Varieties of Heyting algebras and superintuitionistic logics

Varieties of Heyting algebras andsuperintuitionistic logics

Nick BezhanishviliInstitute for Logic, Language and Computation

University of Amsterdamhttp://www.phil.uu.nl/~bezhanishvili

email: [email protected]

http://www.phil.uu.nl/~bezhanishvili

Heyting algebras

A Heyting algebra is a bounded distributive lattice (A,∧,∨,0,1)equipped with a binary operation→, which is a right adjoint of∧. This means that for each a, b, x ∈ A we have

a ∧ x ≤ b iff x ≤ a→ b.

Heyting algebras

Heyting algebras pop up in different areas of mathematics.

1 Logic: Heyting algebras are algebraic models ofintuitionistic logic.

2 Topology: opens of any topological space form a Heytingalgebra.

3 Geometry: open subpolyhydra of any polyhedron form aHeyting algebra.

4 Category theory: subobject classifier of any topos is aHeyting algebra.

5 Universal algebra: lattice of all congruences of any latticeis a Heyting algebra.

Outline

The goal of the tutorial is to give an insight into the complicatedstructure of the lattice of varieties of Heyting algebras.

The outline of the tutorial:

1 Heyting algebras and superintuitionistic logics

2 Representation of Heyting algebras

3 Hosoi classification of the lattice of varieties of Heytingalgebras

4 Jankov formulas and splittings

5 Canonical formulas

Part 1: Heyting algebras and superintuitionistic logics

Constructive reasoning

One of the cornerstones of classical reasoning is the law ofexcluded middle p ∨ ¬p.

Constructive viewpoint: Truth = Proof.

The law of excluded middle p ∨ ¬p is constructivelyunacceptable.

For example, we do not have a proof of Goldbach’s conjecturenor are we able to show that this conjecture does not hold.

Constructive reasoning

On the grounds that the only accepted reasoning should beconstructive, the dutch mathematician L. E. J. Brouwer rejectedclassical reasoning.

Luitzen Egbertus Jan Brouwer (1881 - 1966)

Intuitionistic logic

In 1930’s Brouwer’s ideas led his student Heyting to introduceintuitionistic logic which formalizes constructive reasoning.

Arend Heyting (1898 - 1980)

Intuitionistic logic

Roughly speaking, the axiomatization of intuitionistic logic isobtained by dropping the law of excluded middle from theaxiomatization of classical logic.

CPC = classical propositional calculusIPC = intuitionistic propositional calculus.

The law of excluded middle is not derivable in intuitionisticlogic. So IPC ( CPC.

In fact,CPC = IPC + (p ∨ ¬p).

There are many logics in between IPC and CPC

Superintuitionistic logics

A superintuitionistic logic is a set of formulas containing IPCand closed under the rules of substitution and Modus Ponens.

Superintuitionistic logics contained in CPC are often calledintermediate logics because they are situated between IPC andCPC.

As we will see, intermediate logics are exactly the consistentsuperintuitionistic logics.

Since we are interested in consistent logics, we will mostlyconcentrate on intermediate logics.

Intermediate logics

IPC

CPC

KCKC = IPC + (¬p ∨ ¬¬p)

weak law of excluded middle

LCLC = IPC + (p→ q) ∨ (q→ p)

Godel-Dummett calculus

Equational theories of Heyting algebras

Each formula ϕ in the language of IPC corresponds to anequation ϕ ≈ 1 in the theory of Heyting algebras.

Conversely, each equation ϕ ≈ ψ can be rewritten asϕ↔ ψ ≈ 1, which corresponds to the formula ϕ↔ ψ.

This yields a one-to-one correspondence betweensuperintuitionistic logics and equational theories of Heytingalgebras.

Varieties of Heyting algebras

By the celebrated Birkhoff theorem, equational theoriescorrespond to varieties; that is, classes of algebras closed undersubalgebras, homomorphic images, and products.

Garrett Birkhoff (1911 - 1996)

Varieties of Heyting algebras

Thus, superintuitionistic logics correspond to varieties ofHeyting algebras, while intermediate logics to non-trivialvarieties of Heyting algebras.

Heyt = the variety of all Heyting algebras.

Bool = the variety of all Boolean algebras.

Λ(IPC) = the lattice of superintuitionistic logics.

Λ(Heyt) = the lattice of varieties of Heyting algebras.

Theorem. Λ(IPC) is dually isomorphic to Λ(Heyt).

Consequently, we can investigate superintuitionistic logics bymeans of their corresponding varieties of Heyting algebras.

Part 2: Representation of Heyting algebras

First typical example of a Heyting algebra

Open sets of any topological space X form a Heyting algebra,where for open Y,Z ⊆ X:

Y → Z = Int(Yc ∪ Z), ¬Y = Int(Yc).

R

0 Y¬Y

Y ∨ ¬Y 6= R

Stone Representation

Theorem (Stone, 1937). Every Heyting algebra can beembedded into the Heyting algebra of open sets of sometopological space.

Marshall Stone (1903 - 1989)

Stone representation

For every Heyting algebra A let XA be the set of prime filters of A.

The Stone map ϕ : A→ P(XA) is given by

ϕ(a) = x ∈ XA : a ∈ x.

Let ΩA be the topology generated by the basis ϕ(a) : a ∈ A.

Theorem. ϕ : A→ ΩA is a Heyting algebra embedding.

Second typical example of a Heyting algebra

Up-sets of any poset (X,≤) form a Heyting algebra where forup-sets U,V ⊆ X:

U → V = X − ↓(U − V), ¬U = X − ↓U

Here U is an up-set if x ∈ U and x ≤ y imply y ∈ U and

↓U = x ∈ X : ∃y ∈ U with x ≤ y.

Second typical example of a Heyting algebra

g ¬g

¬¬g ¬¬g→ g0

g ¬g

g ∨ ¬g¬¬g

¬¬g ∨ ¬g ¬¬g→ g

1

Kripke Representation

Theorem (Kripke, 1965). Every Heyting algebra can beembedded into the Heyting algebra of up-sets of some poset.

Saul Kripke

Kripke representation

For every Heyting algebra A, order the set XA of prime filters ofA by set-theoretic inclusion.

For a poset X let Up(X) be the Heyting algebra of up-sets of X.

Theorem. The Stone map ϕ : A→ Up(XA) is a Heyting algebraembedding.

We want to characterize the ϕ-image of A.

For this we will define a topology on XA and characterize thisimage in order-topological terms.

This topology will be the so-called patch topology of ΩA.

Esakia duality

This approach was developed by Esakia in the 1970’s.

Leo Esakia (1934 - 2010)

Esakia duality

An Esakia space is a pair (X,≤), where:

1 X is a Stone space (compact, Hausdorff, zero-dimensional).

2 (X,≤) is a poset.

3 ↑x is closed for each x ∈ X. Here ↑x = y ∈ X : x ≤ y.

4 If U is clopen (closed and open), then so is ↓U. Recall that↓U = x ∈ X : ∃y ∈ U with x ≤ y.

Esakia duality

Given an Esakia space (X,≤) we take the Heyting algebra(CpUp(X),∩,∪,→, ∅,X) of all clopen up-sets of X, where forU,V ∈ CpUp(X):

U → V = X − ↓(U − V).

For each Heyting algebra A we take the set XA of prime filters ofA ordered by inclusion and topologized by the subbasis

ϕ(a) : a ∈ A ∪ ϕ(a)c : a ∈ A.

Alternatively we can take ϕ(a)− ϕ(b) : a, b ∈ A as a basis forthe topology.

Esakia Duality

Theorem.

1 For each Heyting algebra A the map ϕ : A→ CpUp(XA) is aHeyting algebra isomorphism.

2 For each Esakia space X, there is an order-hemeomorphismbetween X and XCpUp(X).

This is the object part of the duality between the category ofHeyting algebras and Heyting algebra homomorphisms and thecategory of Esakia spaces and Esakia morphisms.

Priestley spaces

Order-topological representation of bounded distributive latticeswas developed by Priestley in the 1970s.

Hilary Priestley

Priestley spaces

In each Esakia space the following Priestley separation holds:

x 6≤ y implies there is a clopen up-set U such that x ∈ U andy /∈ U.

Thus, every Esakia space is a Priestley space, but not vice versa.

It follows that Esakia duality is a restricted version of Priestleyduality.

Recap

1 The lattice of superintuitionistic logics is dually isomorphicto the lattice of varieties of Heyting algebras.

2 Stone Representation: Every Heyting algebra can beembedded into the Heyting algebra of open sets of sometopological space.

3 Kripke Representation: Every Heyting algebra can beembedded into the Heyting algebra of up-sets of someposet.

4 Esakia Representation: Every Heyting algebra is isomorphicto the Heyting algebra of clopen up-sets of some Esakiaspace.

Part 3: Depth and Hosoi classification

Depth of Heyting algebrasLet (X,≤) be a poset.

1 We say that X is of depth n > 0, denoted d(X) = n, if thereis a chain of n points in X and no other chain in X containsmore than n points. The poset X is of finite depth ifd(X) = n for some n > 0.

2 We say that X is of infinite depth, denoted d(X) = ω, if forevery n ∈ ω, X contains a chain consisting of n points.

Depth is also referred to as height.

Let A be a Heyting algebra.

The depth d(A) of A = the depth of the dual of A.

Let V be a variety of Heyting algebras.

The depth d(V) of V = supd(A) : A ∈ V.

ChainsLet Cn be the n-element chain.

Cn

0

...

1

Cn is a Heyting algebra, where

a→ b =

1 if a ≤ b,b otherwise.

d(Cn+1) = n

Varieties of depth n

For a class K of Heyting algebras, let Var(K) be the variety ofHeyting algebras generated by K.

Let Lin be the variety generated by all finite chains.

Let also Dn be the class of all Heyting algebras of depth n.

We will see later that each Dn forms a variety.

Rough picture of the lattice

Bool

Var(C3)

D2

Var(C4)

D3...

Lin

Heyt

...

Part 4: Jankov formulas and the cardinality of the lattice ofvarieties

Filters and congruences

As in Boolean algebras, the lattice of filters of a Heyting algebrais isomorphic to the lattice of congruences.

To each filter F corresponds the congruence θF defined by

aθFb if a↔ b ∈ F.

To each congruence θ corresponds the filter

Fθ = a ∈ A : aθ1.

Consequently, the variety of Heyting algebras is congruencedistributive and has the congruence extension property.

Subdirectly irreducible Heyting algebrasBy another theorem of Birkhoff, every variety of algebras isgenerated by its subdirectly irreducible members.

Theorem (Jankov, 1963). A Heyting algebra is subdirectlyirreducible (s.i. for short) if it has a second largest element.

0

s

1

Esakia duals of s.i. Heyting algebras

0

s

1

If a Heyting algebra A is s.i., then the dual of A has a leastelement, a root.

If an Esakia space is rooted and the root is an isolated point,then its dual Heyting algebra is s.i.

Jankov formulas

Let A be a finite subdirectly irreducible Heyting algebra, s thesecond largest element of A.

For each a ∈ A we introduce a new variable pa and define theJankov formula χ(A) as the (∧,∨,→,0,1)-description of thisalgebra.

χ(A) = [∧pa∧b ↔ pa ∧ pb : a, b ∈ A∧∧pa∨b ↔ pa ∨ pb : a, b ∈ A∧∧pa→b ↔ pa → pb : a, b ∈ A∧∧p¬a ↔ ¬pa : a ∈ A]→ ps

If we interpret pa as a, then the Jankov formula of A is equal inA to s, i.e., it is pre-true in A .

Axiomatization of varieties of Heyting algebras

Theorem (Jankov, 1963). Let B a Heyting algebra. Then

B 6|= χ(A) iff A ∈ SH(B).

Dimitri Jankov


Theorem (Jankov, 1963). Let B a Heyting algebra. Then

B 6|= χ(A) iff A ∈ SH(B).

Proof. (Sketch). Suppose B 6|= χ(A). Then there exists a s.i.homomorphic image C of B such that C 6|= χ(A). Moreover χ(A)is pre-true in C. This means that there is a valuation v on C suchthat

v(∧pa∧b ↔ pa ∧ pb : a, b ∈ A∧∧pa∨b ↔ pa ∨ pb : a, b ∈ A∧∧pa→b ↔ pa → pb : a, b ∈ A∧∧p¬a ↔ ¬pa : a ∈ A) = 1C

andv(ps) = sC


Therefore, for all a, b ∈ A we have:

v(pa∧b) = v(pa) ∧ v(pb)v(pa∨b) = v(pa) ∨ v(pb)v(pa→b) = v(pa)→ v(pb)v(p¬a) = ¬v(pa)v(ps) = sC

We consider the map h : A→ C given by h(a) = v(pa).

Then h is a Heyting embedding.

Conversely, as A 6|= χ(A) and A ∈ SH(B) we see that B 6|= χ(A).

Splittings

Jankov formulas are used to axiomatize many varieties ofHeyting algebras.

For example, they axiomatize all splitting varieties of Heytingalgebras.

Splittings started to play an important role in lattice theory inthe 1940s.

A pair (a, b) splits a lattice L if a b and for each c ∈ L:

a ≤ c or c ≤ b

Splittings

R. McKenzie in the 1970’s revisited splittings when he started anextensive study of lattices of varieties.

Ralph McKenzie

Splittings

Var(A)

Bool

Heyt + χ(A)

Heyt

Figure: Splitting of the lattice of varieties of Heyting algebras

Splittings

Theorem. For each subdirectly irreducible Heyting algebra Athe pair (Var(A),Heyt + χ(A)) splits the lattice of varieties ofHeyting algebras.

Proof. (Sketch) Since A 6|= χ(A), we see thatVar(A) 6⊆ Heyt + χ(A).

Suppose V is a variety such that V 6⊆ Heyt + χ(A).

Then there is B ∈ V such that B 6|= χ(A).

By Jankov’s theorem, A ∈ SH(B) and so Var(A) ⊆ V.

The other direction follows from a result of McKenzie (1972).


Bool

Var(C3)

Heyt + χ(C4)

Var(C4)

Heyt + χ(C5)...

Lin

Heyt

...

Continuum of varieties of Heyting algebras

Let A and B be s.i. Heyting algebras. We write A ≤ B ifA ∈ SH(B).

Theorem. If ∆ is an ≤-antichain of finite s.i. algebras, then foreach I, J ⊆ ∆ with I 6= J, we have

Heyt + χ(A) : A ∈ I 6= Heyt + χ(A) : A ∈ J.

Proof. (Sketch) If I 6⊆ J, then there is B ∈ I such that B /∈ J.

Then A 6≤ B for each A ∈ J. Therefore, by Jankov’s theorem,B |= χ(A) for each A ∈ J.

So B ∈ Heyt + χ(A) : A ∈ J.

But B 6|= χ(B). So B /∈ Heyt + χ(A) : A ∈ I.

How can we construct an ≤-antichain of finite s.i. algebras?

Antichains

. . .

Lemma. ∆1 is an ≤-antichain.

Antichains

. . .

Lemma. ∆2 is an ≤-antichain.

Continuum of varieties of Heyting algebras

Corollary.

1 There is a continuum of varieties of Heyting algebras.

2 In fact, there is a continuum of varieties of Heyting algebrasof depth 3.

3 And there is a continuum of varieties of Heyting algebras ofwidth 3.


Bool

Var(C3)

Heyt + χ(C4)

Var(C4)

Heyt + χ(C5)...

Lin

Heyt

...

Varieties axiomatized by Jankov formulas

Is every variety of Heyting algebras axiomatized by Jankovformulas?

A variety V is locally finite if every finitely generated V-algebrais finite.

Theorem Every locally finite variety of Heyting algebras isaxiomatized by Jankov formulas.

Corollary. Varieties of finite depth are locally finite and henceaxiomatized by Jankov formulas.

Finitely generated algebras

However, there are continuum many non-locally finite varietiesof Heyting algebras.

Theorem (Rieger, 1949, Nishimura, 1960). The 1-generatedfree Heyting algebra, also called the Rieger-Nishimura lattice, isinfinite.

The Rieger-Nishimura Lattice

0

g ¬g

g ∨ ¬g¬¬g

¬¬g ∨ ¬g ¬¬g→ g

...

1

1-generated free Heyting algebra

0

g ¬g

g ∨ ¬g¬¬g

¬¬g ∨ ¬g ¬¬g→ g

...

1 g ¬g

¬¬g ¬¬g→ g

......

1


There exist varieties of Heyting algebras that are notaxiomatized by Jankov formulas.

Problem: Can we generalize Jankov’s method to all varieties ofHeyting algebras?

Recap

1 Classification of the lattice of varieties of Heyting algebrasvia their depth.

2 Subdirectly irreducible Heyting algebras and their dualEsakia spaces.

3 Jankov formulas and splitting varieties.

4 Continuum of varieties of Heyting algebras via Jankovformulas.

5 Problem: Can we generalize Jankov’s method to allvarieties of Heyting algebras?

Part 5: Canonical formulas


The affirmative answer was given by Michael Zakharyaschev viacanonical formulas.

Michael Zakharyaschev

Locally finite reducts

We will give an algebraic account of this method.

Although Heyting algebras are not locally finite, they havelocally finite reducts.

Heyting algebras (A,∧,∨,→,0,1).

∨-free reducts (A,∧,→,0,1): implicative semilattices.

→-free reducts (A,∧,∨,0,1): distributive lattices.

Theorem.

(Diego, 1966). The variety of implicative semilattices islocally finite.

(Folklore). The variety of distributive lattices is locallyfinite.

(∧,→)-canonical formulas

We will use these reducts to derive desired axiomatizations ofvarieties of Heyting algebras.

First we will need to extend the theory of Jankov formulas.

Jankov formulas describe the full Heyting signature. We willnow look at ∨-free reducts.

The homomorphisms will now preserve only ∧, 0 and→. Ingeneral they do not preserve ∨. But they may preserve somejoins.

This can be encoded in the following formula.


Let A be a finite subdirectly irreducible Heyting algebra, s thesecond largest element of A, and D a subset of A2.

For each a ∈ A we introduce a new variable pa and define the(∧,→)-canonical formula α(A,D) associated with A and D as

α(A,D) = [∧pa∧b ↔ pa ∧ pb : a, b ∈ A∧∧pa→b ↔ pa → pb : a, b ∈ A∧∧p¬a ↔ ¬pa : a ∈ A∧∧pa∨b ↔ pa ∨ pb : (a, b) ∈ D]→ ps

Note that if D = A2, then α(A,D) = χ(A).


Theorem. Let A be a finite s.i. Heyting algebra, D ⊆ A2, and B aHeyting algebra. Then

B 6|= α(A,D) iff there is a homomorphic image C of B and an(∧,→)-embedding h : A C such that h(a ∨ b) = h(a) ∨ h(b)for each (a, b) ∈ D.

Theorem. Every variety of Heyting algebras is axiomatized by(∧,→,0)-canonical formulas.

We show that for each formula ϕ there exist finitely manyA1, . . . ,Am and Di ⊆ A2

i such that

Heyt + ϕ = Heyt + α(A1,D1) + · · ·+ α(Am,Dm)


Proof idea. Suppose B 6|= ϕ.

Then there exist elements a1, . . . , an ∈ B on which ϕ is refuted.

We generate the implicative semilattice (A,∧,→,0) of B by thesubpolynomials Σ of ϕ(a1, . . . , an).

By Diego’s theorem (A,∧,→,0) is finite.


We define a “fake” ∨ on A by a∨b =∧s ∈ A : s ≥ a, b. Then

(A,∧, ∨,0,→) is a finite Heyting algebra. Also for a, b ∈ A wehave

a ∨ b ≤ a∨b.

Moreover, if a ∨ b ∈ Σ then

a ∨ b = a∨b.

This implies that the algebra (A,∧, ∨,→,0) refutes ϕ.


Now we let D = (a, b) : a ∨ b ∈ Σ.

Then

A i // B

i is a (∧,→,0)-embedding, preserving ∨ on the elements of D.

A may not be s.i.


We take a s.i. homomorphic image A′ of A (such can always befound) via some κ that refutes ϕ. We also let D′ be the κ-imageof D. So

A i //

κ

B

A′

i is a (∧,→,0)-embedding, preserving ∨ on the elements of D,and κ is a Heyting homomorphism.


Implicative semilattices have the congruence extensionproperty. Thus, there is an implicative semillatice C such that

A i //

κ

B

ξ

A′ h // C

Onto (∧,→,0)-homomorphisms are Heyting homomorphisms,so C is a Heyting algebra that is a homomorphic image of B.

Moreover, h preserves ∨ on the elements of D′.

So we found a finite s.i. algebra A′ and a set D′ ⊆ A′2 such thatA′ is (∧,→,0)-embedded into a homomorphic image of Bpreserving ∨ on D′.


So B refutes α(A′,D′).

Let k = |Sub(ϕ)|.

By Diego’s theorem there is M(k) such that every k-generatedimplicative semilattice has less than M(k)-elements.

Let A1, . . . ,Am be the list of all (finitely many) Heyting algebrasof size M(k)-refuting ϕ.

Let Vi be a valuation refuting ϕ in Ai. Set

Σi = Vi(ψ) : ψ ∈ Sub(ϕ).

Let Di = (a, b) : a ∨ b ∈ Σi.

By construction |A′| < M(k). So (A′,D′) = (Ai,Di) for somei ≤ m.

(∧,→)-canonical formulasThus, we proved that B 6|= ϕ implies B 6|= α(Ai,Di) for somei ≤ m.

Conversely, let B 6|= α(Ai,Di) for some i ≤ m.

Then here is a homomorphic image C of B and an(∧,→,0)-embedding h : Ai C such that h(a ∨ b) = h(a) ∨ h(b)for each (a, b) ∈ Di.

By construction of Di we have that C 6|= ϕ.

So B 6|= ϕ.

Thus, we proved

Heyt + ϕ = Heyt + α(A1,D1) + · · ·+ α(Am,Dm)

Therefore, every variety of Heyting algebras is axiomatized by(∧,→)-canonical formulas.

Subframe formulas

α(A,A2) = χ(A).

α(A, ∅) is called a subframe formula.

Subframes play the same role here as submodels in modeltheory.

Theorem. Let A be a finite s.i. algebra and XA its dual space. AHeyting algebra B refutes α(A) iff XA is a subframe XB.

(∧,→)-embeddability means that we take subframes of the dualspace.

There are continuum many logics axiomatized by such formulas.

All subframe logics have the finite model property.

Subframe formulasTheorem: Let A be a s.i. Heyting algebra and XA its dual space.Then

XA has width < n iff n-fork is not a subframe of XA iff α(Fn)is true in A.

. . . n

Fn

A variety of Heyting algebras V is of width < n if the width of XAis < n for each s.i. A ∈ V

V is of width < n iff A |= α(Fn), for each A ∈ V

(∧,∨)-canonical formulas

We can also develop the theory of (∧,∨)-canonical formulasγ(A,D) using the→-free locally finite reducts of Heytingalgebras.

The theory of these formulas is different than that of(∧,→)-canonical formulas.

Theorem. Every variety of Heyting algebras is axiomatized by(∧,∨)-canonical formulas.


Let A be a finite s.i. Heyting algebra, let s be the second largestelement of A, and let D be a subset of A2. For each a ∈ A,introduce a new variable pa, and set

Γ = (p0 ↔ ⊥) ∧ (p1 ↔ >)∧∧pa∧b ↔ pa ∧ pb : a, b ∈ A ∧∧pa∨b ↔ pa ∨ pb : a, b ∈ A ∧∧pa→b ↔ pa → pb : (a, b) ∈ D

and

∆ =∨pa → pb : a, b ∈ A with a 66 b.

Then define the (∧,∨)-canonical formula γ(A,D) associatedwith A and D as

γ(A,D) = Γ→ ∆.


If D = A2, then γ(A,D) = χ(A). If D = ∅, then γ(A, ∅) = γ(A)

Theorem. Let A be a finite s.i. Heyting algebra. A Heytingalgebra B refutes γ(A) iff XA is an order-preserving image of XB.

These formulas are counterparts of subframe formulas.

There are continuum many logics axiomatized by such formulas.

Applications of canonical formulas

In obtaining large classes of logics with the finite modelproperty.

In proving the Blok-Esakia isomorphism between the latticeof varieties of Heyting algebras and the subvrarites of theGrzegorczyk algebras.

In showing that the substructural hierarchy ofCiabattoni-Galatos-Terui collapses over superintuitionisticlogics.

In proving that admissibility is decidable over intuitionisticlogic and in finding a basis for admissible rules.

Open problems

Characterize locally finite varieties of Heyting algebras.

Conjecture: A variety V of Heyting algebras is locally finiteiff FV(2) is finite.

Is every variety of Heyting algebras generated by a class ofHeyting algebras of the form Op(X) for some topologicalspace X (Kuznetsov, 1975).

Heyt is generated by Op(R) (McKinsey and Tarski, 1946).

Generalize the theory of (∧,→) and (∧,∨)-canonicalformulas to other non-classical logics e.g, substracturallogics. For modal logics this has been done already.

Documents

Varieties of Heyting algebras and superintuitionistic logics