Esakia Duality for Sugihara Monoids

Esakia duality for Sugihara monoids

Wesley Fussner (Joint work with Nick Galatos)

University of Denver Department of Mathematics Fall AMS Special Session on Algebraic Logic University of Denver

October 8, 2016

1 / 20

DRAFT 2 Existing dualities (Urquhart, 1996) incorporate the ternary accessibility relation of the Routley-Meyer semantics. Dunn (1976) provided a Kripke-style semantics for R-mingle using only a binary accessibility relation. Kripke semantics is underwritten by Esakia duality for Heyting algebras

Motivating Question: Can we construct an Esakia-style duality that lifts Dunn’s semantics to a categorical equivalence?

Motivation

Why another duality for relevant algebras?

2 / 20

DRAFT 2 Dunn (1976) provided a Kripke-style semantics for R-mingle using only a binary accessibility relation. Kripke semantics is underwritten by Esakia duality for Heyting algebras

Motivating Question: Can we construct an Esakia-style duality that lifts Dunn’s semantics to a categorical equivalence?

Motivation

Why another duality for relevant algebras? Existing dualities (Urquhart, 1996) incorporate the ternary accessibility relation of the Routley-Meyer semantics.

2 / 20

DRAFT 2 Kripke semantics is underwritten by Esakia duality for Heyting algebras

Motivating Question: Can we construct an Esakia-style duality that lifts Dunn’s semantics to a categorical equivalence?

Motivation

Why another duality for relevant algebras? Existing dualities (Urquhart, 1996) incorporate the ternary accessibility relation of the Routley-Meyer semantics. Dunn (1976) provided a Kripke-style semantics for R-mingle using only a binary accessibility relation.

2 / 20

DRAFT 2 Motivating Question: Can we construct an Esakia-style duality that lifts Dunn’s semantics to a categorical equivalence?

Motivation

2 / 20

DRAFT 2 Motivation

Motivating Question: Can we construct an Esakia-style duality that lifts Dunn’s semantics to a categorical equivalence?

2 / 20

DRAFT 2 A Priestley space is called an Esakia space if whenever U is clopen, the down-set ↓U is also clopen. An Esakia function is a map f : X → Y between Esakia spaces that is continuous, order-preserving, and satisﬁes ↑f (x) ⊆ f [↑x] for every x ∈ X .

Priestley and Esakia duality

Deﬁnitions: An ordered topological space (X , ≤, T ) is called a Priestley space if (X , T ) is compact, and whenever x, y ∈ X with x 6≤ y there exists a clopen up-set U ⊆ X with x ∈ U, y ∈/ U.

3 / 20

DRAFT 2 An Esakia function is a map f : X → Y between Esakia spaces that is continuous, order-preserving, and satisﬁes ↑f (x) ⊆ f [↑x] for every x ∈ X .

Priestley and Esakia duality

3 / 20

DRAFT 2 Priestley and Esakia duality

Deﬁnitions: An ordered topological space (X , ≤, T ) is called a Priestley space if (X , T ) is compact, and whenever x, y ∈ X with x 6≤ y there exists a clopen up-set U ⊆ X with x ∈ U, y ∈/ U. A Priestley space is called an Esakia space if whenever U is clopen, the down-set ↓U is also clopen. An Esakia function is a map f : X → Y between Esakia spaces that is continuous, order-preserving, and satisﬁes ↑f (x) ⊆ f [↑x] for every x ∈ X .

3 / 20

DRAFT 2 Theorem (Esakia): The category of Heyting algebras (with morphisms the algebraic homomorphisms) is dually equivalent to the category of Esakia spaces with morphisms the Esakia functions.

The Esakia duality is a restricted form of the Priestley duality.

Priestley and Esakia duality

Theorem (Priestley): The category of bounded distributive lattices (with morphisms the algebraic homomorphisms) is dually equivalent to the category of Priestley spaces with morphisms the continuous, isotone maps.

4 / 20

DRAFT 2 The Esakia duality is a restricted form of the Priestley duality.

Priestley and Esakia duality

Theorem (Esakia): The category of Heyting algebras (with morphisms the algebraic homomorphisms) is dually equivalent to the category of Esakia spaces with morphisms the Esakia functions.

4 / 20

DRAFT 2 Priestley and Esakia duality

Theorem (Esakia): The category of Heyting algebras (with morphisms the algebraic homomorphisms) is dually equivalent to the category of Esakia spaces with morphisms the Esakia functions.

The Esakia duality is a restricted form of the Priestley duality.

4 / 20

DRAFT 2 (A, ·, t) is a commutative monoid, and for all a, b, c ∈ A,

a · b ≤ c ⇐⇒ a ≤ b → c

(A, ∧, ∨) is a lattice,

Sugihara monoids

Deﬁnition: A commutative residuated lattice (CRL) is an algebra (A, ∧, ∨, ·, →, t) such that

5 / 20

DRAFT 2 for all a, b, c ∈ A,

a · b ≤ c ⇐⇒ a ≤ b → c

(A, ·, t) is a commutative monoid, and

Sugihara monoids

Deﬁnition: A commutative residuated lattice (CRL) is an algebra (A, ∧, ∨, ·, →, t) such that (A, ∧, ∨) is a lattice,

5 / 20

DRAFT 2 for all a, b, c ∈ A,

a · b ≤ c ⇐⇒ a ≤ b → c

Sugihara monoids

Deﬁnition: A commutative residuated lattice (CRL) is an algebra (A, ∧, ∨, ·, →, t) such that (A, ∧, ∨) is a lattice, (A, ·, t) is a commutative monoid, and

5 / 20

DRAFT 2 Sugihara monoids

Deﬁnition: A commutative residuated lattice (CRL) is an algebra (A, ∧, ∨, ·, →, t) such that (A, ∧, ∨) is a lattice, (A, ·, t) is a commutative monoid, and for all a, b, c ∈ A,

a · b ≤ c ⇐⇒ a ≤ b → c

5 / 20

DRAFT 2 distributive if its lattice reduct is distributive, idempotent if it satisﬁes the identity a2 = a.

integral if t is the greatest element with respect to the lattice order,

Deﬁnition: The expansion of a CRL by a unary operation ¬ satisfying ¬¬a = a and a → ¬b = b → ¬a is called an involutive CRL. A distributive, idempotent, involutive CRL is called a Sugihara monoid.

For simplicity, we consider expansions of Sugihara monoids by universal bounds ⊥ and >.

Sugihara monoids

Deﬁnition: A CRL is called

6 / 20

DRAFT 2 idempotent if it satisﬁes the identity a2 = a.

distributive if its lattice reduct is distributive,

For simplicity, we consider expansions of Sugihara monoids by universal bounds ⊥ and >.

Sugihara monoids

Deﬁnition: A CRL is called integral if t is the greatest element with respect to the lattice order,

6 / 20

DRAFT 2 idempotent if it satisﬁes the identity a2 = a.

For simplicity, we consider expansions of Sugihara monoids by universal bounds ⊥ and >.

Sugihara monoids

Deﬁnition: A CRL is called integral if t is the greatest element with respect to the lattice order, distributive if its lattice reduct is distributive,

6 / 20

DRAFT 2 Deﬁnition: The expansion of a CRL by a unary operation ¬ satisfying ¬¬a = a and a → ¬b = b → ¬a is called an involutive CRL. A distributive, idempotent, involutive CRL is called a Sugihara monoid.

For simplicity, we consider expansions of Sugihara monoids by universal bounds ⊥ and >.

Sugihara monoids

Deﬁnition: A CRL is called integral if t is the greatest element with respect to the lattice order, distributive if its lattice reduct is distributive, idempotent if it satisﬁes the identity a2 = a.

6 / 20

DRAFT 2 For simplicity, we consider expansions of Sugihara monoids by universal bounds ⊥ and >.

Sugihara monoids

6 / 20

DRAFT 2 Sugihara monoids

For simplicity, we consider expansions of Sugihara monoids by universal bounds ⊥ and >.

6 / 20

DRAFT 2 ¬ satisﬁes the identities ¬¬a = a, ¬⊥ = >, ¬(a ∧ b) = ¬a ∨ ¬b, a ∧ ¬a ≤ b ∨ ¬b.

(A, ∧, ∨, ⊥, >) is a bounded distributive lattice, and

Proposition: The (∧, ∨, ¬, ⊥, >)-reduct of every bounded Sugihara monoid is a Kleene algebra.

Kleene algebras

Deﬁnition: A Kleene algebra is an algebra (A, ∧, ∨, ¬, ⊥, >) such that

7 / 20

DRAFT 2 ¬ satisﬁes the identities ¬¬a = a, ¬⊥ = >, ¬(a ∧ b) = ¬a ∨ ¬b, a ∧ ¬a ≤ b ∨ ¬b.

Proposition: The (∧, ∨, ¬, ⊥, >)-reduct of every bounded Sugihara monoid is a Kleene algebra.

Kleene algebras

Deﬁnition: A Kleene algebra is an algebra (A, ∧, ∨, ¬, ⊥, >) such that (A, ∧, ∨, ⊥, >) is a bounded distributive lattice, and

7 / 20

DRAFT 2 Proposition: The (∧, ∨, ¬, ⊥, >)-reduct of every bounded Sugihara monoid is a Kleene algebra.

Kleene algebras

Deﬁnition: A Kleene algebra is an algebra (A, ∧, ∨, ¬, ⊥, >) such that (A, ∧, ∨, ⊥, >) is a bounded distributive lattice, and ¬ satisﬁes the identities ¬¬a = a, ¬⊥ = >, ¬(a ∧ b) = ¬a ∨ ¬b, a ∧ ¬a ≤ b ∨ ¬b.

7 / 20

DRAFT 2 Kleene algebras

Proposition: The (∧, ∨, ¬, ⊥, >)-reduct of every bounded Sugihara monoid is a Kleene algebra.

7 / 20

DRAFT 2 Proposition (Kalman, 1958): The Kleene algebras are generated as a prevariety by K.

As a consequence, the Sugihara monoids have reducts in a prevariety generated by a single ﬁnite algebra. This suggests that a duality for Sugihara monoids could be obtained by restricting a natural duality for their reducts, just as in the case of Heyting algebra.

Kleene algebras

Example: We may deﬁne a Kleene algebra K = ({−1, 0, 1}, ∧, ∨, ¬, −1, 1), where −1 < 0 < 1 and ¬ is the additive inversion −.

8 / 20

DRAFT 2 As a consequence, the Sugihara monoids have reducts in a prevariety generated by a single ﬁnite algebra. This suggests that a duality for Sugihara monoids could be obtained by restricting a natural duality for their reducts, just as in the case of Heyting algebra.

Kleene algebras

Example: We may deﬁne a Kleene algebra K = ({−1, 0, 1}, ∧, ∨, ¬, −1, 1), where −1 < 0 < 1 and ¬ is the additive inversion −.

Proposition (Kalman, 1958): The Kleene algebras are generated as a prevariety by K.

8 / 20

DRAFT 2 This suggests that a duality for Sugihara monoids could be obtained by restricting a natural duality for their reducts, just as in the case of Heyting algebra.

Kleene algebras

Example: We may deﬁne a Kleene algebra K = ({−1, 0, 1}, ∧, ∨, ¬, −1, 1), where −1 < 0 < 1 and ¬ is the additive inversion −.

Proposition (Kalman, 1958): The Kleene algebras are generated as a prevariety by K.

As a consequence, the Sugihara monoids have reducts in a prevariety generated by a single ﬁnite algebra.

8 / 20

DRAFT 2 Kleene algebras

Example: We may deﬁne a Kleene algebra K = ({−1, 0, 1}, ∧, ∨, ¬, −1, 1), where −1 < 0 < 1 and ¬ is the additive inversion −.

Proposition (Kalman, 1958): The Kleene algebras are generated as a prevariety by K.

8 / 20

DRAFT 2 Example:

We may deﬁne a Kleene space K = ({−1, 0, 1}, v, Q, K0, T ), where −1 v 0 and 1 v 0, −1 ande 1 are incomparable, K0 = {−1, 1}, and Q is the relation of comparability with respect to the order.

Davey-Werner duality

Deﬁnition:

A structure X = (X , ≤, Q, X0, T ) is called a Kleene space if (X , ≤, T ) is a Priestley space, Q is a closed binary relation on X , X0 is a closed subspace, and for all x, y, z ∈ X , xQx,

xQy and x ∈ X0 =⇒ x ≤ y, xQy and y ≤ z =⇒ zQx.

9 / 20

DRAFT 2 Davey-Werner duality

Deﬁnition:

A structure X = (X , ≤, Q, X0, T ) is called a Kleene space if (X , ≤, T ) is a Priestley space, Q is a closed binary relation on X , X0 is a closed subspace, and for all x, y, z ∈ X , xQx,

xQy and x ∈ X0 =⇒ x ≤ y, xQy and y ≤ z =⇒ zQx.

Example:

9 / 20

DRAFT 2 Given a Kleene algebra A, let D(A) be the collection of Kleene algebra homomorphisms A → K endowed with structure inherited pointwise from K. For a morphism h: A → B in K, let D(h): D(B) → eD(A) be deﬁned by D(h)(x) = x ◦ h. Given a Kleene space X, let E(X) be the collection of Kleene space morphisms X → K endowed with structure inherited pointwise from K. For a morphisme ϕ: X → Y in KS, let E(ϕ): E(Y) → E(X) be deﬁned by E(ϕ)(α) = α ◦ ϕ.

Davey-Werner duality

Proposition (Davey-Werner, 1983): The category of Kleene algebras K is dually equivalent to the category of Kleene spaces KS (with morphisms the continuous, structure-preserving maps in the signature (≤, Q, X0)).

10 / 20

DRAFT 2 Given a Kleene space X, let E(X) be the collection of Kleene space morphisms X → K endowed with structure inherited pointwise from K. For a morphisme ϕ: X → Y in KS, let E(ϕ): E(Y) → E(X) be deﬁned by E(ϕ)(α) = α ◦ ϕ.

Davey-Werner duality

10 / 20

DRAFT 2 Davey-Werner duality

Given a Kleene algebra A, let D(A) be the collection of Kleene algebra homomorphisms A → K endowed with structure inherited pointwise from K. For a morphism h: A → B in K, let D(h): D(B) → eD(A) be deﬁned by D(h)(x) = x ◦ h. Given a Kleene space X, let E(X) be the collection of Kleene space morphisms X → K endowed with structure inherited pointwise from K. For a morphisme ϕ: X → Y in KS, let E(ϕ): E(Y) → E(X) be deﬁned by E(ϕ)(α) = α ◦ ϕ.

10 / 20

DRAFT 2 Deﬁnition:

We call a Kleene space X = (X , ≤, Q, X0, T ) a Sugihara space if (X , ≤, T ) is an Esakia space,

X0 is open, Q coincides with ≤ ∪ ≥ (i.e., comparability with respect to the partial order ≤).

We deﬁne operations on the dual of a Sugihara space X in order to make this structure into a Sugihara monoid.

Sugihara spaces

We restrict the Davey-Werner duality to obtain a duality for the Sugihara monoids.

11 / 20

DRAFT 2 We deﬁne operations on the dual of a Sugihara space X in order to make this structure into a Sugihara monoid.

Sugihara spaces

We restrict the Davey-Werner duality to obtain a duality for the Sugihara monoids. Deﬁnition:

We call a Kleene space X = (X , ≤, Q, X0, T ) a Sugihara space if (X , ≤, T ) is an Esakia space,

X0 is open, Q coincides with ≤ ∪ ≥ (i.e., comparability with respect to the partial order ≤).

11 / 20

DRAFT 2 Sugihara spaces

We restrict the Davey-Werner duality to obtain a duality for the Sugihara monoids. Deﬁnition:

We call a Kleene space X = (X , ≤, Q, X0, T ) a Sugihara space if (X , ≤, T ) is an Esakia space,

X0 is open, Q coincides with ≤ ∪ ≥ (i.e., comparability with respect to the partial order ≤).

We deﬁne operations on the dual of a Sugihara space X in order to make this structure into a Sugihara monoid.

11 / 20

DRAFT 2 Let X be a Sugihara space. We deﬁne binary operations · and on the collection of Kleene space morphisms from X to K as follows. e

Kleene space morphisms

Given a Kleene space X, a morphism α: X → K may be represented uniquely as a map of the form e  1, if x ∈/ V  ϕ(U,V )(x) = 0, if x ∈ U ∩ V −1, if x ∈/ U

where U, V are clopen up-sets of X satisfying U ∪ V = X , U ∩ V ⊆ X − X0, and ((X − U) × (X − V )) ∩ Q = ∅.

12 / 20

DRAFT 2 Kleene space morphisms

Given a Kleene space X, a morphism α: X → K may be represented uniquely as a map of the form e  1, if x ∈/ V  ϕ(U,V )(x) = 0, if x ∈ U ∩ V −1, if x ∈/ U

where U, V are clopen up-sets of X satisfying U ∪ V = X , U ∩ V ⊆ X − X0, and ((X − U) × (X − V )) ∩ Q = ∅.

Let X be a Sugihara space. We deﬁne binary operations · and on the collection of Kleene space morphisms from X to K as follows. e

12 / 20

DRAFT 2 We deﬁne a multiplication · on pairs hU1, V1i, hU2, V2i of clopen up-sets of X by hU1, V1i · hU2, V2i = hU3, V3i, where

U3 = [(U1 ⇒ V2) ∩ (U2 ⇒ V1)] → (U1 ∩ U2),

and

V3 =[(U1 ⇒ V2) ∩ (U2 ⇒ V1)] ∗ ∩ [((U1 ⇒ V2) ∩ (U2 ⇒ V1)) → (U1 ∩ U2)]

New operations

The clopen up-sets of X form a Heyting algebra with → deﬁned by

U → V = {x ∈ X : ↑x ∩ U ⊆ V }

Deﬁne also U ⇒ V = [U ∩ (X − X0)] → V , and ∗ U = U → (X − X0).

13 / 20

DRAFT 2 New operations

The clopen up-sets of X form a Heyting algebra with → deﬁned by

U → V = {x ∈ X : ↑x ∩ U ⊆ V }

Deﬁne also U ⇒ V = [U ∩ (X − X0)] → V , and ∗ U = U → (X − X0).

We deﬁne a multiplication · on pairs hU1, V1i, hU2, V2i of clopen up-sets of X by hU1, V1i · hU2, V2i = hU3, V3i, where

U3 = [(U1 ⇒ V2) ∩ (U2 ⇒ V1)] → (U1 ∩ U2),

and

V3 =[(U1 ⇒ V2) ∩ (U2 ⇒ V1)] ∗ ∩ [((U1 ⇒ V2) ∩ (U2 ⇒ V1)) → (U1 ∩ U2)]

13 / 20

DRAFT 2 New operations

Deﬁne also a new implication on pairs hU1, V1i, hU2, V2i of clopen up-sets by hU1, V1i hU2, V2i = hU3, V3i, where

U3 = (U1 → U2) ∩ (V2 ⇒ V1),

and

V3 =[((U1 → U2) ∩ (V2 ⇒ V1)) ⇒ (U1 ∩ (X ⇒ V2)] ∗ ∩ [(U1 → U2) ∩ (V2 ⇒ V1)]

14 / 20

DRAFT 2 New operations

Finally, for morphisms ϕ(U1,V1), ϕ(U2,V2) : X → K, deﬁne e

ϕ(U1,V1) · ϕ(U2,V2) = ϕ(U1,V1)·(U2,V2)

and ϕ ϕ = ϕ (U1,V1) (U2,V2) (U1,V1) (U2,V2)

15 / 20

DRAFT 2 Define a functor F : SM → SS as follows. For a Sugihara monoid A, let F (A) be the Davey-Werner dual of the Kleene algebra reduct of A. For a morphism h: A → B, let F (h): F (B) → F (A) be defined by F (h)(x) = x ◦ h. Define a functor G : SS → SM as follows. For a Sugihara space X, let G(X) be the Davey-Werner dual of X endowed with the additional binary operations · and , and the nullary operation

ϕ(X ,X −X0). For a morphism ϕ: X → Y of SS, deﬁne G(ϕ): G(Y) → G(X) by G(ϕ)(α) = α ◦ ϕ.

The equivalence

16 / 20

DRAFT 2 Deﬁne a functor G : SS → SM as follows. For a Sugihara space X, let G(X) be the Davey-Werner dual of X endowed with the additional binary operations · and , and the nullary operation

ϕ(X ,X −X0). For a morphism ϕ: X → Y of SS, deﬁne G(ϕ): G(Y) → G(X) by G(ϕ)(α) = α ◦ ϕ.

The equivalence

Let SM denote the category of bounded Sugihara monoids with morphisms the algebraic homomorphisms. Let SS denote the category whose objects are Sugihara spaces (X , ≤, ≤ ∪ ≥, X0, T ), and whose morphisms are Esakia functions preserving X0. Deﬁne a functor F : SM → SS as follows. For a Sugihara monoid A, let F (A) be the Davey-Werner dual of the Kleene algebra reduct of A. For a morphism h: A → B, let F (h): F (B) → F (A) be deﬁned by F (h)(x) = x ◦ h.

16 / 20

DRAFT 2 The equivalence

ϕ(X ,X −X0). For a morphism ϕ: X → Y of SS, deﬁne G(ϕ): G(Y) → G(X) by G(ϕ)(α) = α ◦ ϕ.

16 / 20

DRAFT 2 A Sugihara space X is, inter alia, an Esakia space. What happens if instead of taking its Kleene dual, we take its Esakia dual? We obtain the Heyting algebra that forms the negative cone of the Sugihara monoid F (X). Along these lines, the choice of taking the Kleene dual of such an Esakia space can be thought of as a topological construction of the twist product used in paraconsistent logics. This reﬂects recent results by Galatos and Raftery that the Sugihara monoids are (covariantly) equivalent to their negative cones.

Main result

Main Theorem: The functors F and G witness a dual equivalence of categories between SM and SS.

17 / 20

DRAFT 2 We obtain the Heyting algebra that forms the negative cone of the Sugihara monoid F (X). Along these lines, the choice of taking the Kleene dual of such an Esakia space can be thought of as a topological construction of the twist product used in paraconsistent logics. This reﬂects recent results by Galatos and Raftery that the Sugihara monoids are (covariantly) equivalent to their negative cones.

Main result

Main Theorem: The functors F and G witness a dual equivalence of categories between SM and SS.

A Sugihara space X is, inter alia, an Esakia space. What happens if instead of taking its Kleene dual, we take its Esakia dual?

17 / 20

DRAFT 2 Along these lines, the choice of taking the Kleene dual of such an Esakia space can be thought of as a topological construction of the twist product used in paraconsistent logics. This reﬂects recent results by Galatos and Raftery that the Sugihara monoids are (covariantly) equivalent to their negative cones.

Main result

Main Theorem: The functors F and G witness a dual equivalence of categories between SM and SS.

17 / 20

DRAFT 2 This reﬂects recent results by Galatos and Raftery that the Sugihara monoids are (covariantly) equivalent to their negative cones.

Main result

Main Theorem: The functors F and G witness a dual equivalence of categories between SM and SS.

A Sugihara space X is, inter alia, an Esakia space. What happens if instead of taking its Kleene dual, we take its Esakia dual? We obtain the Heyting algebra that forms the negative cone of the Sugihara monoid F (X). Along these lines, the choice of taking the Kleene dual of such an Esakia space can be thought of as a topological construction of the twist product used in paraconsistent logics.

17 / 20

DRAFT 2 Main result

Main Theorem: The functors F and G witness a dual equivalence of categories between SM and SS.

17 / 20

DRAFT 2 Thank you!

Thank you!

18 / 20

DRAFT 2 Bibliography

D.M. Clark and B.A. Davey, Natural Dualities for the Working Algebraist, Cambridge University Press, 1998. B.A. Davey and H. Werner, Dualities and equivalences for varieties of algebras, Contributions to Lattice Theory (Szeged, 1980) (A.P. Huhn and E.T. Schmidt, eds), Colloq. Math. Soc. J´anos Bolyai 33, North-Holland, Amsterdam, New York, 1983, pp. 101–275. J.M. Dunn, A Kripke-style semantics for R-mingle using a binary accessibility relation, Studia Logica 35 (1976), no. 2, 163–172. L. Esakia, Topological Kripke Models, Soviet Math. Dokl., 15:147–151, 1974. N. Galatos and J.G. Raftery, A category equivalence for odd Sugihara monoids and its applications, J. Pure Appl. Algebra 216 (2012), 2177–2192.

19 / 20

DRAFT 2 Bibliography Continued.

N. Galatos and J.G. Raftery, Idempotent residuated structures: Some category equivalences and their applications, Trans. Amer. Math. Soc. 367 (2015), 3189–3223. J.A. Kalman, Lattices with Involution, Trans. Amer. Math. Soc. 87, (1958), 485–491. H. Priestley, Representation of distributive lattices by means of ordered Stone spaces, Bull. London Math. Soc. 2 (1979), 186–190 A. Urquhart, Duality for Algebras of Relevant Logics, Studia Logica, 56 (1996) 253–276.

20 / 20

DRAFT 2