (x))∗ = 1(LM4) ϕi (x∗) = (ϕn−i (x))∗ (LM5) ϕ1(x) ≤ · · · ≤ ϕn−1(x)

(LM6) ϕi (x) = ϕi (y) for any i ∈ 1, n − 1⇒ x = y


n-valued Lukasiewicz-Moisil algebras

LMn-algebra

(L,∨,∧,∗ , ϕ1, . . . , ϕn−1, 1)(LM0)(L,∨,∧,∗ ) De Morgan algebra (LM1) ϕi (ϕj(x)) = ϕj(x)(LM2) ϕi (x ∨ y) = ϕi (x) ∨ ϕi (y) (LM3) ϕi (x) ∨ (ϕi (x))∗ = 1(LM4) ϕi (x∗) = (ϕn−i (x))∗ (LM5) ϕ1(x) ≤ · · · ≤ ϕn−1(x)

(LM6) ϕi (x) = ϕi (y) for any i ∈ 1, n − 1⇒ x = y

L-proper LMn-algebra = LMn-algebra + Fiki ,k + axioms

Postn-algebra = LMn-algebra +cii=1,n−2+ axioms

LMn-algebras, MVn-algebras, Postn-algebras are equational classes.


Examples:

Example 1:

0 1n−1

2n−1 . . . n−2

n−1 1

ϕ1 0 0 0 . . . 0 1ϕ2 0 0 0 . . . 1 1...

......

......

......

ϕn−2 0 0 1 . . . 1 1ϕn−1 0 1 1 . . . 1 1

Example 2:

L = LXn = f | f : X −→ Ln

ϕi (f )(x) =

1, f (x) ≥ n−in−1

0, otherwise


Moisil’s determination principle

L is LMn-algebra, MVn-algebra or Pn-algebraL ⊃ B(L) = x | x ∨ x∗ = 1 (the Boolean reduct)

L ∋ x 7→ ϕ1(x),. . .,ϕn−1(x) ∈ B(L)x = y iff ϕi (x) = ϕi (y) for any i ∈ 1, n − 1

Any element is characterized by (n − 1) Boolean nuances.

The functor B : LukMoisilm → Bool has a right adjoint.


Moisil’s determination principle

L is LMn-algebra, MVn-algebra or Pn-algebraL ⊃ B(L) = x | x ∨ x∗ = 1 (the Boolean reduct)

L ∋ x 7→ ϕ1(x),. . .,ϕn−1(x) ∈ B(L)x = y iff ϕi (x) = ϕi (y) for any i ∈ 1, n − 1

Any element is characterized by (n − 1) Boolean nuances.

The functor B : LukMoisilm → Bool has a right adjoint.

Any element of L can be ”recovered” from its Boolean nuancesiff L is a Postn-algebra


Determination principle for subalgebras

Is it true that for S , T ⊆ A

ϕi (S) = ϕi (T ) for any i ∈ 1, n − 1⇒ S = T ?

Yes, if S and T are proper ideals.Not in general, if S and T are subalgebras.


Determination principle for subalgebras

I.L.,2008.

L - LMn-algebra

S ≤ LI→ J1(L), . . ., Jn−2(L), Jn−1(L) = B(S)

x ∈ L : Ji (x) ∈ Ii∀iS← I1, . . ., In−2, In−1 ≤ B(L)

I injective, S surjective

I(S) = I(T )⇒ S = T

S is L-proper (MVn-algebra) ⇔ I(S) satisfies (MV) (n ≥ 5)(MV): Ii ∩ Ik ⊆ In−i+k−1, 3 ≤ i ≤ n − 2, 1 ≤ k ≤ n − 4, k < i

S is Pn-algebra ⇔ I1 = · · · = In−1 = B(S)

The Boolean nuances of a subalgebra are (n − 1) Boolean ideals


Nuances of truth vs truth degrees

G. Georgescu, A. Popescu, 2006

Moisil logic is derived from the classical logic by the idea of nuancing,mathematically expressed by a categorical adjunction

Starting from a logical system and using the idea of nuance, it ispossible to construct an n-nuanced logical system on the top of thegiven one.

Nuancing the Lukasiewicz logic, they defined the n-nuanced

MV-algebras and they proved that there is a pair of adjoint functorsbetween the two categories.


MV-algebras ↔ Lukasiewicz ∞-valued logic

C.C.Chang, 1958

An MV-algebra is a structure (A,⊕,∗ , 0) such that:

1 (A,⊕, 0) abelian monoid,

2 (x∗)∗ = x ,

3 (x∗ ⊕ y)∗ ⊕ y = (y∗ ⊕ x)∗ ⊕ x ,

4 0∗ ⊕ x = 0∗.

([0, 1],⊕,∗ , 0) MV-algebra, x ⊕ y = min(x + y , 0), x∗ = 1− x

MV-algebras are bounded distributive lattices with1 = 0∗, x ∨ y = (y∗ ⊕ x)∗ ⊕ x , x ∧ y = (x∗ ∨ y)∗

MV-algebras are reziduated lattices withx −→ y = (x∗ ⊕ y), x ⊙ y = (x∗ ⊕ y∗)∗.



C.C.Chang, 1958



2 (x∗)∗ = x ,

3 (x∗ ⊕ y)∗ ⊕ y = (y∗ ⊕ x)∗ ⊕ x ,

4 0∗ ⊕ x = 0∗.

R. Cignoli, I.M.L. D’Ottaviano, D. Mundici,Algebraic foundations of many-valued reasoning, 2000.



C.C.Chang, 1958



2 (x∗)∗ = x ,

3 (x∗ ⊕ y)∗ ⊕ y = (y∗ ⊕ x)∗ ⊕ x ,

4 0∗ ⊕ x = 0∗.

Chang’s completeness theorem

For a formula ϕ TFAE:

(a) ϕ provable, (d) ϕ holds in Ln for any n ≥ 2,(b) ϕ holds in any MV-algebra, (e) ϕ holds in [0, 1] ∩Q,(c) ϕ holds in any MV-chain, (f) ϕ holds in [0, 1].


Mundici’s categorical equivalence

Mundici’s categorical equivalence (1986)

The category of MV-algebras is equivalent with the category of abelianlattice-ordered groups with strong unit. For any MV-algebra A there existsan abelian lattice-ordered group with strong unit (G , u) such that

A ≃ [0, u]G .

u strong unit: u ≥ 0, for any x ∈ G there is n ≥ 1 s.t. x ≤ nu

Γ(G , u) = ([0, u]G ,⊕,∗ , 0): x ⊕ y = (x + y) ∧ 1, x∗ = 1− x

- G0 u


Mundici’s categorical equivalence

Mundici’s categorical equivalence (1986)

The category of MV-algebras is equivalent with the category of abelianlattice-ordered groups with strong unit. For any MV-algebra A there existsan abelian lattice-ordered group with strong unit (G , u) such that

A ≃ [0, u]G .

u strong unit: u ≥ 0, for any x ∈ G there is n ≥ 1 s.t. x ≤ nu

Γ(G , u) = ([0, u]G ,⊕,∗ , 0): x ⊕ y = (x + y) ∧ 1, x∗ = 1− x

strong unit ⇒ logical interpretation⇒ existence maximal ideals


Functional representation

A MV-algebra, ∅ 6= I ⊆ A is an ideal if:(x ∈ I , y ≤ x ⇒ y ∈ I ) and (x , y ∈ I ⇒ x ⊕ y ∈ I )

for any MV-algebra A, the maximal ideal space MaxA with thespectral topology is a compact Hausdorff space(open sets: r(I ) = M ∈ MaxA | I 6⊆ M for some ideal I ).

A is semisimple if⋂

M | M ∈ Max(A) = ∅

C (MaxA) = f : MaxA −→ [0, 1] | f continuous



A MV-algebra, ∅ 6= I ⊆ A is an ideal if:(x ∈ I , y ≤ x ⇒ y ∈ I ) and (x , y ∈ I ⇒ x ⊕ y ∈ I )

for any MV-algebra A, the maximal ideal space MaxA with thespectral topology is a compact Hausdorff space(open sets: r(I ) = M ∈ MaxA | I 6⊆ M for some ideal I ).

A is semisimple if⋂

M | M ∈ Max(A) = ∅

C (MaxA) = f : MaxA −→ [0, 1] | f continuous

L.P.Belluce, 1986

Any semisimple MV-algebra A is isomorphic with a separatingMV-subalgebra of C (MaxA) (with pointwise operations).

ι : A→ C (MaxA) embedding∀ M1 6= M2 ∃ f ∈ ι(A) (f (M1) = 0 and f (M2) 6= 0)



ι : A→ C (MaxA) embedding

A. Di Nola, S.Sessa, 1995

A σ-complete ⇒ MaxA basically disconnected

A complete ⇒ MaxA extremally disconnected



ι : A→ C (MaxA) embedding

A. Di Nola, S.Sessa, 1995

A σ-complete ⇒ MaxA basically disconnected

A complete ⇒ MaxA extremally disconnected

(V. Marra, I.L.)

We characterized those MV-algebras A with the property that A ≃ C (X )for some compact Hausdorff space X . We proved that the category ofcompact Hausdorff spaces and continuous maps is equivalent with a fullsubcategory of MV-algebras.


Semantical and sintactical consequences in L

For a set Θ of formulas, defineΘ⊢ = sintactic consequences of ΘΘ|= = semantic consequences of Θ

Theorem

TFAE:

Θ⊢ = Θ|=

L(Θ) (the Lindenbaum-Tarski algebra of Θ) is semisimple.


Semantical and sintactical consequences in L

For a set Θ of formulas, defineΘ⊢ = sintactic consequences of ΘΘ|= = semantic consequences of Θ

Theorem

TFAE:

Θ⊢ = Θ|=

L(Θ) (the Lindenbaum-Tarski algebra of Θ) is semisimple.

R. Cignoli, I.M.L. D’Ottaviano, D. Mundici, Algebraic foundations ofmany-valued reasoning, 2000.

P. Hajek, Metamathematics of fuzzy logic, 1998.


Di Nola’s representation theorem, 1991

Theorem

Up to isomorphism, every MV-algebra A is an algebra of [0, 1]∗-valuedfunctions over some set, where [0, 1]⋆ is an ultrapower of [0, 1], onlydepending on th cardinatlity of A.

[0, 1]⋆ is the unit interval of R⋆ (non-standard reals)


MV-algebras are twofold structures

generalization of Boolean algebras

intervals in abelian lattice ordered groups with strong unit


MV-algebras are twofold structures

generalization of Boolean algebras

intervals in abelian lattice ordered groups with strong unit

The theory of MV-algebras is a possible answer to Birkhoff’s problem:develop a common abstraction which includes Boolean algebras andlattice-ordered groups as special cases.

G. Birkhoff, Lattice Theory, 1973.


Probability

MV-algebrasւ ց

Boolean algebras Lattice-ordered groupswith strong unit


Probability

MV-algebrasւ ց

Boolean algebras Lattice-ordered groupswith strong unit

↓ ↓measures states↓ ↓

σ-continuity finite additivitym : B → [0,∞] s : G → R

m(0) = 0, s(1) = 1,m(

∨

n an) = Σ∞

1 m(an) s(x + y) = s(x) + s(y),

ak ∧ an = 0, k 6= n s(G+) = R+


Probability vs truth degree


Q1: ”The patient is young”Q2: ”The patient will survive next week”

The sentence Q1 is true to some degree - the lower the age of thepatient, the more the sentence is true.

The sentence Q2 is a crisp sentence (T | F ), but we do not knowwhich is the case; we may have some probability (degree of belief)that the sentence is true.

Most many-valued logics are truth-functional, while the probabilitycalculus is not, since P(a ∧ b) cannot be determined using P(a) andP(b).


Probability vs truth degree


Q1: ”The patient is young”Q2: ”The patient will survive next week”

The sentence Q1 is true to some degree - the lower the age of thepatient, the more the sentence is true.

The sentence Q2 is a crisp sentence (T | F ), but we do not knowwhich is the case; we may have some probability (degree of belief)that the sentence is true.

Most many-valued logics are truth-functional, while the probabilitycalculus is not, since P(a ∧ b) cannot be determined using P(a) andP(b).

Truth of a many-valued sentence is a matter of degree.Probability is not a degree of truth.

Probability is a degree of belief.


States (finitely additive case)

D. Mundici, 1995

Definition

A state on (A,⊕,∗ , 0) is a map s : A→ [0, 1] s.t s(1) = 1 ands(x ⊕ y) = s(x) + s(y) whenever x ≤ y∗ (x ⊙ y = 0).The state s is faithfull if s(x) = 0 implies x = 0.

s(p) is the ”average degree of truth” of p


States (finitely additive case)

D. Mundici, 1995

Definition

A state on (A,⊕,∗ , 0) is a map s : A→ [0, 1] s.t s(1) = 1 ands(x ⊕ y) = s(x) + s(y) whenever x ≤ y∗ (x ⊙ y = 0).The state s is faithfull if s(x) = 0 implies x = 0.

s(p) is the ”average degree of truth” of p

extension results, characterization of the state space, ...


de Finetti’s coherence criterion for MV-algebras

D. Mundici, 2006,

ϕ1, . . ., ϕn are formulas of Lukasiewicz, α1, . . ., αn ∈ [0, 1]. TFAE:

for all λ1, . . ., λn ∈ R there exists a valuation V s.t.∑n

i=1 λi (αi − V (ϕi )) ≥ 0

(the probabilistic assessment (P(ϕi ) = αi )i is coherent),

there exists a state s defined on the Lindenbaum-Tarski algebra of Ls.t. s([ϕi ]) = αi for all i ∈ 1, . . . , n.



D. Mundici, 2006,

ϕ1, . . ., ϕn are formulas of Lukasiewicz, α1, . . ., αn ∈ [0, 1]. TFAE:


i=1 λi (αi − V (ϕi )) ≥ 0


MV-algebra with internal state(T. Flaminio, F. Montagna, ?)

An SMV-algebra is a structure (A, σ) such that A is an MV-algebra andσ : A −→ A s.t.:

σ(0) = 0, σ(x ⊕ y) = σ(x)⊕ σ(y ⊙ (x ⊙ y)∗),σ(x∗) = σ(x)∗, σ(σ(x)⊕ σ(y)) = σ(x)⊕ σ(y).



D. Mundici, 2006,T. Flaminio, F.Montagna, 2009

ϕ1, . . ., ϕn are formulas of Lukasiewicz, α1, . . ., αn ∈ [0, 1]∩Q. TFAE:


i=1 λi (αi − V (ϕi )) ≥ 0


for any 1 ≤ i ≤ n, any non-trivial SMV-algebra satisfies the equations

(mi − 1)xi = x∗i and σ(ϕi ) = nixi ,

where V (ϕi ) = ni

mifor any i .


σ-states

Definition

Let A be a σ-complete MV-algebra (i.e. closed to countable suprema).The state s ∈ St(A) is a σ-state if xn ր x implies lim s(xn) = s(x) for allx , x1, x2, . . . ∈ A.


σ-states

Definition


⇓study of measures on abstract algebra in Lukasiewicz logic


σ-states

Definition


⇓study of measures on abstract algebra in Lukasiewicz logic

B. Riecan and D. Mundici, Probability on MV-algebras,Handbook of Measure Theory, 2002


Finite additivity (linearity) vs σ-continuity?



Riesz representation theorem

Let X be a compact Hausdorff space, θ : C (X ) −→ R a positive linearfunctional. Then there exists a unique regular measure µ defined on theBorel σ-algebra of X such that

θ(f ) =∫

Xfdµ for any f ∈ C (X ).



Riesz representation theorem

Let X be a compact Hausdorff space, θ : C (X ) −→ R a positive linearfunctional. Then there exists a unique regular measure µ defined on theBorel σ-algebra of X such that

θ(f ) =∫

Xfdµ for any f ∈ C (X ).

Kroupa-Panti theorem

Let A be a semisimple MV-algebra. For any state s : A −→ [0, 1], thereexists a unique regular measure µ defined on the Borel σ-algebra ofX = Max(A) such that

s(f ) =∫

Xfdµ for any f ∈ A.


Riesz spaces (vector lattices)

A Riesz space are lattice-ordered real vector spaces A s.t.:

x ≤ y implies x + z ≤ y + z for any z ∈ A,

r ≥ 0 and x ≥ 0 implies r•x ≥ 0,where •: R× A −→ A is the scalar multiplication.

Riesz, Kantorovich, Freundenthal, 1936-1940.

... most of the standard real function spaces are Riesz spaces, and in avery natural way.

G. Birkhoff, Lattice Theory, 1973.


Riesz MV-algebras

L is a Riesz space, •: R× L −→ L scalar multiplication,u ∈ L is a strong unit

•: [0, 1]× [0, u]L −→ [0, u]L


Riesz MV-algebras

L is a Riesz space, •: R× L −→ L scalar multiplication,u ∈ L is a strong unit

•: [0, 1]× [0, u]L −→ [0, u]L

The structure Γ(L, u) = ([0, u]L, •) is called Riesz MV-algebra.

The class of Riesz MV-algebras is equational.

Riesz MV-algebras are categorically equivalent with Riesz spaces withstrong unit.

A. Di Nola, P.Flondor, I.L., MV-modules, 2004.


The logic of Riesz MV-algebras

LRMV can be developed following the algebraic approach of H. RasiowaLLuk ∪ δr : r ∈ [0, 1]Axioms= Axioms of Lukasiewicz logic + Axioms reflecting the equationaldescription of Riesz MV-algebras

LRMV is complete w.r.t. [0, 1]-evaluations.

For any r ∈ [0, 1] the formula r = δr (ϕ −→ ϕ) has the value r underany [0, 1]-evaluation. Hence the truth-constants are represented inlogic, making possible the Pavelka style approach.






F. Esteva, J. Gispert, L. Godo, C. Noguera, 2007.






ϕ formula, Θ theory

truth degree: ‖ ϕ ‖Θ= inf e(ϕ) : e [0, 1]-evaluation, e(Θ) = 1,

provability degree: |ϕ|Θ = supr ∈ [0, 1] : Θ ⊢ r −→ ϕ.

Pavelka completeness. |ϕ|Θ =‖ ϕ ‖Θ


Riesz MV-algebras with faithful states

A Riesz MV-algebra, s : A −→ [0, 1] a faithful state (s(x) = 0⇒ x = 0)ρs(x , y) = s(d(x , y)) = s((x ⊙ y∗) ∨ (y ⊙ x∗))

(A, ρs) metric space


Riesz MV-algebras with faithful states

A Riesz MV-algebra, s : A −→ [0, 1] a faithful state (s(x) = 0⇒ x = 0)ρs(x , y) = s(d(x , y)) = s((x ⊙ y∗) ∨ (y ⊙ x∗))

(A, ρs) metric space

Example

If (Y ,Ω, µ) a measure space then denote L1(µ) the Riesz space of allintegrable functions. The constant function 1 is a strong unit of L1(µ), soΓ(L1(µ), 1) is a Ries MV-algebra. Define the state

s : Γ(L1(µ), 1) −→ [0, 1], s(f ) =∫

Yfdµ.

Then (Γ(L1(µ), 1), ρs) is a complete metric space.


A logical approach to metric spaces?

Theorem

For any pair (A, s) s.t. A is a Riesz MV-algebra, s : A −→ [0, 1] is afaithful state and (A, ρs) is a complete metric space, there exists ameasure space (Y ,Ω, µ) s.t.:(a) Y is an extremally disconnected compact Hausdorff space,(b) Ω is the Borel σ-algebra of Y ,(c) A ≃ Γ(L1(µ), 1).



Theorem


Y is the maximal ideal space of the complete Boolean algebra of allclosed ideals of A

MV-algebraic expresssion of Kakutani’s representation for abstractL-spaces (1942)



Theorem


Y is the maximal ideal space of the complete Boolean algebra of allclosed ideals of A

MV-algebraic expresssion of Kakutani’s representation for abstractL-spaces (1942)

Problem

Is it possible to extend the above connection to a categorical equivalence ?


Thank you!


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