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The Finite Embeddability Property for Topological Quasi-Boolean Algebra 5 Zhe Lin 1 and Mihir Kumar Chakraborty 2 Southwest University, Chongqing, China. 1 School of Cognitive Science, Jadavpur University, India 2 2019/03/03 Zhe Lin1 and Mihir

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The Finite Embeddability Property for Topological Quasi-Boolean Algebra 5

Zhe Lin1 and Mihir Kumar Chakraborty2

Southwest University, Chongqing, China.1 School of Cognitive Science, Jadavpur University, India2

2019/03/03

Zhe Lin1 and Mihir Kumar Chakraborty2 (Southwest University, Chongqing, China.1School of Cognitive Science, Jadavpur University, India2) The Finite Embeddability Property for Topological Quasi-Boolean Algebra 5 2019/03/03 1 / 27

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Quasi-Boolean algebra

Definition (qBa) A quasi-Boolean algebra (qBa) is an algebra A = (A, ∧, ∨, ¬, 0, 1) where (A, ∧, ∨, 0, 1) is a bounded distributive lattice, and ¬ is an unary operation on A such that the following conditions hold for all a, b ∈ A: (DN) ¬¬a = a, (DM) ¬(a ∨ b) = ¬a ∧ ¬b

Zhe Lin1 and Mihir Kumar Chakraborty2 (Southwest University, Chongqing, China.1School of Cognitive Science, Jadavpur University, India2) The Finite Embeddability Property for Topological Quasi-Boolean Algebra 5 2019/03/03 2 / 27

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Topological quasi-Boolean algebra

Definition (tqBa) A topological quasi-Boolean algebra (tqBa) is an algebra A = (A, ∧, ∨, ¬, 0, 1, ) where (A, ∧, ∨, ¬, 0, 1) is a quasi-Boolean algebra, and is an unary operation on A such that for all a, b ∈ A: (K) (a ∧ b) = a ∧ b, (N) ⊤ = ⊤ (T) a ≤ a, (4) a ≤ a

Zhe Lin1 and Mihir Kumar Chakraborty2 (Southwest University, Chongqing, China.1School of Cognitive Science, Jadavpur University, India2) The Finite Embeddability Property for Topological Quasi-Boolean Algebra 5 2019/03/03 3 / 27

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Topological quasi-Boolean algebra 5

Definition (tqBa5) A topological quasi-Boolean algebra 5 is a topological quasi-Boolean algebra A = (A, ∧, ∨, ¬, , 0, 1) such that for all a ∈ A: (5) ♦a ≤ ♦a, where ♦ is an unary operation on A defined by ♦a := ¬¬a. (1) Banerjee, M., Chakraborty, M.: Rough algebra. Bulletin of Polish Academy of Sciences (Mathematics) 41(4), 293–297 (1993) (2) Banerjee, M., Chakraborty, M.: Rough sets through algebraic logic. Fundamenta Informaticae 28(3-4), 211–221 (1996)

Zhe Lin1 and Mihir Kumar Chakraborty2 (Southwest University, Chongqing, China.1School of Cognitive Science, Jadavpur University, India2) The Finite Embeddability Property for Topological Quasi-Boolean Algebra 5 2019/03/03 4 / 27

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Example (Banerjee, M.) Let us consider the lattice whose Hasse diagram is shown in Fig.2 and ¬, are defined as follows: 1 a b ¬ 1 a b 1 a b

  • 1

a b

Figure: Fig.2

Zhe Lin1 and Mihir Kumar Chakraborty2 (Southwest University, Chongqing, China.1School of Cognitive Science, Jadavpur University, India2) The Finite Embeddability Property for Topological Quasi-Boolean Algebra 5 2019/03/03 5 / 27

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Finite embeddability property

An embedding from a partial algebra B into an algebra C, we mean an injection h : B → C such that if b1, . . . , bn, f B(b1, . . . , bn) ∈ B, then h(f B(b1, . . . , bn)) = f C(h(b1), . . . , h(bn)). If B and C are ordered, then h is required to be an order embedding i.e. a ≤B b ⇔ h(a) ≤C h(b). Definition (FEP) A class K of algebras has the finite embeddability property (FEP), if every finite partial subalgebra of a member of K can be embedded into a finite member of K.

Zhe Lin1 and Mihir Kumar Chakraborty2 (Southwest University, Chongqing, China.1School of Cognitive Science, Jadavpur University, India2) The Finite Embeddability Property for Topological Quasi-Boolean Algebra 5 2019/03/03 6 / 27

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Variety

Lemma (Birkhoff) A class K of algebras is a variety if and only if K is an equational class (3) G. Birkhoff, On the structure of abstract algebras 31 (1935), 433-454. Lemma The class of tqBa5 is a variety. (4) Saha, A., Sen, J., Chakraborty, M.: Algebraic structures in the vicinity of pre-rough algebra and their logics. Information Science 282, 296C320 (2014). (2) Banerjee, M., Chakraborty, M.: Rough sets through algebraic logic. Fundamenta Informati- cae 28(3-4), 211C221 (1996).

Zhe Lin1 and Mihir Kumar Chakraborty2 (Southwest University, Chongqing, China.1School of Cognitive Science, Jadavpur University, India2) The Finite Embeddability Property for Topological Quasi-Boolean Algebra 5 2019/03/03 7 / 27

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Strong Finite Model Property

Definition (SFMP) The strong finite model property (SFMP) i.e. every quasi-equation (quasi-identity) which fails to hold in a class K of algebras can be falsified in a finite member of K.

Zhe Lin1 and Mihir Kumar Chakraborty2 (Southwest University, Chongqing, China.1School of Cognitive Science, Jadavpur University, India2) The Finite Embeddability Property for Topological Quasi-Boolean Algebra 5 2019/03/03 8 / 27

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FEP and SFMP

Lemma ((5)—Lemma 6.40) For variety K of finite type the following are equivalent: (1) K has FEP (2) K have SFMP (5) Galatos, N., Jipsen, P ., Kowalski, T., Ono, H.: Residuated Lattices: An Algebraic Glimpse at Substructural Logics. Springer (2007). If a formal system S is strongly complete with respect to a class K of algebras SFMP for S with respect to K yields SFMP for K.

Zhe Lin1 and Mihir Kumar Chakraborty2 (Southwest University, Chongqing, China.1School of Cognitive Science, Jadavpur University, India2) The Finite Embeddability Property for Topological Quasi-Boolean Algebra 5 2019/03/03 9 / 27

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Sequent system for tqBa5 I

Definition (Formula) The language of the logic of tqBa5 is defined as follows α ::= p | ⊥ | ⊤ | α ∧ β | α ∨ β | ¬α | ♦α | α, where p ∈ Prop, the set of propositional variables. Definition (Formula structures) Formula structure are defined as follows with a unary structural operation : α is a formula structure if α is a formula Γi is a formula structure if Γ is a formula structure Definition (Sequent) Sequent is an expression of the form αi ⇒ β where i ≥ 0 for some formulae α and β.

Zhe Lin1 and Mihir Kumar Chakraborty2 (Southwest University, Chongqing, China.1School of Cognitive Science, Jadavpur University, India2) The Finite Embeddability Property for Topological Quasi-Boolean Algebra 5 2019/03/03 10 / 27

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Sequent system for tqBa5 II

The Gentzen sequent calculus G5 consists of the following axioms and inference rules: (1) Axioms: (Id) ϕ ⇒ ϕ (⊥) ⊥i ⇒ ϕ (⊤) ϕi ⇒ ⊤ (D) ϕ ∧ (ψ ∨ χ) ⇒ (ϕ ∧ ψ) ∨ (ϕ ∧ χ) (DN) ϕ ⇔ ¬¬ϕ (2) Connective rules: ϕi ⇒ χ ϕ ∧ ψi ⇒ χ(∧L) χi ⇒ ϕ χi ⇒ ψ χi ⇒ ϕ ∧ ψ (∧R) ϕi ⇒ χ ψi ⇒ χ ϕ ∨ ψi ⇒ χ (∨L) χi ⇒ ψ χi ⇒ ψ ∨ ϕ(∨R)

Zhe Lin1 and Mihir Kumar Chakraborty2 (Southwest University, Chongqing, China.1School of Cognitive Science, Jadavpur University, India2) The Finite Embeddability Property for Topological Quasi-Boolean Algebra 5 2019/03/03 11 / 27

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(3) Modal rules ϕi+1 ⇒ ψ ♦ϕi ⇒ ψ (♦L) ϕi ⇒ ψ ϕi+1 ⇒ ♦ψ (♦R) ϕi ⇒ ψ ϕi+1 ⇒ ψ (L) ϕi+1 ⇒ ψ ϕi ⇒ ψ (R) ϕi+1 ⇒ ψ ϕi ⇒ ψ (T) ϕi+1 ⇒ ψ ϕi+2 ⇒ ψ (4) ϕi ⇒ ψ ¬ψi ⇒ ¬ϕ(♦) (4) Cut rule ϕi ⇒ χ χj ⇒ ψ ϕi+j ⇒ ψ (Cut)

Zhe Lin1 and Mihir Kumar Chakraborty2 (Southwest University, Chongqing, China.1School of Cognitive Science, Jadavpur University, India2) The Finite Embeddability Property for Topological Quasi-Boolean Algebra 5 2019/03/03 12 / 27

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Algebraic models

An algebraic model of G5 is a pair (G, σ) such that G is a tqBa5, and σ is a mapping from Prop into G, called a valuation, which is extended to formulae and formula trees as follows: σ(α) = σ(α), σ(♦α) = ♦σ(α) σ(α ∧ β) = σ(α) ∧ σ(β), σ(α ∨ β) = σ(α) ∨ σ(β), σ(¬α) = ¬σ(α), σ(αi+1) = ♦σ(αi). Definition (True) (G, σ) | = αi ⇒ β, if σ(αi) ≤ σ(β) (here ≤ is the lattice order in G). Φ | = αi ⇒ β with respect to tqBa5s, if G, σ ⊢ αi ⇒ β in all models (G, σ) such that G ∈ tqBa5 and for any sequent ϕj ⇒ ψ ∈ Φ, G, σ ⊢ ϕj ⇒ ψ

Zhe Lin1 and Mihir Kumar Chakraborty2 (Southwest University, Chongqing, China.1School of Cognitive Science, Jadavpur University, India2) The Finite Embeddability Property for Topological Quasi-Boolean Algebra 5 2019/03/03 13 / 27

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Strongly complete and SFMP

Theorem (Strongly completeness) G5 is strongly complete with respect to tqBa5s: for any set of sequents Φ and any sequent αi ⇒ β , Φ ⊢G5 αi ⇒ β if and only if Φ | = αi ⇒ β with respect to tqBa5. Definition (SFMP) For any finite set of sequents Φ, if Φ ⊢G5 αi ⇒ β, then there exists a finite G ∈ tqBa5s and a valuation σ such that all sequents from Φ are true in (G, σ), but αi ⇒ β is not.

Zhe Lin1 and Mihir Kumar Chakraborty2 (Southwest University, Chongqing, China.1School of Cognitive Science, Jadavpur University, India2) The Finite Embeddability Property for Topological Quasi-Boolean Algebra 5 2019/03/03 14 / 27

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Interpolant I

Definition (T-sequent) A sequent αi ⇒ β is called a T sequent if α, β ∈ T. Definition (T-derivation) A derivation from Φ in G5 of a T-sequent αi ⇒ β is called a T-derivation if all sequents appearing in the derivation are T-sequents, which is denoted by Φ ⊢G5 αi ⇒T β.

Zhe Lin1 and Mihir Kumar Chakraborty2 (Southwest University, Chongqing, China.1School of Cognitive Science, Jadavpur University, India2) The Finite Embeddability Property for Topological Quasi-Boolean Algebra 5 2019/03/03 15 / 27

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Interpolant II

Definition (Interpolant) Assume that Φ ⊢G5 ϕi+j ⇒T ψ. A formula γ is called a T interpolant of ϕi if γ ∈ T, Φ ⊢G5 ϕi ⇒T γ and Φ ⊢G5 γj ⇒T ψ and additionally Φ ⊢G5 γ ⇒T γ if i ≥ 1. Lemma (Interpolant) If Φ ⊢G5 ϕi+j ⇒T ψ, then ϕi has a T interpolant. T is closed under ¬, ∧, ∨ and subformulas.

Zhe Lin1 and Mihir Kumar Chakraborty2 (Southwest University, Chongqing, China.1School of Cognitive Science, Jadavpur University, India2) The Finite Embeddability Property for Topological Quasi-Boolean Algebra 5 2019/03/03 16 / 27

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Definitions

Definition (≤T) Let α1i, α2j ∈ T s, we say α1i ≤T α2j if Φ ⊢G5 α2jt ⇒T β implies Φ ⊢G5 α1it ⇒T β for any context t where t ≥ 0 and T formula β. Definition (≈T) Let α1i ≈T α2j if α1i ≤T α2j and α2j ≤T α1i. Obviously ≈T is a equivalence relation on T formula structures.

Zhe Lin1 and Mihir Kumar Chakraborty2 (Southwest University, Chongqing, China.1School of Cognitive Science, Jadavpur University, India2) The Finite Embeddability Property for Topological Quasi-Boolean Algebra 5 2019/03/03 17 / 27

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Definitions

We define {αi}≈

T = {βj|βj ≈T αi}(i, j ≥ 0)

Obviously {α}≈

T = {βj|βj ≈T α}(j ≥ 0)

T/≈T denote the set of all {αi}≈

T where αi ∈ T s and i ≥ 0.

T •/≈T denote the set of all {α}≈

T where αi ∈ T s and i ≥ 0.

Define {αi}≈

T T {βj}≈ T if αi ≤T βj.

Zhe Lin1 and Mihir Kumar Chakraborty2 (Southwest University, Chongqing, China.1School of Cognitive Science, Jadavpur University, India2) The Finite Embeddability Property for Topological Quasi-Boolean Algebra 5 2019/03/03 18 / 27

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Closure operation

We define a closure operation C on T/≈T as follows: C({αi}≈

T ) = {

  • 1≤j≤n

βj}≈

T

for any {βj}≈

T ∈ T •/≈T s.t. {αi}≈ T T {βj}≈ T

Lemma For any {αi}≈

T , {βj}≈ T ∈ T/≈T , the following hold:

(1) {αi}≈

T T C({αi}≈ T ).

(2) if {αi}≈

T T {βj}≈ T , then C({αi}≈ T ) T C({βj}≈ T ).

(3) C(C({αi}≈

T )) T C({αi}≈ T )

Zhe Lin1 and Mihir Kumar Chakraborty2 (Southwest University, Chongqing, China.1School of Cognitive Science, Jadavpur University, India2) The Finite Embeddability Property for Topological Quasi-Boolean Algebra 5 2019/03/03 19 / 27

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Interitor operation

We defined a interitor operation I on T/≈T as follows: I({αi}≈

T ) = {

  • 1≤j≤n

βj}≈

T

for any {βj}≈

T ∈ T •/≈T s.t. {βj}≈ T T {αi}≈ T .

Lemma For any {αi}≈

T , {βj}≈ T ∈ T/≈T , the following hold:

(1) I({αi}≈

T ) T {αi}≈ T .

(2) if {αi}≈

T T {βj}≈ T , then I({αi}≈ T ) T I({βj}≈ T ).

(3) I({αi}≈

T ) T I(I({αi}≈ T ))

Zhe Lin1 and Mihir Kumar Chakraborty2 (Southwest University, Chongqing, China.1School of Cognitive Science, Jadavpur University, India2) The Finite Embeddability Property for Topological Quasi-Boolean Algebra 5 2019/03/03 20 / 27

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Modal operations

We define a unary operations ♦ and on T/≈T : ♦{αi}≈

T = {αi+1}≈ T

{αi}≈

T = {ϕj}≈ T

s.t ϕj+1 ≈T αi We define two unary operation on T/≈T as follows: ({ϕ}≈

T ) = C(♦({ϕ}≈ T ))

({ϕ}≈

T ) = I(({ϕ}≈ T ))

Zhe Lin1 and Mihir Kumar Chakraborty2 (Southwest University, Chongqing, China.1School of Cognitive Science, Jadavpur University, India2) The Finite Embeddability Property for Topological Quasi-Boolean Algebra 5 2019/03/03 21 / 27

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Main Lemmas

Lemma For any {αi}≈

T , {βj}≈ T ∈ T/≈T , ♦C({αi}≈ T ) T C(♦{αi}≈ T ).

Lemma For any {ϕ}≈

T ∈ T/≈T , the following hold:

(1) ({ϕ}≈

T ) T ({ϕ}≈ T )

(2) ({ϕ}≈

T ) T ({ϕ}≈ T )

(3) If (({ϕ}≈

T )) T {ψ}≈ T , then (({¬ψ}≈ T )) T {¬ϕ}≈ T

Lemma For any {ϕ}≈

T , {ψ}≈ T ∈ T/≈T , ({ϕ}≈ T ) T {ψ}≈ T iff {ϕ}≈ T T {ψ}≈ T .

Zhe Lin1 and Mihir Kumar Chakraborty2 (Southwest University, Chongqing, China.1School of Cognitive Science, Jadavpur University, India2) The Finite Embeddability Property for Topological Quasi-Boolean Algebra 5 2019/03/03 22 / 27

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Finite algebras

For any {ϕ}≈

T , {ψ}≈ T ∈ T/≈T one defines:

{ϕ}≈

T ∧ {ψ}≈ T = {ϕ ∧ ψ}≈ T ,

{ϕ}≈

T ∨ {ψ}≈ T = {ϕ ∨ ψ}≈ T

¬{ϕ}≈

T = {¬ϕ}≈ T

Lemma A(T, Φ) = (T •/≈T , ∧, ∨, ¬, , ) is a finite tqBa5.

Zhe Lin1 and Mihir Kumar Chakraborty2 (Southwest University, Chongqing, China.1School of Cognitive Science, Jadavpur University, India2) The Finite Embeddability Property for Topological Quasi-Boolean Algebra 5 2019/03/03 23 / 27

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SFMP for tqBa5 I

We define a assignment σ from T-formulae to A(T, Φ) as follows: σ(p) = {p}≈

T for any p ∈ T.

Lemma σ(ϕ) = {ϕ}≈

T

Lemma ♦i{ϕ}≈

T T ♦i c{ϕ}≈ T

Lemma If Φ ⊢G5 ϕi ⇒T ψ, then Φ | =A(T, ) σ(ϕi) T σ(ψ)

Zhe Lin1 and Mihir Kumar Chakraborty2 (Southwest University, Chongqing, China.1School of Cognitive Science, Jadavpur University, India2) The Finite Embeddability Property for Topological Quasi-Boolean Algebra 5 2019/03/03 24 / 27

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SFMP fot tqBa 5 II

Theorem If Φ ⊢G5 ϕi ⇒ ψ then there exists a model (G, σ) s.t. G is finite tqBa5 such that all sequents in Φ is true while ϕi ⇒ ψ is not. Theorem The variety tqBa5 has SFMP

Zhe Lin1 and Mihir Kumar Chakraborty2 (Southwest University, Chongqing, China.1School of Cognitive Science, Jadavpur University, India2) The Finite Embeddability Property for Topological Quasi-Boolean Algebra 5 2019/03/03 25 / 27

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FEP

Theorem The variety tqBa5 has FEP

Zhe Lin1 and Mihir Kumar Chakraborty2 (Southwest University, Chongqing, China.1School of Cognitive Science, Jadavpur University, India2) The Finite Embeddability Property for Topological Quasi-Boolean Algebra 5 2019/03/03 26 / 27

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Zhe Lin1 and Mihir Kumar Chakraborty2 (Southwest University, Chongqing, China.1School of Cognitive Science, Jadavpur University, India2) The Finite Embeddability Property for Topological Quasi-Boolean Algebra 5 2019/03/03 27 / 27