diatetagmena synola kai algebres logikis

8/15/2019 Diatetagmena Synola Kai Algebres Logikis

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supremum

infimum

Boole

Boole


4/85

J. Hintikka, The

principles of Mathematics revisited , Cambridge 1996

supremum

infimum

Dedekind - MacNeille

Boole

Stone

Stone

supremum

infimum

Heyting

Boole


5/85

Stone


6/85


7/85

supremum

infimum

suprema

infima

P

≤

P


8/85

i ∀x x ≤ x

ii ∀x∀y x ≤ y & y ≤ x ⇒ x = y

iii ∀x∀y∀z x ≤ y & y ≤ z ⇒ x ≤ z

N

⊆ ⊆ ⊆

N

n ≤ m k ∈ N m = n.k

0

1

s1 ≤ s2 s1

s2 s1 = 00101 s2 = 001011 s3 = 0010100

s1 ≤ s2 s1 ≤ s3

p ≤ q p → q

≤

⊆

(P, ≤) x y x y y ≤ x

P

P

P op

x

y

x ≤ y y ≤ x

(P, ≤)


9/85

x ≤ y y < x

P op P

P op

Hasse

{a, b}

{a, b}

{a} {b}

∅

D2 =< a,b | a2 = b2 =

1, ab = ba >

D2 |

{1, a} {1, ab} {1, b} |

{1}


10/85

Suprema

infima

S ⊆ P

x ∈ P S s ∈ S s ≤ x

x ∈ P S s ∈ S x ≤ s

x ∈ P supremum S ⊆ P S

y

S

x ≤ y supremum S ⊆ P supremum S ⊆ P

S supS supremum

{x, y} x ∨ y

x ∈ P

infimum

S ⊆ P

S

y

S

y ≤ x

infimum

S ⊆ P

infimum

S ⊆ P

S

infS

infimum {x, y} x ∧ y

P

I

P {xi|i ∈ I } supremum

infimum

i∈I xi

i∈I xi

R

A = {1, 12

, ..., 1n

,...}

B = {2, 3 − 12

, ..., 3 − 1n

,...}

A = 1

A = 0

B = 3

B = 2

0

A

x

0 ≥ x R 0 < x

1N

1N

< x

N = [ 1x

] + 1)

x

A

su-premum

infimum

supremum

maximum

1 A infimum

minimum

supremum

infimum

infimum


11/85

SUPREMA INFIMA

supremum

supremum

supremum

infimum

N

supremum

infimum

supremum

p ∨ q = p ∨ q infimum p ∧ q = p ∧ q

supremum

infimum

P

supremum

P

infimum

S ⊆ P

infimum

X = {x ∈P | ∀s ∈ S x ≤ s} supremum

X

infimum

S

•

X

S

s S X

S

s

s

X

X ≤ s

• X S x ∈ P

S

s ∈ S x ≤ s x ∈ X

x ≤

X

P

infimum

P

supremum

P op

supremum

P

infimum

P op

infimum

supremum

P op


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P supremum

S ⊆ P

supremum

P

supremum

K = {

F | F ⊆ S, F } supre-ma

S

K

supremum

P

K

supremum

S

•

F

F

K

F ∪ F

F ≤

(F ∪ F )

F ≤

(F ∪ F )

A ⊆ B

A ≤

B

a ∈ A a ∈ B a ≤

B

a ∈ A A ≤

B

K

• supremum K supremum

S

s ∈ S s =

{s} ∈ K

K

S x

S

F K F ≤x K ≤ x

(P, ≤) supremum

infimum

supremum

infimum

P

0

1

(P, ≤)

supremum

supremum


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x, y

supremum x ∨ y

supremum

(P, ≤)

infimum

infimum

x, y

infimum x ∧ y infimum

(P, ≤)

supremum

infimum

supremum

infimum

0

1

x, y

supremum

x ∨ y infimum x ∧ y

supremum

x ∨ x = x, x ∨ y = y ∨ x, (x ∨ y) ∨ z = x ∨ (y ∨ z ), x ∨ 0 = x.

infimum

x ∧ x = x, x ∧ y = y ∧ x (x ∧ y) ∧ z = x ∧ (y ∧ z ), x ∧ 1 = x.

(A, +, 0)

A +

supremum

a ≤ b a + b = b


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a ∨ b = a + b supremum

a ∨ b = a + b

≤

a ≤ a

a + a = a,

a ≤ b & b ≤ a

a + b = b & b + a = a

a = b,

a ≤ b & b ≤ c

a + b = b & b + c = c

a + c = a + (b + c) = (a + b) + c = b + c = c

a ≤ c.

a+b a, b a+(a+b) = (a+a)+b = a+b ⇒ a ≤ a+b

b ≤ a + b c a, b a + c = c b + c = c

(a + b) + c = a + (b + c) = a + c = c ⇒ a + b ≤ c.

supremum a b

a b ⇔ a b = b ⇔ a + b = b ⇔ a ≤ b,


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(A,., 1)

A

.

infimum

a ≤ b a.b = a

a ∧ b = a.b infimum

a ∧ b = a.b

supremum

infimum

a ∧ (a ∨ b) = a a ∨ (a ∧ b) = a.

supremum

infimum

(A, +, O) (A,., 1)

A

a ≤ b a + b = b

a ≤ b a.b = a

a.(a + b) = a

a + (a.b) = a

supremum

infimum

a + b = b

a.b = a


17/85

f : P → Q

x, y ∈ P x ≤ y f (x) ≤ f (y)

f : P → Q suprema

f (

S ) =

{f (s)| s ∈ S }

infima

f (

S ) =

{f (s)| s ∈ S }.

suprema

infima

f (d)|

d g

b c, f (b) f (c)

a e|

f (a)

f

f (b) ∨ f (c) = g = f (d) = f (b ∨ c)

f (b) ∧ f (c) = e = f (a) = f (b ∧ c).

f (a) = e f (d) = g

infima

suprema

suprema

infima

suprema

infima


18/85

{f (s)| s ∈ S } ≤ f (S ) f (

S ) ≤

{f (s)| s ∈ S }

s ∈ S

S ≤ s ≤

S

f

s ∈ S f (

S ) ≤ f (s) ≤ f (

S )

f : P → Q

x,y ∈ P, x ≤ y f (x) ≤ f (y) f (x) = f (y) f (x) ≤f (y)

x ≤ y f (y) ≤ f (x) y ≤ x x = y

P

Q

QP

P

Q

S

P

Q

supremum

S : P → Q

(

S )(x) =

{s(x)|s ∈ S }

x ≤ y ∈ P s ∈ S s(x) ≤ s(y)

(

S )(x) =

{s(x)|s ∈ S } ≤

{s(y)|s ∈ S } = (

S )(y)

S

S

x ∈ P

s(x) ≤

{s(x)|s ∈ S } = (

S )(x)

f

S

x ∈ P s(x) ≤ f (x) {s(x)|s ∈ S } ≤ f (x)

S ≤ f

S supremum

S

2P

P

2 = {0, 1}


19/85

YONEDA

2 = {0, 1}

QX X

Q

x ≤ y x = y

Q

Yoneda

{0, 1} S

χS

χS (x) =

1, x ∈ S 0, x /∈ S

f : P → 2 { p ∈ P | f ( p) = 1} P

P

P → 2 = {0, 1}

P

S ⊆ P

p ∈ S & p ≤ q ⇒ q ∈ S

p ∈ S χS ( p) = 1 χS (q ) ≥ χS ( p) = 1

p ≤ q ∈ P

χS ( p) = 1 χS (q ) = 0

p

S

q ≥ p S

P {0, 1} P


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S ⊆ S χS ≤ χS p ≤ q χS ( p) = 0 χS ( p) ≤ χS ( p) χS ( p) = 1 p ∈ S p ∈ S p ∈ P χS ( p) ≤ χS ( p) p ∈ S χS ( p) = 1 χS ( p) = 1 p ∈ S S ⊆ S

{S i| i ∈ I } p ∈

i∈I S i q ≤ p i ∈ I p ∈ S i

q ≤ p q ∈ S i S i q ∈

i∈I S i

suprema

infima

suprema

supremum

infimum 2P

P

2P

P 2 = {0, 1} ↑ (P ) P

S

P

p ∈ S & p ≥ q ⇒ q ∈ S

2P op

P

2 = {0, 1} ↓ (P ) P


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y: P −→ 2P op

∼=↓ (P )

p ∈ P ↓ ( p) = {x ∈ P |x ≤ p}

y: P −→ 2P op

• y p ≤ q

x ∈↓ ( p) ⇔ x ≤ p ⇒ x ≤ q ⇔ x ∈↓ (q ),

y( p) ≤ y(q ) • y( p) ≤ y(q )

p ∈↓ ( p) ⊆↓ (q ) = {x ∈ P |x ≤ q },

p ≤ q

y

P

Yoneda

P

f : P → X

f

f̂ : 2P op

→ X f̂ ◦ y = f

suprema

2P op

supremum

y( p)


22/85

2P op

P ↓ ( p) S S =

{↓ ( p)| p ∈ S }

2P op

y( p) ≤

S

s ∈ S

y( p) ≤ s

↓ ( p) ⊆{S i|i ∈ I } S i ↓ ( p) ⊆ S i p ∈

{S i|i ∈ I },

S i p ∈ S i S i ↓ ( p) ⊆ S i

{S i|i ∈ I }

P

(i∈I

S i) =i∈I

S i

supremum

{

S i|i ∈ I } i∈I S i supremum {S i|i ∈ I }

supremum

s ∈ S i

s ≤

S i

S i ⊆

i∈I S i i ∈ I S i ≤

(

i∈I S i) i ∈ I {

S i|i ∈ I } ≤

(i∈I

S i).

f̂ : 2P op

→ X

f̂ (s) =

{f ( p) ∈ X | y( p) ≤ s}

f̂

s ≤ s y( p) ≤ s y( p) ≤ s

{f ( p) ∈ X |y( p) ≤ s} ⊆ {f ( p) ∈ X | y( p) ≤ s},


23/85

f̂ (s) =

{f ( p) ∈ X | y( p) ≤ s} ≤

{f ( p) ∈ X |y( p) ≤ s} = f̂ (s).

f̂

suprema

f̂ (

S ) =

{f ( p) ∈ X | y( p) ≤

S }

=

{f ( p) ∈ X | ∃s ∈ S y( p) ≤ s}

=

s∈S {f ( p) ∈ X | y( p) ≤ s}

=s∈S

{f ( p) ∈ X | y( p) ≤ s}

=

{f̂ (s)| s ∈ S }

f̂

f̂ ◦ y = f

( f̂ ◦ y)( p) = f̂ (y( p))

= {f (q ) ∈ X | y(q ) ≤ y( p)}= f ( p)

f ( p)

supremum

y( p) ≤ y( p)

maximum

f (q ) y(q ) ≤ y( p) f (q ) y q ≤ p

f

f (q ) ≤ f ( p)

f̂

g: 2P op

→ X suprema g ◦ y = f

g = f̂ .

s =

{y( p)| y( p) ≤s}

g(s) = g(

{y( p) ∈ 2P op

| y( p) ≤ s}


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= {g(y( p)) ∈ X | y( p) ≤ s}=

{(g ◦ y)( p) ∈ X | y( p) ≤ s}

=

{f (s)| y( p) ≤ s}

= f̂ (s)

supremum

in-fimum

f : X → Y

g: Y → X g ◦ f = idX f ◦ g = idY

{(X, Y ) ∈ P (N) × P (N)| X ∩ Y = ∅}

suprema

suprema

suprema

x ≤ y {x, y}

(x ∧ y) ∨ (x ∧ z ) ≤ x ∧ (y ∨ z ) x ∨ (y ∧ z ) ≤ (x ∨ y) ∧ (x ∨ z )

x ≤ z

x ∨ (y ∧ z ) ≤ (x ∨ y) ∧ z


25/85

supremum

H ∨ K = H.K = {h.k| h ∈ H k ∈ K }

supremum

P

K

i

P =< P (N), ⊆> K = {X ⊆ N| X }

ii

P =< P (N), ⊆> K = {X ⊆ N| N − X }

iii

P =< N2,

> K = {(0, n)| n ∈ N} ∪ {(n, 0)| n ∈N}

iv P = K = {f i| i ≥ 0 f i(i) = 1 f i( j) =0

i = j}

X

f : X → X

x̂ =

{x ∈ X | x ≤ f (x)}

i

f (x̂) {x ∈ X | x ≤ f (x)}

ii

f (x̂) = x̂

{(X, Y ) ∈ P (N) × P (N)| X ∩ Y =∅} P N

. .P =

.


26/85

L

− ∧ −: L2 → L − ∨−: L2 → L

x,y,z ∈ L,

x ≤ y ⇒ x ∧ z ≤ y ∧ z, x ≤ y ⇒ x ∨ z ≤ y ∨ z.

S

P

p ∈ P ↓ ( p) ⊆ S

S =

{↓ ( p)| p ∈ S }


27/85

suprema

U

U ⊆

i∈I U i {i1,...,in} I U ⊆ U i1 ∪ ... ∪ U in

X ∈


28/85

O(X )

O(X )

u u ≤

S

s1,...,sn S u ≤ s1 ∨ ... ∨ sn

X

x ∈ X

S ⊆ X x ≤

S

s ∈ S x ≤ s

x ∈ X x ≤

S

S

x

x ≤ s1∨...∨sn S

S

s j1 s1, s2 s j2 s j1, s3 s j3

s j3, s4

s ∈ S s ≥ s1, ... s ≥ sn x ≤ s1 ∨ ... ∨ sn ≤ s

• x

x ≤

S

S

S =

{

F | F ⊆S, F

} supremum x ≤ F F S

x

X

⊆ S ⊆ X

S

S = {x1,...,xn} S ⊆

i∈I Y i xi ∈ S Y i xi ∈ Y i xi S ⊆ Y i1 ∪ ... ∪ Y in S

S

S


29/85

suprema

{(X, Y ) ∈ P (N)×P (N)| X ∩Y = ∅}

suprema

supremum (2N, 2N + 1)

(2N + 1, 2N)

suprema

X x, y supremum x ∨ y

S ⊆ X

x ∨ y ≤ S x ≤ S y ≤ S

x

y

s

s

S

x ≤ s

y ≤ s S s

s ≤ s s ≤ s s x y

x ∨ y ≤ s

K (X )

X

suprema

suprema

K

supremum

I ⊆ K

K


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• 0 K supremum I

• x I y ≤ x y I • x y I supremum x ∨ y

I

{I j| j ∈ J } K

j∈J I j

x y

supremum

x

y

x

y

Idl(K )

K ⊆

suprema

K

e

c d

b|a

e

c d

a


31/85

c d

b

b| |a a,

c d

b

|a

supremum

c

d

supremum

Idl(K )

K

supremum

x

y ≤ x x

y

x, y

j, k ∈ J

x ∈ I j, y ∈ I k

l ∈ J I j ⊆ I l I k ⊆ I l x, y ∈ I l

x ∨ y ∈ I l x ∨ y

suprema

X

• X


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• X supremum

K

Idl(K )

Idl(K )

Idl(K )

Idl(K )

supremum

↓(x) x ∈ K

Idl(K ) {I j| j ∈ J }

↓ (x) ⊆ j

I j = j

I j

j ∈ J, x ∈ I j ↓(x) ⊆ I j I

I =

{↓(x)|x ∈ I }

K

infimum

F ⊆ K K

• 1 K infimum

F

• x F y ≥ x y F • x y F infimum x ∧ y F

P (N)


33/85

F = {X ⊆ N| N − X }

L

L

L

Filt(L) = Idl(Lop)

L Boole

Idl(K )

X

Yoneda

y: K → K

Idl(K ) ↓(x)

y: K → Idl(K )

K

f : K → X

suprema

f

f̂ : Idl(K ) → X

f̂ ◦ y = f suprema

f̂ : Idl(K ) →X


34/85

f̂ (I ) =

{f ( p) ∈ X | y( p) ⊆ I }

=

{f ( p) ∈ X | ↓ ( p) ⊆ I }

=

{f ( p) ∈ X | p ∈ I }

f̂

f̂

suprema

f̂ (

I k) =

{f ( p) ∈ X | p ∈

I k}

=

{f ( p) ∈ X | ∃k p ∈ I k}

=

k

{f ( p) ∈ X | p ∈ I k}

=k

{f ( p) ∈ X | p ∈ I k}

= k{f̂ (I k)| k}

f̂

f̂ ◦ y = f

( f̂ ◦ y)( p) = f̂ (y( p))

=

{f (q ) ∈ X | q ∈↓ ( p)}

= f ( p)

f̂

suprema

suprema

suprema


35/85

Idl(K ) K

X

X

g: Idl(K (X )) → X

i: K (X ) → X

K (X )

i: Idl(K (X )) → X

suprema

i

i(I ) =

{i( p)| p ∈ I, p } i

i

• i supre-mum

x ∈

X

x =

{ p| p ≤ x, p }

=

{i( p)| p ∈↓ (x), p }

= i(↓ (x)).

• i i(I ) ≤ i(J )

{ p| p ∈ I , p } ≤ {s| s ∈ J, s }

p ∈ I s ∈ J p ≤s

J p ∈ I p ∈ J I ⊆ J i


36/85

: (

)

(

):

,

x ∧ (y ∨ z ) = (x ∧ y) ∨ (x ∧ z )

x ∨ (y ∧ z ) = (x ∨ y) ∧ (x ∨ z )

(x ∨ y) ∧ (x ∨ z ) = ((x ∨ y) ∧ x) ∨ ((x ∨ y) ∧ z )

= x ∨ ((x ∨ y) ∧ z )

= x ∨ ((x ∧ z ) ∨ (y ∧ z ))

= (x ∨ (x ∧ z )) ∨ (y ∧ z )= x ∨ (y ∧ z )

infi-mum

suprema

x ∧ (

S ) =

{x ∧ s|s ∈ S },


37/85

S

S

{x ∧ s|s ∈ S } ≤ x ∧ (

S )

x ∧ s x ∧ (

S )

supremum

x ∧ s

Φ ≤ Ψ,

Φ

Ψ

infima

supre-ma

t

t ≤ Φ ⇒ t ≤ Ψ,

supremum

t t ≤ Φ ⇒ t ≤ Ψ.

X

x ∧ (S ) = {x ∧ s|s ∈ S },

x

S

X

{x ∧ s|s ∈ S } ≤ x ∧ (

S )

t ∈ X


38/85

t ≤ x ∧ (S ).

t ≤ x t ≤

S

t

S

s ∈ S t ≤ s s ∈ S t ≤ x ∧ s t ≤

{x ∧ s|s ∈ S }.

x ∧ (S ) = {x ∧ s|s ∈ S },

S

suprema

supremum

suprema

infimum

suprema

D

Idl(D)

I ∧ (

S ) =

{I ∧ J |J ∈ S },

I

S

Idl(D)

Idl(D)

suprema

D

I ∩ (J ∨ K ) = (I ∩ J ) ∨ (I ∩ K )

I ∩ (J ∨ K ) ⊆ (I ∩ J ) ∨ (I ∩ K ),


39/85

supre-

mum supremum

supremum

J

K

J ∨ K = {x ∈ D| y ∈ J, z ∈ K x ≤ y ∨ z }

x, x ∈ J ∨ K y ∈ J, z ∈ K x ≤ y ∨ z y ∈ J, z ∈ K x ≤ y ∨ z . x ∨ x ≤ (y ∨ y ) ∨ (z ∨ z ), y ∨ y ∈ J, z ∨ z ∈ K J ⊆ J ∨ K K ⊆ J ∨ K x ∈ J x = x ∨ 0 ∈ J ∨ K L

J

K

x ∈ J ∨ K x ≤ y ∨ z y ∈ J y ∈ L z ∈ K z ∈ L

x ∈ L

x ∈ I ∩ (J ∨ K ) I y ∈ J z ∈ K x ≤ y ∨ z

x = x ∧ (y ∨ z )

= (x ∧ y) ∨ (x ∧ z ),

x ∧ y ≤ x ∈ I x ∧ y ≤ y ∈ J x ∧ z ≤ x ∈ I

x ∧ z ≤ z ∈ K x ∧ y ∈ I ∩ J x ∧ z ∈ I ∩ K

x = (x ∧ y) ∨ (x ∧ z ) ∈ (I ∩ J ) ∨ (I ∩ K ),

frame

Idl(D)

D


40/85

infima

infimum

D ∈ Idl(D) ↓(a) ↓(a)∩ ↓(b) =↓(a ∧ b)

I k = D 1 ∈ I k

k

I k = D

X

infima

K (X )

X

K (X )

X

infima

(

) ∼= ( )

(

) ∼= ( )

Boole

Boole B

x

x

x ∧ x = 0 x ∨ x = 1 (∗)

(∗)

x

x ∧ x = 0 x ∨ x = 1


41/85

BOOLE

x = x ∧ 1

= x ∧ (x ∨ x)

= (x ∧ x) ∨ (x ∧ x)

= 0 ∨ (x ∧ x)

= x ∧ x

x ≤ x

x = x ∨ 0

= x ∨ (x ∧ x)

= (x ∨ x) ∧ (x ∨ x)

= 1 ∧ (x ∨ x)

= x ∨ x

x ≤ x

x = x

(∗) x

¬x ¬1 = 0 ¬0 = 1

Boole

p ≤ q p → q Boole

Lindenbaum Boole

U ⊆ X U = I nt(U )


42/85

Boole

infima

supremum

Boole

supremum ↓(x)

x ∈ B

↓(x)∩ ↓(¬x) = {0} ↓(x)∨ ↓(¬x) = B

y ≤ x y ≤ ¬x y ≤ x ∧ ¬x = 0

1 = x ∨ ¬x ∈↓(x)∨ ↓(¬x

supremum

D

x, y

x ∨ y, x ∧ y ¬x

x ∨ y ¬x ∧ ¬y

(x ∨ y) ∨ (¬x ∧ ¬y) = (x ∨ y ∨ ¬x) ∧ (x ∨ y ∨ ¬y)

= 1 ∧ 1

= 1,

(x ∨ y) ∧ (¬x ∧ ¬y) = (x ∧ ¬x ∧ ¬y) ∨ (y ∧ ¬x ∧ ¬y)

= 0 ∨ 0

= 0

x ∧ y ¬x ∨ ¬y


43/85

BOOLE

¬x ∨ ¬¬x = ¬(x ∧ ¬x)

= ¬0

= 1,

¬x ∧ ¬¬x = ¬(x ∨ ¬x)

= ¬1

= 0,

¬x

Boole

Boole

De Morgan

¬(x ∨ y) = ¬x ∧ ¬y ¬(x ∧ y) = ¬x ∨ ¬y,

¬¬x = x.

b ≤

xi {xi| i ∈ I }

1 = ¬b ∨ b = ¬b ∨

xi =

(¬b ∨ xi),

i

1 = ¬b ∨ xi

b = b ∧ 1 = b ∧ (¬b ∨ xi) = b ∨ xi,

b ≤ xi


44/85

Boole

(

) ∼= ( )

(

) ∼= ( )

(

Boole) ∼= ( )

Boole

Lindembaum

Boole

a → b = ¬a ∨ b

F

Boole

• 1 ∈ F

• a ∈ F a → b ∈ F b ∈ F b = a ∧ (¬a ∨ b) = 0 ∨ b

F ⊆ B Boole a ∈ F a ≤ b a → b = 1

b ∈ F a, b ∈ F

a → (a ∧ b) = ¬a ∨ (a ∧ b)

= (¬a ∨ a) ∧ (¬a ∨ b)

= ¬a ∨ b

≥ b,

a → (a ∧ b) ∈ F a ∧ b ∈ F F ⊆ B

B

Lindembaum

modus ponens


45/85

P

P

P

2

Idl(P )

P

f : P → 2 suprema infima

X j: X → X

idX ≤ j = j2

j

X j = {x ∈ X | j(x) = x}

j

suprema

X j X suprema

p

X

j( p)

X j

X

X j

i {(X, Y ) ∈ P (N) × P (N)| X ∩ Y = ∅}

suprema

ii

R

P

I ⊆ P • I • x I y ≤ x y I

• x y I z I x ≤ z y ≤ z

Idl(P ) P

suprema

Idl(P )

P

supre-ma

L

Idl(L) ↓(a) a ∈ L


46/85

X

{A ⊆ X | A X − A }

Boole

Boole

a ≤ b a → b = 1

¬: Bop → B

x ∧ (y ∨ z ) ≤ (x ∧ y) ∨ z.

Boole

< N

, | >

diatetagmena synola kai algebres logikis

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