Web-geometric approach to models of fuzzy logic Milan Petr k Peter - - PowerPoint PPT Presentation

Web-geometric approach to models of fuzzy logic

Milan Petr´ ık Peter Sarkoci

Department of Mathematics and Descriptive Geometry Faculty of Civil Engineering, Slovak University of Technology Radlinsk´ eho 11, 813 68 Bratislava, Slovakia

Mathematical Structures for Non-standard Logics, Prague, Czech Republic, 2009

1 / 17

Outline

1

Definitions

2

Rectangles and relations

3

Associativity of local togmas

4

Corollaries

2 / 17

Definitions

magma (G, ∗) . . . a set G with an operation ∗: G × G → G togma (G, ∗, ≤) (totally ordered magma)

(G, ∗) ... magma (G, ≤) ... chain ∀x, y, z ∈ G: x ≤ y ⇒ x ∗ z ≤ y ∗ z

monoid (G, ∗, 1, ≤)

(G, ∗) ... magma ∗ ... associative 1 ... neutral element

tomonoid (G, ∗, 1, ≤) (totally ordered monoid)

(G, ∗, 1) ... monoid (G, ≤) ... chain

3 / 17

Definitions

(A, ≤A), (B, ≤B) . . . chains f : A → B . . . non-decreasing surjection we define inverse of f: f −1 : B → P(A): y → {x ∈ A | f(x) = y} we extend f to sets: f(M) =

• x∈M

f(x) for some M ⊆ A f(f −1(x)) = x ∀x ∈ A

4 / 17

Definitions

(G, ∗, ≤) . . . togma left section at v ∈ G:

vg: G → G: x → x ∗ v

right section at u ∈ G: gu : G → G: y → u ∗ y local operation ⊛ at (u, v) ∈ G2: ⊛: (x, y) → vg−1(x) ∗ g−1

u (y)

⊛ is defined on H ⊆ G ⇔ ∀x, y ∈ H: x ⊛ y makes sense ⊛ is closed on H ⊆ G ⇔ ∀x, y ∈ H: x ⊛ y ⊆ H ⊛ is single-valued on H ⊆ G ⇔ ∀x, y ∈ H: x ⊛ y is a singleton

5 / 17

Definitions

Definition of local togma (G, ∗, ≤) . . . togma (H, ⊛, ≤) . . . local togma of (G, ∗, ≤) at (u, v) ∈ G2 on H ⊆ G if ⊛ is the local operation at (u, v) and ⊛ is defined, closed, and single-valued on H

6 / 17

Computing x ⊛ y = vg−1(x) ∗ gu−1(y):

• G

G

7 / 17

Computing x ⊛ y = vg−1(x) ∗ gu−1(y):

• u

v G G

7 / 17

Computing x ⊛ y = vg−1(x) ∗ gu−1(y):

• u

v

vg

gu G G

7 / 17

Computing x ⊛ y = vg−1(x) ∗ gu−1(y):

• u

v x y

vg

gu G G

7 / 17

Computing x ⊛ y = vg−1(x) ∗ gu−1(y):

• u

v x y

vg

gu

vg−1(x)

gu

−1(y)

G G

7 / 17

Computing x ⊛ y = vg−1(x) ∗ gu−1(y):

• u

v x y x ⊛ y

vg

gu

vg−1(x)

gu

−1(y)

G G

7 / 17

Computing x ⊛ y = vg−1(x) ∗ gu−1(y):

• u

v x y x ⊛ y x y

vg

gu

vg−1(x)

gu

−1(y)

G G

7 / 17

Computing x ⊛ y = vg−1(x) ∗ gu−1(y):

• u

v x y x y x ⊛ y

vg

gu

vg−1(x)

gu

−1(y)

G G

7 / 17

Computing x ⊛ y = vg−1(x) ∗ gu−1(y):

• u

v x y

vg

gu G G

8 / 17

Computing x ⊛ y = vg−1(x) ∗ gu−1(y):

• u

v x y

vg

gu

vg−1(x)

gu

−1(y)

G G

8 / 17

Computing x ⊛ y = vg−1(x) ∗ gu−1(y):

• u

v x y x ⊛ y

vg

gu

vg−1(x)

gu

−1(y)

G G

8 / 17

Properties of local togma

Observations: (G, ∗, ≤) . . . togma (H, ⊛, ≤) . . . local togma of (G, ∗, ≤) at (u, v) ∈ G2 on H ⊆ G u ∗ v is the neutral element of ⊛ ∗ is commutative ⇒ ⊛ is commutative ∗ is associative ⇒ ⊛ is associative

9 / 17

Outline

1

Definitions

2

Rectangles and relations

3

Associativity of local togmas

4

Corollaries

10 / 17

Rectangles and relations

(G, ∗, ≤) . . . togma rectangle P = {a, b} × {c, d} ⊂ G2 pair of rectangles P, R is (u, v)-local if: P = {a, b} × {c, v} R = {e, u} × {g, h} (where a, b, c, e, g, h ∈ G) P, R are equivalent . . . P ≅∗ R

functional values at the corresponding pairs of vertices are equal

P, R are strongly aligned according to (u, v) ∈ G2 . . . P ≃uv

∗ R

functional values at the corresponding pairs of vertices are equal, except of the pair (a, c), (e, g)

intention: to show that P ≃uv

∗ R ⇒ P ≅∗ R is related to the

associativity of local togmas

11 / 17

Rectangles and relations

Rectangle P = {a, b} × {c, d} ⊂ G2 a b c d P G G

11 / 17

Rectangles and relations

(G, ∗, ≤) . . . togma rectangle P = {a, b} × {c, d} ⊂ G2 pair of rectangles P, R is (u, v)-local if: P = {a, b} × {c, v} R = {e, u} × {g, h} (where a, b, c, e, g, h ∈ G) P, R are equivalent . . . P ≅∗ R

functional values at the corresponding pairs of vertices are equal

P, R are strongly aligned according to (u, v) ∈ G2 . . . P ≃uv

∗ R

functional values at the corresponding pairs of vertices are equal, except of the pair (a, c), (e, g)

intention: to show that P ≃uv

∗ R ⇒ P ≅∗ R is related to the

associativity of local togmas

11 / 17

Rectangles and relations

Pair of (u, v)-local rectangles P, R u v P R G G

11 / 17

Rectangles and relations

(G, ∗, ≤) . . . togma rectangle P = {a, b} × {c, d} ⊂ G2 pair of rectangles P, R is (u, v)-local if: P = {a, b} × {c, v} R = {e, u} × {g, h} (where a, b, c, e, g, h ∈ G) P, R are equivalent . . . P ≅∗ R

functional values at the corresponding pairs of vertices are equal

P, R are strongly aligned according to (u, v) ∈ G2 . . . P ≃uv

∗ R

functional values at the corresponding pairs of vertices are equal, except of the pair (a, c), (e, g)

intention: to show that P ≃uv

∗ R ⇒ P ≅∗ R is related to the

associativity of local togmas

11 / 17

Rectangles and relations

Equivalent rectangles P, R . . . P ≅∗ R P R G G ∗

11 / 17

Rectangles and relations

(G, ∗, ≤) . . . togma rectangle P = {a, b} × {c, d} ⊂ G2 pair of rectangles P, R is (u, v)-local if: P = {a, b} × {c, v} R = {e, u} × {g, h} (where a, b, c, e, g, h ∈ G) P, R are equivalent . . . P ≅∗ R

functional values at the corresponding pairs of vertices are equal

P, R are strongly aligned according to (u, v) ∈ G2 . . . P ≃uv

∗ R

functional values at the corresponding pairs of vertices are equal, except of the pair (a, c), (e, g)

intention: to show that P ≃uv

∗ R ⇒ P ≅∗ R is related to the

associativity of local togmas

11 / 17

Rectangles and relations

Rectangles P, R, strongly aligned according to (u, v) . . . P ≃uv

∗ R

u v P R G G ∗

11 / 17

Rectangles and relations

(G, ∗, ≤) . . . togma rectangle P = {a, b} × {c, d} ⊂ G2 pair of rectangles P, R is (u, v)-local if: P = {a, b} × {c, v} R = {e, u} × {g, h} (where a, b, c, e, g, h ∈ G) P, R are equivalent . . . P ≅∗ R

functional values at the corresponding pairs of vertices are equal

P, R are strongly aligned according to (u, v) ∈ G2 . . . P ≃uv

∗ R

functional values at the corresponding pairs of vertices are equal, except of the pair (a, c), (e, g)

intention: to show that P ≃uv

∗ R ⇒ P ≅∗ R is related to the

associativity of local togmas

11 / 17

Outline

1

Definitions

2

Rectangles and relations

3

Associativity of local togmas

4

Corollaries

12 / 17

Main result

Theorem: (G, ∗, ≤) . . . togma (H, ⊛, ≤) . . . local togma of (G, ∗, ≤) at (u, v) ∈ G2 Ω = vg−1(H) × g−1

u (H)

Then the following statements are equivalent:

⊛ is associative ... and thus (H, ⊛, ≤) is a tomonoid every pair of (u, v)-local rectangles P, R satisfies P ≃uv

∗ R ⇒ P ≅∗ R

13 / 17

(x ⊛ y) ⊛ z = x ⊛ (y ⊛ z) x ⊛ y = vg−1(x) ∗ gu−1(y) u v G G

14 / 17

(x ⊛ y) ⊛ z = x ⊛ (y ⊛ z) x ⊛ y = vg−1(x) ∗ gu−1(y)

• u

v

vg

gu G G

14 / 17

(x ⊛ y) ⊛ z = x ⊛ (y ⊛ z) x ⊛ y = vg−1(x) ∗ gu−1(y)

• u

v x y y z x y z

vg

gu G G

14 / 17

(x ⊛ y) ⊛ z = x ⊛ (y ⊛ z) x ⊛ y = vg−1(x) ∗ gu−1(y)

• u

v x y y z x y z

vg

gu G G

14 / 17

(x ⊛ y) ⊛ z = x ⊛ (y ⊛ z) x ⊛ y = vg−1(x) ∗ gu−1(y)

• u

v x y y z x ⊛ y x y z x ⊛ y

vg

gu G G

14 / 17

(x ⊛ y) ⊛ z = x ⊛ (y ⊛ z) x ⊛ y = vg−1(x) ∗ gu−1(y)

• u

v x y y z x ⊛ y y ⊛ z x y z x ⊛ y y ⊛ z

vg

gu G G

14 / 17

(x ⊛ y) ⊛ z = x ⊛ (y ⊛ z) x ⊛ y = vg−1(x) ∗ gu−1(y)

• u

v x y y z x ⊛ y y ⊛ z x y z x ⊛ y y ⊛ z

vg

gu G G

14 / 17

(x ⊛ y) ⊛ z = x ⊛ (y ⊛ z) x ⊛ y = vg−1(x) ∗ gu−1(y)

• u

v x y y z x ⊛ y y ⊛ z x y z x ⊛ y y ⊛ z (x ⊛ y) ⊛ z x ⊛ (y ⊛ z)

vg

gu G G

14 / 17

Outline

1

Definitions

2

Rectangles and relations

3

Associativity of local togmas

4

Corollaries

15 / 17

Corollaries

Proposition: (G, ∗, 1, ≤) . . . togma with neutral element 1 local togma at (1, 1) is actually the togma (G, ∗, 1, ≤) itself Corollary: (G, ∗, 1, ≤) is tomonoid if and only if: P ≃11

∗ R

⇒ P ≅∗ R for every pair of (1, 1)-local rectangles P, R

16 / 17

Corollaries

Proposition: (G, ∗, 1, ≤) . . . integral togma (u, v) ∈ G2 . . . point

vg: x → x ∗ v is surjection to ↓{v}

gu : y → u ∗ y is surjection to ↓{u}

a local togma can be defined at (u, v) ∈ G2 on H =↓{u ∗ v} Corollary: (G, ∗, 1, ≤) is integral tomonoid if and only if: P ≃uv

∗ R

⇒ P ≅∗ R for every pair of (u, v)-local rectangles P, R ⊂ G2 in Ω = vg−1(H) × g−1

u (H) = ↓{u}×↓{v}

17 / 17