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Piero Pagliani Università di Milano Bicocca maggio 2009
Piero Pagliani
Approximation and Pretopologies
Piero Pagliani Università di Milano Bicocca maggio 2009
Approximation Spaces
(U), lR, uR
•given a set U
•given a relation R which structures U
•R is an equivalence relation
•lR is an interior operator of a 0-dimensional topological space
•uR is a closure operator of a 0-dimensional topological space
Piero Pagliani Università di Milano Bicocca maggio 2009
First part:
Observations and Pre-topological Approximation Operators
Second part:
Dynamic Property Systems and Pre-topological Approximation
Operators
Third part:
Formal Topological Systems and Pre-topological Approximation
Operators
Approximation and Pretopologies
Piero Pagliani Università di Milano Bicocca maggio 2009
The basic ingredients to represent objects through properties:
a map from objects to properties
a map from properties to objects
result of the factorisation
through f and g
A set of properties
A set of objects
G G
M
f g
Same set of objects
Gegenstand Objekt
Piero Pagliani Università di Milano Bicocca maggio 2009
a
a’
a’’
b
b’
b’’
b’’’
An “ideal” situation: f is an injective function
G M
a’’’
Objects
Observable
properties
G G
M
1G
RETRACTIONf
Piero Pagliani Università di Milano Bicocca maggio 2009
G G
M
1G
RETRACTION
The best model of a horse is a horse?
Modelling means abstraction
Abstraction means loosing something
(it must be injective)
Horse-ness property
f
Piero Pagliani Università di Milano Bicocca maggio 2009
a
a’
a’’
b
b’’’
G M
a’’’
ObjectsObservable
properties
M M
G
1M
SECTION
Sorts
(fibres, stalks)
An almost ideal situation: f is a surjective function
(it must be surjective)f
Piero Pagliani Università di Milano Bicocca maggio 2009
a
a’
a’’’
b
b’
b’’
b’’’
G M
a’’’
Objects and properties are linked
by means of an arbitrary fulfillment relation
⊩ G M
We call
G, M, ⊩
a Property System or a Basic Pair
The basic picture
Piero Pagliani Università di Milano Bicocca maggio 2009
Observations, Property Systems and Attribute Systems
R p1 p2 p3 p4
a 1 0 1 1
b 0 0 1 1
c 0 0 1 1
d 1 0 0 1
e 0 1 1 0
a Property system
R a1 a2 a3 a4 a5
a 3 b 1/3 6.5 0
b 3 b 1/3 6.5 0
c 1 a 2 7 1
d 1 c 1/2 3.3 1
e 3 b 1 6 0
an Attribute system
Observation systems
Piero Pagliani Università di Milano Bicocca maggio 2009
Observation and substantiation
a
a’
a’’
b
b’
b’’
b’’’
sub (substance)
G M
a’’’
sub(b’’’)
obs(a’)
obs
(observations)
Substance
neighbourhoods
Observation
neighbourhoods
m obs(g) if and only if g⊩ m if and only if g sub(m)
Piero Pagliani Università di Milano Bicocca maggio 2009
(G)
(M)
(G)
A subset of objects O
fulfils a
subset of properties P
A subset of properties P’
describes a
subset of objects O’
Usually, sub(obs(O)) O
Approximations and best approximations. 1
However, is sub(obs(O)) O an approximation of O by means
of the set of properties obs(O)?
No, because obs(sub(obs(O))) obs(O). That is,
O’ is defined by means of properties which do not have
any “essential” relationship with the set obs(O).
obs sub
Piero Pagliani Università di Milano Bicocca maggio 2009
(G) (G)
(M)
( )
la ( ua )
φ ψ
Approximations and best approximations. 2
How to define two operators φ and ψ
by means of obs and sub (i. e. ⊩ )
such that the following holds:
Piero Pagliani Università di Milano Bicocca maggio 2009
Towards a solution. 1
Operators must fulfil maximality (minimality),
i. e. limit, properties
X
best (least)
upper
approximation
ua(X)
best (largest)
lower
approximation
la(X)
{boundary
region
Piero Pagliani Università di Milano Bicocca maggio 2009
Towards a solution. 2
Maximality (minimality), implies isotonicity
If the approximations are not defined by means
of isotonic operators,
then the results could be incomparable:
we could hardly speak of a “best” approximation
Piero Pagliani Università di Milano Bicocca maggio 2009
Continuity and approximations: 1
Let ψ be isotonic and YG.
If ψ–1(Y) belongs to ψ–1(Y), then
it is the least X s. t. ψ(X) Y
i. e. the best approximation from above of Y via ψ.
But if this is the case, then
ψ–1(Y)= X, for X=ψ–1(Y)
that is, the pre-image of a principal filter is a principal filter
Otherwise stated: ψ is lower-residuated
Dually, if ψ is upper-residuated, then
the pre-image of a principal ideal is a principal ideal
and ψ–1( Y)) is the largest X such that
ψ (X)Y
i. e. the best approximation from below of Y via ψ.
Piero Pagliani Università di Milano Bicocca maggio 2009
Continuity and approximations: 2
If ψ is isotonic and lower-residuated, then set
ψ– =min(ψ–1(Y))
ψ– is called the lower residual of ψ
If ψ is isotonic and upper-residuated, then set
ψ+ =max(ψ–1( Y))
ψ+ is called the upper residual of ψ
Piero Pagliani Università di Milano Bicocca maggio 2009
Continuity and approximations: 3
(G) (G)
(M)
( )
la ( ua )
φ ψ
If ψ is isotonic and lower residuated, set φ=ψ–.
Then ψφ(Y) Y
and for any other X such that ψ(X) Y,
φ(Y) X
If ψ is isotonic and upper-residuated, set φ = ψ+ .
Then ψφ(Y) Y
and for any other X such that ψ(X) Y,
φ(Y) X
Piero Pagliani Università di Milano Bicocca maggio 2009
Approximations and Galois Adjunctions. 1
Let O and O’ be two ordered sets.
Let : O O’ and : O’ O, such that
p O, p’O’, (p’) p if and only if p’ ’ (p)
Then is called the lower adjoint of
and is called the upper adjoint of
O’ ┫ O
The pair , is called a Galois adjunction or an axiality and the
following holds:
is a closure operator
is an interior operator
If O’ ┫ Oop,
then , is called a Galois connection or a polarity.
In this case, both and are closure operators.
Piero Pagliani Università di Milano Bicocca maggio 2009
Approximations and Galois Adjunctions. 2
If φ = ψ– , then ψ = φ+ and we have
(G) φ ┫ψ (M)
(G) (G)
(M)
( )
la ( ua )
φ ψ
Problem:
How one can define a Galois adjunction on an
Observation System, in a natural way?
Piero Pagliani Università di Milano Bicocca maggio 2009
Let us extend the constructors obs and sub from tokens to types
e : (M) (G) ; e (Y)={g G : m(m Y g sub(m))}
existential extension of sub
[[e]]: (M) (G) ; [[e]](Y)={g G : m(m Y g sub(m))}
universal extension of sub
[e]: (M) (G) ; [e](Y)={g G : m(g sub(m) m Y)}
co-universal extension of sub
Extensional
operators
i : (G) (M) ; i (X)={m M : g(g X m obs(g))}
existential extension of obs
[[i]]: (G) (M) ; [[i]](X)={m M : g(g X m obs(g))}
universal extension of obs
[i]: (G) (M) ; [i](X)={m M : g(m obs(g) g X)}
co-universal extension of obs
Intensional
operators
Basic perception constructors. 1
Piero Pagliani Università di Milano Bicocca maggio 2009
Basic perception constructors. 2
aa’
a’’
b
b’
b’’
b’’’
G M
i ({a, a’})
e ({b, b’, b’’’})
[e]({b, b’, b’’’})
[[e]]({b, b’})
a’’’
[[i]]({a, a’})
[i]({a, a’})
i : (G) (M) ; i (X)={m M : g(g X m obs(g))}
existential extension of obs
[[i]]: (G) (M) ; [[i]](X)={m M : g(g X m obs(g))}
universal extension of obs
[i]: (G) (M) ; [i](X)={m M : g(m obs(g) g X)}
co-universal extension of obs
Piero Pagliani Università di Milano Bicocca maggio 2009
<i >(X)
X
b
[i](X)
X
b
[[i]](X) b
X
Intuitive and modal reading
of the basic perception constructors
it is possible for elements
of X to fulfil b
to fulfil b it is necessary
to be in Xto fulfil b it is sufficient
to be in X
at least one
there is an example of
elements of X
which fulfils b
at most all
there are not examples of
elements outside X
which fulfill b
at least all
there are not examples of
elements in X
which do not fulfill b
Piero Pagliani Università di Milano Bicocca maggio 2009
Adjunction properties of the basic perception constructors
i (X) Y if and only if X [e](Y )
e (Y) X if and only if Y [i](X )
[[e]] (Y) X if and only if [[i]](X ) Y
for all X G, for all Y M
adjunction
Galois
connection
i ┫ [e]
e ┫ [i]upper adjointlower adjoint
multiplicative
isotonic
additive
isotonic
[[e]] ┫op [[i]] anti-multiplicative
anti-isotonic
anti-multiplicative
anti-isotonic
[ ] is an interior operator
[ ] is an closure operator
Piero Pagliani Università di Milano Bicocca maggio 2009
a
a’
a’’
b
b’
b’’
b’’’
G M
ITS({b’, b’’})
est({a, a’})
cl({a, a’})
int({a,a’})
a’’’
C ({b’’, b’’’})
A ({b’, b’’})
Combining adjoint constructors: perception operators
int : (G) (G) ; int (X)= e [i](X)
cl : (G) (G) ; cl (X)= [e] i (X)
C : (M) (M) ; C (Y)= i [e] (Y)
A : (M) (M) ; A (Y)= [i] e (Y)
est : (G) (G) ; est (X)= [[e]][[i]] (X)
ITS : (M) (M) ; ITS (Y)= [[i]][[e]] (Y)
modal-style operators
Qualitative Data Analysis
?
intent-extent operators
Formal Concept Analysis
Piero Pagliani Università di Milano Bicocca maggio 2009
But what do int, cl, C and A represent, actually?
Preliminary question: what does M represent?
Each element m of M represents the set e (m) of elements of G which fulfil property m.
Hence m collects elements of G by means of the nearness relation “to fulfil m”.
m is a formal neighbourhood a proxy for a subset of G
[i](X) is the set of formal neighbourhoods m “included” in X, i. e. such that e (m) X
e [i](X) is the extension of such neighbourhoods.
Therefore: e [i](X) denotes in a formal way the interior of X.
i (X) is the set of formal neighbourhoods m with non void “intersection” with X. i. e.
such that X e (m)
[e] i (X) is the extension of all and only such neighbourhoods.
Therefore: [e] i (X) denotes in a formal way the closure of X.
Piero Pagliani Università di Milano Bicocca maggio 2009
FORMAL
(Pre) topological properties of the perception operators
We obtain G. Sambin’s
results
on Formal Topology
(point-free topology)
Straightforwardly
from the
adjunction properties
between the basic
constructors
closed
CONCRETE
: <e>[i] int <i>[e] C :
: [e]<i> cl [i]<e> A :
CONCRETE
open closed
symmetric
symmetric
: <e> <i> :
open
[i] : : [e]
du
al d
ua
l
FORMAL
Piero Pagliani Università di Milano Bicocca maggio 2009
Algebraic properties of the perception operators
Given the following set of fixpoints of the operators
one obtains the following lattices
Piero Pagliani Università di Milano Bicocca maggio 2009
Example
P b b1 b2 b3
a 1 1 0 0
a1 0 1 0 1
a2 0 1 1 1
a3 0 0 0 1
Piero Pagliani Università di Milano Bicocca maggio 2009
Approximation properties of perception operators
e [i](X) X [e] i (X )
i. e.
int (X) X cl (X)
with maximality, respectively, minimality property with respect to inclusion
for all X G
(immediate from the adjunction properties)
int is a pre-topological lower approximation operator
cl is a pre-topological upper approximation operator
Piero Pagliani Università di Milano Bicocca maggio 2009
Choosing the initial perception act.
Information quanta
g’ Qg if and only if g’ fulfils at least all the properties fulfilled by g:
i ({g}) i ({g’})
J. L. Bell’s approach to Quantum Logic: any object g is perceived together with all
the elements which are indiscernible from g in the “field of perception”.
Hence any object is a location and it is perceived embedded in a quantum at that
location: minimum perceptibilium at that location.
“Orthologic, forcing and the manifestation of attributes”.
We take this phenomenological approach and claim that any object g is perceived
together with all the elements which manifest at least all the same properties as g:
information quantum at g.
Piero Pagliani Università di Milano Bicocca maggio 2009
g, g’ RS if and only if g’ Qg
Information quantum relation
and
Information quantum systems
information quantum relation
• reflexive
• transitive
Given a Property System S
Q(S) = G, G, RS
Information quantum relation system
• it is a property system!!
If g’, g RS then g’ Qg , thus i (g) i (g’).
Hence g is less defined than g’ in S.
Indeed, RS is a (anti)specialisation preorder in the usual topological sense.
Piero Pagliani Università di Milano Bicocca maggio 2009
Adjuncion and algebraic properties of i-quantum operators
In Q(S) the sets of fixpoints of adjoint i-quantum operators coincide
Let cl, int, C and A be induced by Q(S). Then
cl = i int = [i]
C = [e] A = e
Since cl is lower adjoint, cl is additive, hence topological
Since int is upper adjoint, int is multiplicative, hence topological
Similarly for A and C
cl C
A int
Piero Pagliani Università di Milano Bicocca maggio 2009
Example
Satcl(Q(P))
P b b1 b2 b3
a 1 1 0 0
a1 0 1 0 1
a2 0 1 1 1
a3 0 0 0 1
RP a a1 a2 a3
a 1 0 0 0
a1 0 1 1 0
a2 0 0 1 0
a3 0 1 1 1
Piero Pagliani Università di Milano Bicocca maggio 2009
Let cl, int, C and A in Q(S). Then
cl = i : inverse upper approximationcl(X) = set of elements approximated by some member of X
int = [i]: inverse lower approximationint(X) = set of elements approximated just by members of X
C = [e]: direct lower approximationC (X) = set of elements specialised just by members of X
A = e : direct upper approximationA(X) = set of elements specialised by some member of X
Topological approximation operators
Piero Pagliani Università di Milano Bicocca maggio 2009
If S is an Attribute System, its nominalisation
is a Perception System N(S) such that:
RN(S) is an equivalence relation
and in Q(N(S))
cl(X) = upper approximation of X
int(X) = lower approximation of X
in the sense of Pawlak’s classical Rough Set Theory
Indeed, in an Attribute System “to fulfill at least the same properties”
is tantamount to saying “to have the same attribute value”, which
induces an equivalence relation
The same is true of dichotomic Property Systems
A particular case
Piero Pagliani Università di Milano Bicocca maggio 2009
Second part:
Dynamic Property Systems
and
Pre-topological Approximation Operators
Piero Pagliani Università di Milano Bicocca maggio 2009
•Z. Pawlak
Approximation Spaces
<U, E>
for E an equivalence relation
•T. Y. Lin & Y. Y. Yao
Neighbourhood Spaces
<U, R>
for R any binary relation
•A. Skowron & J. Stepaniuk
•P. Pagliani
Approximation of relations
<U, {Ei}iI >
for {Ei}iI a family of equivalence relations
•P. Pagliani
Dynamic Approximation Spaces
<U, {Ri}iI >
for {Ri}iI a family of binary relations
Piero Pagliani Università di Milano Bicocca maggio 2009
R
context 1R1
time mRmtime 2R2time 1
1 context 2R12context nR1n
Sources of dynamics
1) Evolution of the observation process over time
2) Change of context
3) 1+2
Piero Pagliani Università di Milano Bicocca maggio 2009
Object 1
Object 2
Object 3
Object n
Tim
e
Dynamic Rough
Sets
A layered observation system
Piero Pagliani Università di Milano Bicocca maggio 2009
f(x)f(y)
f(w)
f(z)
U ’: applicable criteria (metrics) of type TContext Context
xy
w z
U: object level
w’
Nx
q
z’z’’
More than 1 context More than 1 neighbourhood
xy
w
w’ z q
z’z’’
Neighbourhood
of x
Why more than one neighbourhood?
Piero Pagliani Università di Milano Bicocca maggio 2009
More than 1 neighbourhood Neighbourhood families
More neighbourhood families Neighbourhood systems
Neighbourhood
of x
Neighbourhood
of y
What if more than one neighbourhood?
Piero Pagliani Università di Milano Bicocca maggio 2009
A
NzNy
x, y, z G(A)
Nx
Let us denote n(x) with Nx
Core maps and vicinity maps
Piero Pagliani Università di Milano Bicocca maggio 2009
For the sake of semplicity, let U’=U and let f be the identity function.
One can distinguish the following properties, which define a topological space,
for any xU, AU:
1. UNx
0. Nx
Id. if xG(A) then G(A) Nx
N1. xN, for all NNx
N2. if NNx and N N’ then N’Nx
N3. if N, N’Nx then NN’Nx
N4. there is an N such that Nx =N
Sectioning topological systems
Piero Pagliani Università di Milano Bicocca maggio 2009
Conditions and properties of neighbourhood systems
Conditions 0, 1, Id, N1, N2 and N3 characterise topological spaces
Piero Pagliani Università di Milano Bicocca maggio 2009
Neighbourhood systems and systems of relations.
Piero Pagliani Università di Milano Bicocca maggio 2009
x
yz
c
b
a
x
y zc
b
af(A)
x
y
zg(A)
Expansion and contraction processes
A
Piero Pagliani Università di Milano Bicocca maggio 2009
c
A
c
b
y
c
w
a
y
x
z
w
b
f(A)y
x
z
a
f(B)
x
z
a
b
B
c
w
y
x
z
Expansion and contraction are
not necessarily isotonic
Piero Pagliani Università di Milano Bicocca maggio 2009
Pre-topological spaces and neighbourhood systems
A pre-topological space is a triple U, ε, κ such that:
(i) U is a set, (ii) ε: (U)(U) is an expansion map such that ε()= ,
(iii) κ : (U)(U) is a contraction map dual of ε.
The core map (vicinity map) induced by a neighbourhood system of type
(at least) N1 is a contraction map (an expansion map)
Piero Pagliani Università di Milano Bicocca maggio 2009
Dynamic Approximation Spaces
U, {Ri}1 i n, m, m
for {Ri}1 i n a family of binary relations, 1 m n
n use case involving the expansion process,
and n use cases involving the contraction process
Piero Pagliani Università di Milano Bicocca maggio 2009
Pre-topological spaces and Dynamic Spaces
Piero Pagliani Università di Milano Bicocca maggio 2009
Piero Pagliani Università di Milano Bicocca maggio 2009
Associating systems of relations with pre-topological spaces
Piero Pagliani Università di Milano Bicocca maggio 2009
Piero Pagliani
Approximation and pre-topologies
Third part
Formal Topological Systems
and
Pre-topological Approximation Operators
Piero Pagliani Università di Milano Bicocca maggio 2009
Concrete and formal pre-topology
Satint(G)=int (G), , , , G where iI Xi=int(iI Xi)
is a complete lattice
SatA(M)=A (M), , , , M where iI Xi=A(iI Xi)
is a complete lattice
Let P=G, M, R, be any property system, for R GM any binary relation.
We remind that
int(G)={X G :int(X)=X} is the family of concrete open subsets of P
A(M)={YM: A(X)=X} is the family of formal open subsets of P
e [i]
Piero Pagliani Università di Milano Bicocca maggio 2009
A formal semi-covering relation
between neighbourhoods
bY
bY
bY YY’
b Y’
YY
reflexivity
transitivity
identity
Let G, M, R be a basic pair. Then, for any bM and Y, Y’M,
a formal semi-cover relation is defined as follows:
(basis) bY iff bA(Y), (step) YY’ iff y Y, yY’
From this definition, it follows:
Piero Pagliani Università di Milano Bicocca maggio 2009
Combination of formal neighbourhoods
and semi-covering relation
Properties may be combined by means of a formal meet operation
that we call “fusion” and denote with “”
The operation “” is inherited by subsets of properties in the following way:
XY=xy: xX & yY (point-meet)
Formal opens inherit the operation “” in the following way:
X•Y=A(XY)
Piero Pagliani Università di Milano Bicocca maggio 2009
For all Y, Y’A(M), and b, b’ M
Properties which can
be added to a semi-cover
b Y
b b’Y
bY bY’
b Y Y’
left
right
stabilitybY b’Y’
b b’ Y Y’
Piero Pagliani Università di Milano Bicocca maggio 2009
Let G, M, R be a basic pair (i. e. property system),
let M, , 1 be commutative monoid, set =mM : R(m)=. Then:
M, , 1, , is called a semi-topological formal system induced by
the basic pair G, M, R
A semi-topological formal system in which stability holds, is called
a pre-topological formal system, and is called a precover
A semi-topological formal system in which left and right hold, is called
a topological formal system, and is called a cover.
If M, , 1, , is a topological formal system, then
A(M), •, , M, A() , for any subset of M,
is a complete lattice with complete distributivity and ordering .
In any semi-topological formal system in which · is idempotent
and stability holds, right is derivable.
Any semi-topological formal system in which · is idempotent and
stability and left hold, is a topological formal system.
Property systems and formal systems
Piero Pagliani Università di Milano Bicocca maggio 2009
A, (A), R is called a basic neighborhood pair
and, therefore,
(A), , A, , will be called a formal neighborhood system
Since is idempotent, any formal neighborhood system in which
stability and left hold, is a topological formal system.
A bridge between
concrete neighborhood systems and formal systems
Piero Pagliani Università di Milano Bicocca maggio 2009
Let (A), , A, , be induced by A, (A), R. Then:
a) If R(x): xA fulfils N3, then right holds. That is:
for all X, X’ (A), x A and Y, Y’(A),
(X, X’ R(x) XX’ R(x)) (X Y and X Y’ X Y·Y’)
b) R(x): xA fulfils N2 if and only if left holds. That is:
for all X, X’ (A), x A and Y(A),
(X R(x) & X X’ X’ R(x)) (X Y X ·X’ Y)
Towards the main connection between
concrete pre-topological systems and formal systems
Piero Pagliani Università di Milano Bicocca maggio 2009
Let (A), , A, , be induced by A, (A), R. Then, for all X(A),
G(X)=int(X)
if and only if R(x): xA is a neighborhood system of type N2Id.
A formal neighborhood system (A), , A, , such that for all X(A),
G(x)=int(X), will be called a pseudo-topological formal system.
Main connection between
concrete pre-topolgical systems and formal systems
Piero Pagliani Università di Milano Bicocca maggio 2009
Let (A), , A, , be a pseudo-topological formal system. Then:
A ((A)),
is a meet semilattice with ordering .
However, in general does not distribute over in SatA ((A)).
To obtain distributivity of over , we need stability:
In any pseudo-topological formal system, if stability holds, then
A ((A)), ,
is a complete distributive lattice, hence a Heyting algebra.
But this is not a surprise, because:
In any pseudo-topological formal system N2 holds, by definition.
Hence left holds, too. Moreover, the monoidal operation is , which is
idempotent. It follows that if we add stability we obtain right.
But we know that a pre-topological formal system in which right and left hold,
is a topological formal system.
From concrete pre-topolgical systems to
concrete topological systems, through formal systems
Piero Pagliani Università di Milano Bicocca maggio 2009
Piero Pagliani Università di Milano Bicocca maggio 2009
Piero Pagliani Università di Milano Bicocca maggio 2009
Piero Pagliani Università di Milano Bicocca maggio 2009
Pretopological formal systems provide models for limited resource logics
Logic formula Pretopological interpretation
a
a b ABa b AB
A logical interpretation
multiplicative conjunction
a & b AB additive conjunction
AB
additive conjunction
a ⊢ b
left &
right &
(left)
(right)
AC
A ·BCΓ, a ⊢ c
Γ, ab ⊢ c
AB AC
A B ·C
Γ⊢ a Γ⊢ b
Γ⊢ ab
A A (M)
Piero Pagliani Università di Milano Bicocca maggio 2009
Categorical
modal logic
[Meloni, Ghilardi]
Ortholattices
[Bell]
Quantum logic
Lukasiewicz
Nelson
Post
Stone
ALGEBRAS
[Pomikała,
Düntsch,
Pagliani]
Topological Bilattices
[Fitting]
Semi-Post Algebras
[Rasiowa, Cat Ho]
Formal Concept
Analysis
[Wille, Kent]
Linear Logic
Program semantics
[Parikh, He, Hoare]
Formal
(pre) topology
[Sambin]
Quanta of
information
Modal Logic
Rough
Sets
Intermediate
(non-standard)
constructive
logics
Topological quasi
Boolean algebras
[Chakraborty,
Banerje]
Chu Spaces
Information Flow
[Pratt, Barwise,
Seligman]
Galois adjunctions
[Ore, Grothendieck,
Lowvere, Scott]
Logics with
strong negation
Duality
theory
[Stone-Birkfoff]
Kripke
models
Relational
modal logic
[Orlowska, Düntsch,
Pagliani]
Neighborhood systems
[Lin, Yao, Pagliani]
Heyting
co-Heyting
ALGEBRAS
[Rauszer, Lawvere,
Pagliani, Iturrioz]
Intuitionistic logic
Modal-style
operators
[Yao, Düntsch,
Gegida, Pagliani]