Poset Product and BL-chains Conrado Gomez (joint work with Manuela - - PowerPoint PPT Presentation

Poset Product and BL-chains

Conrado Gomez (joint work with Manuela Busaniche)

Instituto de Matemática Aplicada del Litoral Santa Fe, Argentina

Syntax Meets Semantics 2016 Barcelona, 5th September

Hoops and BL-algebras

A hoop is an algebra H ❂ ❤❍❀ ✁❀ ✦❀ ✶✐ of type ❤✷❀ ✷❀ ✵✐ such that ❤❍❀ ✁❀ ✶✐ is a commutative monoid satisfying (i) ① ✦ ① ❂ ✶ (ii) ① ✁ ✭① ✦ ②✮ ❂ ② ✁ ✭② ✦ ①✮ (iii) ① ✦ ✭② ✦ ③✮ ❂ ✭① ✁ ②✮ ✦ ③ for all ①❀ ②❀ ③ ✷ ❍. If H is a hoop, then ✭❍❀ ✁❀ ✶✮ is a naturally ordered residuated commutative monoid, where ① ✔ ② if and only if ① ✦ ② ❂ ✶ and the residuation is ① ✁ ② ✔ ③ if and only if ① ✔ ② ✦ ③✿

Hoops and BL-algebras

A hoop is called bounded if it is an algebra H ❂ ❤❍❀ ✁❀ ✦❀ ✵❀ ✶✐ such that ❤❍❀ ✁❀ ✦❀ ✶✐ is a hoop and ✵ ✔ ① for all ① ✷ ❍. basic if it is a hoop satisfying the identity ✭✭✭① ✦ ②✮ ✦ ③✮ ✁ ✭✭② ✦ ①✮ ✦ ③✮✮ ✦ ③ ❂ ✶✿ a Wajsberg hoop if it satisfies ✭① ✦ ②✮ ✦ ② ❂ ✭② ✦ ①✮ ✦ ①✿ The prelineariry equation ✭① ✦ ②✮ ❴ ✭② ✦ ①✮ ❂ ✶ holds in every basic hoop.

Hoops and BL-algebras

A BL-algebra is a bounded basic hoop and a BL-chain is a totally ordered BL-

• algebra. We will mainly work with two subvarieties of BL-algebras

the subvariety of MV-algebras, characterized by ✿✿① ❂ ① (where ✿① ❂ ① ✦ ✵). the subvariety of product algebras, characterized by ✭✿✿③ ✁ ✭✭① ✁ ③✮ ✦ ✭② ✁ ③✮✮✮ ✦ ✭① ✦ ②✮ ❂ ✶ ① ❫ ✿① ❂ ✵ An MV-chain is a totally ordered MV-algebra and a product chain is a totally

• rdered product algebra.

Classical examples

The standard MV-chain ❬✵❀ ✶❪ ❬✵❀ ✶❪ ❬✵❀ ✶❪MV is the MV-algebra whose universe is the real unit interval ❬✵❀ ✶❪, where ①✁② ❂ ♠❛①✭✵❀ ①✰②✶✮ and ① ✦ ② ❂ ♠✐♥✭✶❀ ✶①✰②✮. For ♥ ✕ ✷, Ł♥ is the subalgebra of ❬✵❀ ✶❪ ❬✵❀ ✶❪ ❬✵❀ ✶❪MV with domain Ł♥ ❂

♥

✵ ♥ ✶❀ ✶ ♥ ✶❀ ✷ ♥ ✶❀ ✿ ✿ ✿ ❀ ♥ ✶ ♥ ✶

♦

✿ The standard product chain is the algebra ❬✵❀ ✶❪ ❬✵❀ ✶❪ ❬✵❀ ✶❪Π ❂ ❤❬✵❀ ✶❪❀ ✁❀ ✦❀ ✵❀ ✶✐ where ✁ is the usual product over the real interval ❬✵❀ ✶❪ and ✦ is given by ① ✦ ② ❂

✚

②❂① if ① ❃ ②❀ ✶ if ① ✔ ②✿

Ordinal sum

Let ❢H✐ ✿ ✐ ✷ ■❣ be a family of hoops indexed by a totally ordered set ✭■❀ ✔✮. Let us assume that H✐ ❭ H❥ ❂ ❢✶❣ whenever ✐ ✻❂ ❥ ✷ ■. The ordinal sum of this family is the hoop

▼

✐✷■

H✐ ❂ ❤

❬

✐✷■

❍✐❀ ✁❀ ✦❀ ✶✐❀ where the operations are given by ① ✁ ② ❂

✽ ❁ ✿

① ✁✐ ② if ①❀ ② ✷ ❍✐❀ ① if ① ✷ ❍✐ ♥ ❢✶❣❀ ② ✷ ❍❥❀ ✐ ❁ ❥❀ ② if ② ✷ ❍✐ ♥ ❢✶❣❀ ① ✷ ❍❥❀ ✐ ❁ ❥✿ ① ✦ ② ❂

✽ ❁ ✿

✶ if ① ✷ ❍✐ ♥ ❢✶❣❀ ② ✷ ❲❥❀ ✐ ❁ ❥❀ ① ✦✐ ② if ①❀ ② ✷ ❍✐❀ ② if ② ✷ ❍✐❀ ① ✷ ❍❥❀ ✐ ❁ ❥✿

BL-chain decomposition

Decomposition theorem for BL-chains (Aglianò-Montagna) Each non-trivial BL-chain admits a unique decomposition into an ordinal sum of non-trivial totally ordered Wajsberg hoops.

BL-chain decomposition

Decomposition theorem for BL-chains (Aglianò-Montagna) Each non-trivial BL-chain admits a unique decomposition into an ordinal sum of non-trivial totally ordered Wajsberg hoops. Remarks If ▲

✐✷■ W✐ is the decomposition of a BL-chain into Wajsberg hoops, then the

index set ■ has a minimum element ✐✵ and the resulting constant bottom in the

• rdinal sum is the bottom of W✐✵.

BL-chain decomposition

Decomposition theorem for BL-chains (Aglianò-Montagna) Each non-trivial BL-chain admits a unique decomposition into an ordinal sum of non-trivial totally ordered Wajsberg hoops. Remarks Totally ordered Wajsberg hoops can be either lower bounded or not. If bounded, they are bottom free reducts of MV-chains. If unbounded, they are cancellative Wajsberg hoops, i.e. they satisfy the identity ① ✦ ✭① ✁ ②✮ ❂ ②. Example: ✭✵❀ ✶❪ ✭✵❀ ✶❪ ✭✵❀ ✶❪Π.

BL-chain decomposition

Decomposition theorem for BL-chains (Aglianò-Montagna) Each non-trivial BL-chain admits a unique decomposition into an ordinal sum of non-trivial totally ordered Wajsberg hoops. Remarks ❬✵❀ ✶❪ ❬✵❀ ✶❪ ❬✵❀ ✶❪Π ✘ ❂ Ł✷ ✟ ✭✵❀ ✶❪ ✭✵❀ ✶❪ ✭✵❀ ✶❪Π. In general, if A is a product chain, then A ✘ ❂ Ł✷ ✟ W❀ where W is a cancellative hoop. In addition, for each cancellative totally ordered hoop W, the ordinal sum Ł✷ ✟ W is a product chain.

Poset product

Given a poset P ❂ ❤P❀ ✔✐ and a collection ❢A♣ ✿ ♣ ✷ P❣ of BL-algebras sharing the same neutral element ✶ and the same minimum element ✵, the poset product

◆

♣✷P A♣ is the residuated lattice A ❂ ❤❆❀ ✁❀ ✦❀ ❴❀ ❫❀ ❄❀ ❃✐ defined as follows:

① ✷ ◗

♣✷P ❆♣

✐ ✷ P ①✐ ✻❂ ✶ ①❥ ❂ ✵ ❥ ❃ ✐ ❃ ✶ ❄ ✭① ✦ ②✮✐ ❂

✚

①✐ ✦

✐ ②✐

①❥ ✔ ②❥ ❥ ❁ ✐❀ ✵

Poset product

Given a poset P ❂ ❤P❀ ✔✐ and a collection ❢A♣ ✿ ♣ ✷ P❣ of BL-algebras sharing the same neutral element ✶ and the same minimum element ✵, the poset product

◆

♣✷P A♣ is the residuated lattice A ❂ ❤❆❀ ✁❀ ✦❀ ❴❀ ❫❀ ❄❀ ❃✐ defined as follows:

The domain of A is the set of all maps ① ✷ ◗

♣✷P ❆♣ such that for all

✐ ✷ P, if ①✐ ✻❂ ✶, then ①❥ ❂ ✵ provided that ❥ ❃ ✐. ❃ ✶ ❄ ✭① ✦ ②✮✐ ❂

✚

①✐ ✦

✐ ②✐

①❥ ✔ ②❥ ❥ ❁ ✐❀ ✵

Poset product

Given a poset P ❂ ❤P❀ ✔✐ and a collection ❢A♣ ✿ ♣ ✷ P❣ of BL-algebras sharing the same neutral element ✶ and the same minimum element ✵, the poset product

◆

♣✷P A♣ is the residuated lattice A ❂ ❤❆❀ ✁❀ ✦❀ ❴❀ ❫❀ ❄❀ ❃✐ defined as follows:

The domain of A is the set of all maps ① ✷ ◗

♣✷P ❆♣ such that for all

✐ ✷ P, if ①✐ ✻❂ ✶, then ①❥ ❂ ✵ provided that ❥ ❃ ✐. ❃ is the map whose value in each coordinate is ✶. Analogously for the symbol ❄ to denote the minimum element. ✭① ✦ ②✮✐ ❂

✚

①✐ ✦

✐ ②✐

①❥ ✔ ②❥ ❥ ❁ ✐❀ ✵

Poset product

Given a poset P ❂ ❤P❀ ✔✐ and a collection ❢A♣ ✿ ♣ ✷ P❣ of BL-algebras sharing the same neutral element ✶ and the same minimum element ✵, the poset product

◆

♣✷P A♣ is the residuated lattice A ❂ ❤❆❀ ✁❀ ✦❀ ❴❀ ❫❀ ❄❀ ❃✐ defined as follows:

The domain of A is the set of all maps ① ✷ ◗

♣✷P ❆♣ such that for all

✐ ✷ P, if ①✐ ✻❂ ✶, then ①❥ ❂ ✵ provided that ❥ ❃ ✐. ❃ is the map whose value in each coordinate is ✶. Analogously for the symbol ❄ to denote the minimum element. Monoid and lattice operations are defined pointwise. ✭① ✦ ②✮✐ ❂

✚

①✐ ✦

✐ ②✐

①❥ ✔ ②❥ ❥ ❁ ✐❀ ✵

Poset product

Given a poset P ❂ ❤P❀ ✔✐ and a collection ❢A♣ ✿ ♣ ✷ P❣ of BL-algebras sharing the same neutral element ✶ and the same minimum element ✵, the poset product

◆

♣✷P A♣ is the residuated lattice A ❂ ❤❆❀ ✁❀ ✦❀ ❴❀ ❫❀ ❄❀ ❃✐ defined as follows:

The domain of A is the set of all maps ① ✷ ◗

♣✷P ❆♣ such that for all

✐ ✷ P, if ①✐ ✻❂ ✶, then ①❥ ❂ ✵ provided that ❥ ❃ ✐. ❃ is the map whose value in each coordinate is ✶. Analogously for the symbol ❄ to denote the minimum element. Monoid and lattice operations are defined pointwise. The residual is ✭① ✦A ②✮✐ ❂

✚

①✐ ✦A✐ ②✐ if ①❥ ✔ ②❥ for all ❥ ❁ ✐❀ ✵

• therwise.

Properties and examples

If P is finite and totally ordered, then ◆

✐✷P A✐ ✘

❂ ▲

✐✷P A✐.

Let P ❂ ❢❛ ❁ ❜❣, A❛ ❂ Ł✸ and A❜ ❂ Ł✷, then Ł✸ ✡ Ł✷ ✘ ❂ Ł✸ ✟ Ł✷.

✵Ł✸

✶ ✷

✵Ł✷ ✶ ❄ ❂ ✭✵❀ ✵✮

• ✶

✷ ❀ ✵✁

✭✶❀ ✵✮ ❃ ❂ ✭✶❀ ✶✮

• ✶

✷ ❀ ✵✁

✁

✶ ✷ ❀ ✵✁

❂ ❄ ✭✶❀ ✵✮ ✦

✶ ✷ ❀ ✵✁

❂

✶ ✷ ❀ ✵✁

P

◆

✐✷P ✐ ❂ ◗ ✐✷P ✐

P ❂ ❢❛ ❦ ❜❣

❛ ❂ ❜ ❂ ✷ ✷ ✡ ✷ ❂ ✷ ✂ ✷

Properties and examples

If P is finite and totally ordered, then ◆

✐✷P A✐ ✘

❂ ▲

✐✷P A✐.

Let P ❂ ❢❛ ❁ ❜❣, A❛ ❂ Ł✸ and A❜ ❂ Ł✷, then Ł✸ ✡ Ł✷ ✘ ❂ Ł✸ ✟ Ł✷.

✵Ł✸

✶ ✷

✵Ł✷ ✶ ❄ ❂ ✭✵❀ ✵✮

• ✶

✷ ❀ ✵✁

✭✶❀ ✵✮ ❃ ❂ ✭✶❀ ✶✮

• ✶

✷ ❀ ✵✁

✁

✶ ✷ ❀ ✵✁

❂ ❄ ✭✶❀ ✵✮ ✦

✶ ✷ ❀ ✵✁

❂

✶ ✷ ❀ ✵✁

If P is an antichain, then ◆

✐✷P A✐ ❂ ◗ ✐✷P A✐.

Let P ❂ ❢❛ ❦ ❜❣ and A❛ ❂ A❜ ❂ Ł✷, then Ł✷ ✡ Ł✷ ❂ Ł✷ ✂ Ł✷.

Properties and examples

If Λ ❂ ❤✄❀ ❁✐ ❂ ❤❢❛❀ ❜❀ ❝❣❀ ❢✭❜❀ ❛✮❀ ✭❝❀ ❛✮❣✐ and A❛ ❂ A❜ ❂ A❝ ❂ Ł✷, then

❜ ❛ ❝

❄ ❂ ✭✵❀ ✵❀ ✵✮ ① ❂ ✭✵❀ ✶❀ ✵✮ ② ❂ ✭✵❀ ✵❀ ✶✮ ✭✵❀ ✶❀ ✶✮ ❃ ❂ ✭✶❀ ✶❀ ✶✮

The poset product of the family is

❖

✄

Ł✷ ❂ ❢✭✵❀ ✵❀ ✵✮❀ ✭✵❀ ✵❀ ✶✮❀ ✭✵❀ ✶❀ ✵✮❀ ✭✵❀ ✶❀ ✶✮❀ ✭✶❀ ✶❀ ✶✮❣✿

Properties and examples

If Λ ❂ ❤✄❀ ❁✐ ❂ ❤❢❛❀ ❜❀ ❝❣❀ ❢✭❜❀ ❛✮❀ ✭❝❀ ❛✮❣✐ and A❛ ❂ A❜ ❂ A❝ ❂ Ł✷, then

❜ ❛ ❝

❄ ❂ ✭✵❀ ✵❀ ✵✮ ① ❂ ✭✵❀ ✶❀ ✵✮ ② ❂ ✭✵❀ ✵❀ ✶✮ ✭✵❀ ✶❀ ✶✮ ❃ ❂ ✭✶❀ ✶❀ ✶✮

◆

Λ Ł✷ is not a BL-algebra because

✭① ✦ ②✮ ❴ ✭② ✦ ①✮ ❂ ✭✵❀ ✵❀ ✶✮ ❴ ✭✵❀ ✶❀ ✵✮ ❂ ✭✵❀ ✶❀ ✶✮ ❁ ✭✶❀ ✶❀ ✶✮ ❂ ❃✿

Forests

From now on, we will consider posets that do not contain as a subposet the configuration ✄. They are known as forests. Thus, a forest is a poset P ❂ ❤P❀ ✔✐ such that for each ✐ ✷ P, the downset ★ ✐ ❂ ❢❥ ✷ P ✿ ❥ ✔ ✐❣ is totally ordered.

Forests

From now on, we will consider posets that do not contain as a subposet the configuration ✄. They are known as forests. Thus, a forest is a poset P ❂ ❤P❀ ✔✐ such that for each ✐ ✷ P, the downset ★ ✐ ❂ ❢❥ ✷ P ✿ ❥ ✔ ✐❣ is totally ordered. ✐

Connected forest

✐

Not connected forest

Forests

From now on, we will consider posets that do not contain as a subposet the configuration ✄. They are known as forests. Thus, a forest is a poset P ❂ ❤P❀ ✔✐ such that for each ✐ ✷ P, the downset ★ ✐ ❂ ❢❥ ✷ P ✿ ❥ ✔ ✐❣ is totally ordered. Theorem If P is a forest and A♣ is a BL-chain for all ♣ ✷ P, then ◆

♣✷P A♣ is a BL-

algebra.

Idempotent free BL-algebras

An algebra A is said to be poset product indecomposable if A is non-trivial and if A is a poset product of two algebras A✶ and A✷, then either A✶ or A✷ is trivial. We will say that a BL-chain A is idempotent free if Id✭A✮ ✘ ❂ Ł✷. ✭ ✮ ♥ ✕ ✷

♥ ✟ ✭✵❀ ✶❪✆

✭✵❀ ✶❪✆ ✭✵❀ ✶❪✆

Idempotent free BL-algebras

An algebra A is said to be poset product indecomposable if A is non-trivial and if A is a poset product of two algebras A✶ and A✷, then either A✶ or A✷ is trivial. We will say that a BL-chain A is idempotent free if Id✭A✮ ✘ ❂ Ł✷. Proposition Let A be a non-trivial BL-chain. Then A is idempotent free ✭ ✮ A is poset product indecomposable. For all ♥ ✕ ✷, Ł♥ ✟ ✭✵❀ ✶❪✆ ✭✵❀ ✶❪✆ ✭✵❀ ✶❪✆ is indecomposable in the sense of poset product.

Representability

Given a BL-chain A, if there are a totally ordered set P and a family of idem- potent free BL-chains ❢A✐ ✿ ✐ ✷ P❣ such that A ✘ ❂ ◆

✐✷P A✐, we will say that

A is representable. If the family only contains MV-chains and product chains, we will say that A is ✆MV-representable.

✸ ✟ ✭✵❀ ✶❪✆

✭✵❀ ✶❪✆ ✭✵❀ ✶❪✆ ✆

Representability

Given a BL-chain A, if there are a totally ordered set P and a family of idem- potent free BL-chains ❢A✐ ✿ ✐ ✷ P❣ such that A ✘ ❂ ◆

✐✷P A✐, we will say that

A is representable. If the family only contains MV-chains and product chains, we will say that A is ✆MV-representable. MV-chains and product chains are representable BL-chains. Ł✸ ✟ ✭✵❀ ✶❪✆ ✭✵❀ ✶❪✆ ✭✵❀ ✶❪✆ is representable ✆

Representability

Given a BL-chain A, if there are a totally ordered set P and a family of idem- potent free BL-chains ❢A✐ ✿ ✐ ✷ P❣ such that A ✘ ❂ ◆

✐✷P A✐, we will say that

A is representable. If the family only contains MV-chains and product chains, we will say that A is ✆MV-representable. MV-chains and product chains are representable BL-chains. Ł✸ ✟ ✭✵❀ ✶❪✆ ✭✵❀ ✶❪✆ ✭✵❀ ✶❪✆ is representable but is not ✆MV-representable.

Representability

Given a BL-chain A, if there are a totally ordered set P and a family of idem- potent free BL-chains ❢A✐ ✿ ✐ ✷ P❣ such that A ✘ ❂ ◆

✐✷P A✐, we will say that

A is representable. If the family only contains MV-chains and product chains, we will say that A is ✆MV-representable. MV-chains and product chains are representable BL-chains. Ł✸ ✟ ✭✵❀ ✶❪✆ ✭✵❀ ✶❪✆ ✭✵❀ ✶❪✆ is representable but is not ✆MV-representable. Jipsen-Montagna’s generalization for Di Nola-Lettieri’s result Every finite BL-algebra is isomorphic to the a poset product of a collection of MV-chains.

Poset product of idempotent free BL-chains

Theorem Let ❤P❀ ✔✐ be a totally ordered set and ❢A♣ ✿ ♣ ✷ P❣ be a family of idempotent free BL-chains. Then ▲

♣✷P A♣ ✘

❂ ◆

♣✷P A♣ if and only if P is well-ordered.

✭✮✮

▲

♣✷P ♣ ✘

❂ ◆

♣✷P ♣

■❞✭

♣✮ ❂ ❢✵❀ ✶❣ ✽♣ ✷ P

▼

P ✷ ✘

❂

❖

P ✷✿

◆

P ✷

P ✭✭✮ P ❢ ✿ ▲

♣✷P ♣ ✦ ◆ ♣✷P ♣

❢✭✶✮ ❂ ❃ ❢✭❛✮♣ ❂

✽ ❁ ✿

✶ ♣ ❁ ❥❀ ❛ ♣ ❂ ❥❀ ✵ ♣ ❃ ❥✿ ❛ ✷ ❆❥ ♥ ❢❃❣

Poset product of idempotent free BL-chains

Theorem Let ❤P❀ ✔✐ be a totally ordered set and ❢A♣ ✿ ♣ ✷ P❣ be a family of idempotent free BL-chains. Then ▲

♣✷P A♣ ✘

❂ ◆

♣✷P A♣ if and only if P is well-ordered.

✭✮✮ If ▲

♣✷P A♣ ✘

❂ ◆

♣✷P A♣, since

■❞✭A♣✮ ❂ ❢✵❀ ✶❣ ✽♣ ✷ P,

▼

P

Ł✷ ✘ ❂

❖

P

Ł✷✿ Given that ◆

P Ł✷ is complete, P can

be seen as a complete poset which actually is a well-ordered set. ✭✭✮ P ❢ ✿ ▲

♣✷P ♣ ✦ ◆ ♣✷P ♣

❢✭✶✮ ❂ ❃ ❢✭❛✮♣ ❂

✽ ❁ ✿

✶ ♣ ❁ ❥❀ ❛ ♣ ❂ ❥❀ ✵ ♣ ❃ ❥✿ ❛ ✷ ❆❥ ♥ ❢❃❣

Poset product of idempotent free BL-chains

Theorem Let ❤P❀ ✔✐ be a totally ordered set and ❢A♣ ✿ ♣ ✷ P❣ be a family of idempotent free BL-chains. Then ▲

♣✷P A♣ ✘

❂ ◆

♣✷P A♣ if and only if P is well-ordered.

✭✮✮ If ▲

♣✷P A♣ ✘

❂ ◆

♣✷P A♣, since

■❞✭A♣✮ ❂ ❢✵❀ ✶❣ ✽♣ ✷ P,

▼

P

Ł✷ ✘ ❂

❖

P

Ł✷✿ Given that ◆

P Ł✷ is complete, P can

be seen as a complete poset which actually is a well-ordered set. ✭✭✮ If P is a well-ordered set, the map ❢ ✿ ▲

♣✷P A♣ ✦ ◆ ♣✷P A♣ de-

fined by ❢✭✶✮ ❂ ❃ and ❢✭❛✮♣ ❂

✽ ❁ ✿

✶ if ♣ ❁ ❥❀ ❛ if ♣ ❂ ❥❀ ✵ if ♣ ❃ ❥✿ if ❛ ✷ ❆❥ ♥ ❢❃❣ is an isomorphism.

Some issues

Unfortunately, not all BL-chain can be written as an ordinal sum of idempotent free BL-chains. If it were the case, the index set would not always be a well-

• rdered set.

❂ ▲

■ ✷

❂ ❤❢❜❣❬❩❀ ✔✐ ■ ✘ ❂ ◆

❩

▲

❩ ✷ ✢ ◆ ❩ ✷

❂ ▲

✐✷■ ✐

❂ ❤◆ ❬ ❢t❣❀ ✔✐

♥ ❂ ✷

♥ ✷ ◆

t ❂ ✭✵❀ ✶❪✆

✭✵❀ ✶❪✆ ✭✵❀ ✶❪✆

t

Some issues

Unfortunately, not all BL-chain can be written as an ordinal sum of idempotent free BL-chains. If it were the case, the index set would not always be a well-

• rdered set.

Representable BL-chain without a well-ordered index set Let A ❂ ▲

■ Ł✷, where I ❂ ❤❢❜❣❬❩❀ ✔✐. Although ■ is not a well-ordered set,

A ✘ ❂ ◆

❩ Ł2. Observe that ▲ ❩ Ł✷ ✢ ◆ ❩ Ł✷.

❂ ▲

✐✷■ ✐

❂ ❤◆ ❬ ❢t❣❀ ✔✐

♥ ❂ ✷

♥ ✷ ◆

t ❂ ✭✵❀ ✶❪✆

✭✵❀ ✶❪✆ ✭✵❀ ✶❪✆

t

Some issues

Unfortunately, not all BL-chain can be written as an ordinal sum of idempotent free BL-chains. If it were the case, the index set would not always be a well-

• rdered set.

Representable BL-chain without a well-ordered index set Let A ❂ ▲

■ Ł✷, where I ❂ ❤❢❜❣❬❩❀ ✔✐. Although ■ is not a well-ordered set,

A ✘ ❂ ◆

❩ Ł2. Observe that ▲ ❩ Ł✷ ✢ ◆ ❩ Ł✷.

In addition, a well-ordered index set in a decomposition of a BL-chain does not guarantee a representation in terms of idempotent free BL-chains. ❂ ▲

✐✷■ ✐

❂ ❤◆ ❬ ❢t❣❀ ✔✐

♥ ❂ ✷

♥ ✷ ◆

t ❂ ✭✵❀ ✶❪✆

✭✵❀ ✶❪✆ ✭✵❀ ✶❪✆

t

Some issues

Unfortunately, not all BL-chain can be written as an ordinal sum of idempotent free BL-chains. If it were the case, the index set would not always be a well-

• rdered set.

Representable BL-chain without a well-ordered index set Let A ❂ ▲

■ Ł✷, where I ❂ ❤❢❜❣❬❩❀ ✔✐. Although ■ is not a well-ordered set,

A ✘ ❂ ◆

❩ Ł2. Observe that ▲ ❩ Ł✷ ✢ ◆ ❩ Ł✷.

In addition, a well-ordered index set in a decomposition of a BL-chain does not guarantee a representation in terms of idempotent free BL-chains. Non-representable BL-chain indexed by a well-ordered set Let A ❂ ▲

✐✷■ W✐, where I ❂ ❤◆ ❬ ❢t❣❀ ✔✐, W♥ ❂ Ł✷ for all ♥ ✷ ◆ and

Wt ❂ ✭✵❀ ✶❪✆ ✭✵❀ ✶❪✆ ✭✵❀ ✶❪✆. Then A is not representable. Note that Wt is not a BL-chain.

A sufficient (but strong) condition for representability

Proposition If each prime filter in a BL-chain A is a principal filter, then A is representable. If A ✘ ❂ ▲

✐✷■ W✐, it turns out that the index set ■ is well-ordered and every W✐

is a bounded hoop (MV-chain). Thus A ✘ ❂ ◆

✐✷■ W✐.

♥ ✕ ✷ ✭✵❀ ✶❪

♥ ✟ ✭✵❀ ✶❪✆

✭✵❀ ✶❪✆ ✭✵❀ ✶❪✆

A sufficient (but strong) condition for representability

Proposition If each prime filter in a BL-chain A is a principal filter, then A is representable. If A ✘ ❂ ▲

✐✷■ W✐, it turns out that the index set ■ is well-ordered and every W✐

is a bounded hoop (MV-chain). Thus A ✘ ❂ ◆

✐✷■ W✐.

Since in a finite BL-algebra all filters are principal, this is a proposition that (for the case of BL-chains) enhances the Jipsen and Montagna’s result we cited before. ♥ ✕ ✷ ✭✵❀ ✶❪

♥ ✟ ✭✵❀ ✶❪✆

✭✵❀ ✶❪✆ ✭✵❀ ✶❪✆

A sufficient (but strong) condition for representability

Proposition If each prime filter in a BL-chain A is a principal filter, then A is representable. If A ✘ ❂ ▲

✐✷■ W✐, it turns out that the index set ■ is well-ordered and every W✐

is a bounded hoop (MV-chain). Thus A ✘ ❂ ◆

✐✷■ W✐.

Since in a finite BL-algebra all filters are principal, this is a proposition that (for the case of BL-chains) enhances the Jipsen and Montagna’s result we cited

• before. However, it must be said that the hypothesis is still too restrictive, since

in general idempotent free BL-chains contain a non-prime principal filter. For all ♥ ✕ ✷, the set ✭✵❀ ✶❪ is a prime filter in the representable BL-chain Ł♥ ✟ ✭✵❀ ✶❪✆ ✭✵❀ ✶❪✆ ✭✵❀ ✶❪✆ which is not a principal filter.

Saturated BL-chains

Let A be a BL-chain. A pair of sets ✭❳❀ ❨ ✮ is called a cut in A if ❳ ❬ ❨ ❂ ❆, ① ✔ ② for all ① ✷ ❳ and all ② ✷ ❨ , ❨ is closed under ✁ and ① ✁ ② ❂ ① for all ① ✷ ❳ and all ② ✷ ❨ . A is called saturated if for every cut ✭❳❀ ❨ ✮ there exists ✉ ✷ ■❞✭A✮ such that ① ✔ ✉ ✔ ② for all ① ✷ ❳ and all ② ✷ ❨ .

Representation of saturated BL-chains

MV-chains and product chains are the only idempotent free BL-chains with the property of being saturated chains. The Gödel chain ▲

❬✵❀✶❪ Ł✷ is a saturated chain that is not representable.

✘ ❂ ▲

✐✷P ✐ ❥

❥ ✷ P ❥✵ ✷ P ❥

❥✵ ✘

❂

✷

▲

✐✷P ✐

P P ✵ ✘ ❂ ▲

✐✷P ✵ ✐ ✐

✆

Representation of saturated BL-chains

MV-chains and product chains are the only idempotent free BL-chains with the property of being saturated chains. The Gödel chain ▲

❬✵❀✶❪ Ł✷ is a saturated chain that is not representable.

Lemma Let A ✘ ❂ ▲

✐✷P W✐ be a saturated BL-chain. If W❥ is an unbounded hoop for

some ❥ ✷ P, then there exists ❥✵ ✷ P preceding ❥ such that W❥✵ ✘ ❂ Ł✷.

▲

✐✷P ✐

P P ✵ ✘ ❂ ▲

✐✷P ✵ ✐ ✐

✆

Representation of saturated BL-chains

MV-chains and product chains are the only idempotent free BL-chains with the property of being saturated chains. The Gödel chain ▲

❬✵❀✶❪ Ł✷ is a saturated chain that is not representable.

Lemma Let A ✘ ❂ ▲

✐✷P W✐ be a saturated BL-chain. If W❥ is an unbounded hoop for

some ❥ ✷ P, then there exists ❥✵ ✷ P preceding ❥ such that W❥✵ ✘ ❂ Ł✷. Theorem Let A be a saturated BL-chain and let ▲

✐✷P W✐ be its unique decomposition

into non-trivial Wajsberg hoops. If P is a well-ordered set, then there is a well-

• rdered set P ✵ such that A ✘

❂ ▲

✐✷P ✵ A✐, with A✐ an MV-chain or a product

• chain. Consequently, A is ✆MV-representable.

Representation of saturated BL-chains

We know that A ✘ ❂ ▲

✐✷P W✐ and P is a well-ordered set. As remarked, a

hoop W✐ in the decomposition of a BL-chain A can be unbounded. For in- stance, let us assume that W❥ and W❦ are unbounded hoops for some ❥❀ ❦ ✷ P. A ✘ ❂

▼

✐✷P

W✐ ❂ W✶ ✟ ✿ ✿ ✿ ✟ W❥ ✟ ✿ ✿ ✿ ✟ W❦ ✟ ✿ ✿ ✿ ✟ W❧ ✟ ✿ ✿ ✿

Representation of saturated BL-chains

We know that A ✘ ❂ ▲

✐✷P W✐ and P is a well-ordered set. As remarked, a

hoop W✐ in the decomposition of a BL-chain A can be unbounded. For in- stance, let us assume that W❥ and W❦ are unbounded hoops for some ❥❀ ❦ ✷ P. A ✘ ❂

▼

✐✷P

W✐ ❂ W✶ ✟ ✿ ✿ ✿ ✟ W❥ ✟ ✿ ✿ ✿ ✟ W❦ ✟ ✿ ✿ ✿ ✟ W❧ ✟ ✿ ✿ ✿ Let ❥✵❀ ❦✵ ✷ P be the elements below ❥ and ❦, respectively. Then A ✘ ❂ W✶ ✟ ✿ ✿ ✿ ✟ ✭W❥✵ ✟ W❥✮ ✟ ✿ ✿ ✿ ✟ ✭W❦✵ ✟ W❦✮ ✟ ✿ ✿ ✿ ✟ W❧ ✟ ✿ ✿ ✿

Representation of saturated BL-chains

We know that A ✘ ❂ ▲

✐✷P W✐ and P is a well-ordered set. As remarked, a

hoop W✐ in the decomposition of a BL-chain A can be unbounded. For in- stance, let us assume that W❥ and W❦ are unbounded hoops for some ❥❀ ❦ ✷ P. A ✘ ❂

▼

✐✷P

W✐ ❂ W✶ ✟ ✿ ✿ ✿ ✟ W❥ ✟ ✿ ✿ ✿ ✟ W❦ ✟ ✿ ✿ ✿ ✟ W❧ ✟ ✿ ✿ ✿ Let ❥✵❀ ❦✵ ✷ P be the elements below ❥ and ❦, respectively. Then A ✘ ❂ W✶ ✟ ✿ ✿ ✿ ✟ ✭W❥✵ ✟ W❥✮ ✟ ✿ ✿ ✿ ✟ ✭W❦✵ ✟ W❦✮ ✟ ✿ ✿ ✿ ✟ W❧ ✟ ✿ ✿ ✿ Moreover, since W❥✵ ✘ ❂ W❦✵ ✘ ❂ Ł✷, A ✘ ❂ W✶ ✟ ✿ ✿ ✿ ✟ ✭Ł✷ ✟ W❥✮ ✟ ✿ ✿ ✿ ✟ ✭Ł✷ ✟ W❦✮ ✟ ✿ ✿ ✿ ✟ W❧ ✟ ✿ ✿ ✿

Representation of saturated BL-chains

We know that A ✘ ❂ ▲

✐✷P W✐ and P is a well-ordered set. As remarked, a

hoop W✐ in the decomposition of a BL-chain A can be unbounded. For in- stance, let us assume that W❥ and W❦ are unbounded hoops for some ❥❀ ❦ ✷ P. A ✘ ❂ W✶ ✟ ✿ ✿ ✿ ✟ ✭Ł✷ ✟ W❥✮ ✟ ✿ ✿ ✿ ✟ ✭Ł✷ ✟ W❦✮ ✟ ✿ ✿ ✿ ✟ W❧ ✟ ✿ ✿ ✿ Following the above suggested idea we define P ✵ as a rearrangement of P. P ✵ will index the summands A✐ ❂

✚

Ł✷ ✟ W✐ if W✐ is unbounded❀ W✐ if W✐ is bounded.

Representation of saturated BL-chains

We know that A ✘ ❂ ▲

✐✷P W✐ and P is a well-ordered set. As remarked, a

hoop W✐ in the decomposition of a BL-chain A can be unbounded. For in- stance, let us assume that W❥ and W❦ are unbounded hoops for some ❥❀ ❦ ✷ P. A ✘ ❂ W✶ ✟ ✿ ✿ ✿ ✟ ✭Ł✷ ✟ W❥✮ ✟ ✿ ✿ ✿ ✟ ✭Ł✷ ✟ W❦✮ ✟ ✿ ✿ ✿ ✟ W❧ ✟ ✿ ✿ ✿ Following the above suggested idea we define P ✵ as a rearrangement of P. P ✵ will index the summands A✐ ❂

✚

Ł✷ ✟ W✐ if W✐ is unbounded❀ W✐ if W✐ is bounded. Then A ✘ ❂ ▲

✐✷P ✵ A✐ and each summand is an MV-chain or a product chain.

Note that P ✵ is a well-ordered set because P so is. Thus A ✘ ❂

❖

✐✷P ✵

A✐✿

Representation of saturated BL-chains

The next result provides an alternative definition for ✆MV-representability. It also reveals the link between the notions of representability and ✆MV-representability. Corollary A BL-chain A is representable and saturated if and only if it is ✆MV-repre- sentable.

Further readings on the poset product construction

Busaniche, M., and F. Montagna, ‘Hájek’s logic BL and BL-algebras’, in Handbook

• f Mathematical Fuzzy Logic, vol. 1 of Studies in Logic, Mathematical Logic and

Foundations, chap. V, College Publications, London, 2011, pp. 355–447. Jipsen, P., ‘Generalizations of boolean products for lattice-ordered algebras’, An- nals of Pure and Applied Logic, 161 (2009), 228–234 Jipsen, P., and F. Montagna, ‘On the structure of generalized BL-algebras’, Algebra Universalis, 55 (2006), 227–238. Jipsen, P., and F. Montagna, ‘The Blok-Ferreirim theorem for normal GBL-algebras and its applications’, Algebra Universalis, 60 (2009), 381–404. Jipsen, P., and F. Montagna, ‘Embedding theorems for classes of GBL-algebras’, Journal of Pure and Applied Algebra, 214 (2010), 1559–1575.

Thank you