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 ordered 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 ✚ ②❂① if ① ❃ ② ❀ 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 if ①❀ ② ✷ ❍ ✐ ❀ ❁ ① ✁ ② ❂ ① if ① ✷ ❍ ✐ ♥ ❢ ✶ ❣ ❀ ② ✷ ❍ ❥ ❀ ✐ ❁ ❥❀ ✿ ② if ② ✷ ❍ ✐ ♥ ❢ ✶ ❣ ❀ ① ✷ ❍ ❥ ❀ ✐ ❁ ❥✿ ✽ ✶ if ① ✷ ❍ ✐ ♥ ❢ ✶ ❣ ❀ ② ✷ ❲ ❥ ❀ ✐ ❁ ❥❀ ❁ 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 ordinal 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.

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:

❃ ✶ ❄ ✚ 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.

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 ✐ if ① ❥ ✔ ② ❥ for all ❥ ❁ ✐ ❀ ✭ ① ✦ A ② ✮ ✐ ❂ ✵ otherwise.

◆ ✐ ❂ ◗ P ✐ ✐ ✷ P ✐ ✷ P P ❂ ❢ ❛ ❦ ❜ ❣ ❛ ❂ ❜ ❂ ✷ ✡ ✷ ❂ ✷ ✂ ✷ ✷ Properties and examples ✐ ✷ P A ✐ ✘ If P is finite and totally ordered, then ◆ ❂ ▲ ✐ ✷ P A ✐ . Let P ❂ ❢ ❛ ❁ ❜ ❣ , A ❛ ❂ Ł ✸ and A ❜ ❂ Ł ✷ , then Ł ✸ ✡ Ł ✷ ✘ ❂ Ł ✸ ✟ Ł ✷ . ❃ ❂ ✭✶ ❀ ✶✮ ✶ � ✷ ❀ ✵ ✁ ✁ � ✷ ❀ ✵ ✁ ✵ Ł ✷ ✶ ✶ ✭✶ ❀ ✵✮ ❂ ❄ � ✷ ❀ ✵ ✁ ✭✶ ❀ ✵✮ ✦ � ✷ ❀ ✵ ✁ ❂ � ✷ ❀ ✵ ✁ ✶ ✶ ✶ ✶ ✷ ✵ Ł ✸ ❄ ❂ ✭✵ ❀ ✵✮