nhóm con mờ tự do và nhóm con mờ của nhóm abel

X, Y, Z
X µ : X −→ [0, 1]
X
X FP(X)
µ ∈ FP(X) {µ(x)|x ∈ X}
µ µ(X) Im(µ)
µ
∗
= {x ∈ X|µ(x) > 0}
µ µ
µ
∗
Y X a ∈ [0, 1]
a
Y
∈ FP(X)
a
Y
(x) =

a x ∈ Y
0 x ∈ X\Y
Y Y = {y} a
{y}
a

y
1
Y
Y
µ, ν ∈ FP(X) µ(x) ≤ ν(x), ∀x ∈ X µ
ν ν µ µ ⊆ ν ν ⊇ µ
µ(x) = ν(x), ∀x ∈ X µ = ν
µ, ν ∈ FP(X)
(µ ∪ ν)(x) = µ(x) ∨ ν(x) := max{µ(x), ν(x)}
(µ ∩ ν)(x) = µ(x) ∧ ν(x) := min{µ(x), ν(x)}, ∀x ∈ X
µ ∪ ν µ ∩ ν µ ν
ν µ ν(x) = 1 − µ(x), ∀x ∈ X
{µ
i
|i ∈ I}
X I
(

i∈I
µ
i
)(x) =

i∈I
µ
i
(x) := sup
i∈I
µ
i

(x)
(

i∈I
µ
i
)(x) =

i∈I
µ
i
(x) := inf
i∈I
µ
i
(x)
µ ∈ FP(X) a ∈ [0, 1]
µ
a
= {x ∈ X|µ(x) ≥ a}
µ
a
a− a− µ
∀µ, ν ∈ FP(X)
1) µ ⊆ ν, a ∈ [0, 1] =⇒ µ
a
⊆ ν
a
2) a ≤ b, a, b ∈ [0, 1] =⇒ µ
b

⊆ µ
a
3) µ = ν ⇐⇒ µ
a
= ν
a
, ∀a ∈ [0, 1]
f X Y µ ∈ FP(X) ν ∈ FP(Y )
f(µ) ∈ FP(Y ) f
−1
(ν) ∈ FP(X)
∀y ∈ Y
f(µ)(y) :=

∨{µ(x)|x ∈ G, f(x) = y} f
−1
(y) = Ø
0
∀x ∈ X, f
−1
(ν)(x) = ν(f(x)) f(µ) µ f
f
−1
(ν) ν f
f g X Y Y Z
1) µ
i
∈ FP(X), i ∈ I, f(∪
i∈I
µ

i
) = ∪
i∈I
f(µ
i
)
µ
1
⊆ µ
2
=⇒ f(µ
1
) ⊆ f(µ
2
), ∀µ
1
, µ
2
∈ FP(X)
2) ν
j
∈ FP(Y ), j ∈ J J
f
−1
(∪
j∈J
ν
j
) = ∪
j∈J

f
−1
(ν
j
) f
−1
(∩
j∈J
ν
j
) = ∩
j∈J
f
−1
(ν
j
)
ν
1
⊆ ν
2
=⇒ f
−1
(ν
1
) ⊆ f
−1
(ν
2
), ∀ν

1
, ν
2
∈ FP(Y )
3) f
−1
(f(µ)) ⊇ µ, ∀µ ∈ FP(X) f
f
−1
(f(µ)) = µ, ∀µ ∈ FP(X) µ −→ f(µ)
FP(X) FP(Y ) ν −→ f
−1
(ν) FP(Y ) FP(X)
4) f(f
−1
(ν)) ⊆ ν, ∀ν ∈ FP(Y ) f
f(f
−1
(ν)) = ν, ∀ν ∈ FP(Y ) µ −→ f(µ) FP(X)
FP(Y ) ν −→ f
−1
(ν) FP(Y ) FP(X)
5) f(µ) ⊆ ν ⇐⇒ µ ⊆ f
−1
(ν), ∀µ ∈ FP(X), ∀ν ∈ FP(Y )
6) g(f(µ)) = (g ◦ f)(µ), ∀µ ∈ FP(X) f
−1
(g
−1
(ξ)) = (g ◦ f)

−1
(ξ), ∀ξ ∈
FP(Z)
G
e
µ ∈ FP(G) µ
G µ ∀x, y ∈ G
1) µ(xy) ≥ µ(x) ∧ µ(y)
2) µ(x
−1
) ≥ µ(x)
G F(G)
µ ∈ F(G) H G µ|
H
∈ F(H)
Z µ
µ(x) =

a x ∈ 2Z
b x ∈ 2Z + 1.
a, b ∈ [0, 1] b ≤ a µ Z
µ ∈ F(G) x ∈ G
1) µ(e) ≥ µ(x)
2) µ(x) = µ(x
−1
)
µ ∈ FP(G)
1) µ ∈ F(G)
2) µ(x
−1

y) ≥ µ(x) ∧ µ(y)
3) µ
a
G a ∈ µ(G) ∪ [0, µ(e)]
1) =⇒ 3). A = µ(G) ∪ [0, µ(e)] a ∈ A a ≤ µ(e) e ∈ µ
a
µ
a
= Ø ∀x, y ∈ µ
a
µ(x
−1
y) ≥ µ(x
−1
) ∧ µ(y) ≥ µ(x) ∧ µ(y) ≥ a ∧ a = a
x
−1
y ∈ µ
a
µ
a
G
3) =⇒ 2). x, y ∈ G a = µ(x), b = µ(y) c = a ∧ b c ∈ µ(G)
µ
c
G x, y ∈ µ
c
x
−1
y ∈ µ

c
µ(x
−1
y) ≥ c = a ∧ b = µ(x) ∧ µ(y)
2) =⇒ 1). x, y ∈ G µ(e) = µ(x
−1
x) ≥ µ(x) ∧ µ(x) = µ(x)
µ(x
−1
) = µ(x
−1
e) ≥ µ(x
−1
) ∧ µ(e) ≥ µ(x) ∧ µ(e) = µ(x),
µ(xy) = µ((x
−1
)
−1
y) ≥ µ(x
−1
) ∧ µ(y) ≥ µ(x) ∧ µ(y)
µ ∈ F(G)
µ ∈ F(G) µ
∗
µ
∗
G
µ
∗
= {x ∈ G|µ(x) = µ(e)} µ

∗
= {x ∈ G|µ(x) > 0}
∀µ, ν ∈ FP(G) ∀x ∈ G
(µ ◦ ν)(x) = ∨{µ(y) ∧ ν(z)|y, z ∈ G, yz = x} µ
−1
(x) = µ(x
−1
)
µ ◦ ν µ
−1
µ ν µ
◦
µ ◦ ν µ
−1
G
µ, ν, µ
i
∈ FP(G), i ∈ I a = ∨{µ(x)|x ∈ G}
1) µ ◦ ν(x) = ∨
y∈G
(µ(y) ∧ ν(y
−1
x)) = ∨
y∈G
(µ(xy
−1
) ∧ ν(y)), ∀x ∈ G
2) (a
y
◦ µ)(x) = µ(y

−1
x), ∀x, y ∈ G
3) (µ ◦ a
y
)(x) = µ(xy
−1
), ∀x, y ∈ G
4) (µ
−1
)
−1
= µ
5) µ ⊆ µ
−1
⇐⇒ µ
−1
⊆ µ ⇐⇒ µ = µ
−1
⇐⇒ µ(x) ≤ µ(x
−1
), ∀x ∈ G
⇐⇒ µ(x
−1
) ≤ µ(x), ∀x ∈ G ⇐⇒ µ(x) = µ(x
−1
), ∀x ∈ G.
6) µ ⊆ ν ⇐⇒ µ
−1
⊆ ν
−1

7) (

i∈I
µ
i
)
−1
=

i∈I
µ
−1
i
8) (

i∈I
µ
i
)
−1
=

i∈I
µ
−1
i
9) (µ ◦ ν)
−1
= ν
−1

◦ µ
−1
(a
y
◦ µ)(x) = ∨
z∈G
{a
y
(z) ∧ µ(z
−1
x)} = ∨
z∈G
{a ∧ µ(y
−1
x), 0}
= a ∧ µ(y
−1
x) = µ(y
−1
x).
∀x ∈ G (µ ◦ ν)
−1
(x) = (µ ◦ ν)(x
−1
) = ∨
y∈G
{µ(x
−1
y
−1

) ∧ ν(y)}
= ∨
z∈G
{ν(z
−1
) ∧ µ(x
−1
z)} = ∨
z∈G
{ν(z
−1
) ∧ µ((z
−1
x)
−1
)}
= ∨
z∈G
{ν
−1
(z) ∧ µ
−1
(z
−1
x)} = (ν
−1
◦ µ
−1
)(x)
(µ ◦ ν)

−1
= ν
−1
◦ µ
−1
µ ∈ FP(G) µ ∈ F(G)
µ ◦ µ ⊆ µ
µ
−1
⊇ µ
µ ∈ F(G) µ
−1
(x) = µ(x
−1
) ≥ µ(x), ∀x ∈ G
µ
−1
⊇ µ x ∈ G
µ(x) = µ(yz) ≥ µ(y) ∧ µ(z)
y, z ∈ G yz = x
µ(x) ≥ ∨{µ(y) ∧ µ(z)|y, z ∈ G, yz = x} = (µ ◦ µ)(x)
µ ◦ µ ⊆ µ
µ ◦ µ ⊆ µ µ
−1
⊇ µ
µ(x
−1
) = µ
−1
(x) ≥ µ(x)

µ(xy) ≥ (µ ◦ µ)(xy) = ∨{µ(z) ∧ µ(t)|z, t ∈ G, zt = xy} ≥ µ(x) ∧ µ(y)
z = x, t = y
µ ∈ F(G)
µ, ν ∈ F(G) µ ◦ ν ∈ F(G) ⇐⇒ µ ◦ ν = ν ◦ µ
µ, ν ∈ F(G) µ ◦ ν ∈ F(G) µ ∈ F(G)
µ
−1
⊇ µ µ
−1
= µ
ν
−1
= ν (µ ◦ ν)
−1
= µ ◦ ν µ ◦ ν = (µ ◦ ν)
−1
= ν
−1
◦ µ
−1
= ν ◦ µ
µ, ν ∈ F(G) µ ◦ ν = ν ◦ µ
(µ ◦ ν)
−1
= (ν ◦ µ)
−1
= µ
−1
◦ ν
−1

= µ ◦ ν
(µ ◦ ν) ◦ (µ ◦ ν) = µ ◦ (ν ◦ µ) ◦ ν = µ ◦ (µ ◦ ν) ◦ ν = (µ ◦ µ) ◦ (ν ◦ ν) ⊆ µ ◦ ν
µ ◦ ν ∈ F(G)
f : G −→ H µ ∈ F(G)
ν ∈ F(H) f(µ) ∈ F(H) f
−1
(ν) ∈ F(G)
{µ
i
|i ∈ I} ⊆ F(G)

i∈I
µ
i
∈ F(G)
µ ∈ FP(G)
< µ >=

{ν|µ ⊆ ν, ν ∈ F(G)}
G µ
< µ > G µ
µ ∈ F(G) µ
G µ G µ(xy) = µ(yx), ∀x, y ∈ G
G N F(G)
G G
G
µ ∈ F(G)
1) µ ∈ N F(G)
2) µ(xyx
−1

) = µ(y), ∀x, y ∈ G
3) µ
a
G ∀a ∈ µ(G) ∪ [0, µ(e)]
4) µ(xyx
−1
) ≥ µ(y), ∀x, y ∈ G
5) µ(xyx
−1
) ≤ µ(y), ∀x, y ∈ G
6) µ ◦ ν = ν ◦ µ, ∀ν ∈ FP(G)
1) =⇒ 2). ∀x, y ∈ G µ(xyx
−1
) = µ((xy)x
−1
) = µ(x
−1
(xy)) = µ(y)
2) =⇒ 3). ∀x ∈ G ∀y ∈ µ
a
µ(xyx
−1
) = µ(y) ≥ a xyx
−1
∈ µ
a
µ
a
G
3) =⇒ 4). x, y ∈ G a = µ(y) y ∈ µ

a
µ
a
G xyx
−1
∈ µ
a
µ(xyx
−1
) ≥ a = µ(y)
4) =⇒ 5). ∀x, y ∈ G µ(xyx
−1
) ≤ µ((x
−1
)(xyx
−1
)(x
−1
)
−1
) = µ(y)
5) =⇒ 1). ∀x, y ∈ G µ(xy) = µ(x(yx)x
−1
) ≤ µ(yx)
µ(yx) = µ(y(xy)y
−1
) ≤ µ(xy)
µ(xy) = µ(yx) µ ∈ N F(G)
1) =⇒ 6). x ∈ G
(µ ◦ ν)(x) = ∨

y∈G
{µ(xy
−1
) ∧ ν(y)} = ∨
y∈G
{µ(y
−1
x) ∧ ν(y)}
= ∨
y∈G
{ν(y) ∧ µ(y
−1
x)} = (ν ◦ µ)(x).
µ ◦ ν = ν ◦ µ
(6) =⇒ 1)) : y ∈ G 1
{y
−1
}
◦ µ = µ ◦ 1
{y
−1
}
x ∈ G (1
{y
−1
}
◦ µ)(x) = (µ ◦ 1
{y
−1
}

)(x)
∨
z∈G
{1
{y
−1
}
(z) ∧ µ(z
−1
x)} = ∨
z∈G
{µ(xz
−1
) ∧ 1
{y
−1
}
(z)} (∗)
1
{y
−1
}
(∗)
1
{y
−1
}
(y
−1
) ∧ µ((y

−1
)
−1
x) = µ(x(y
−1
)
−1
) ∧ 1
{y
−1
}
(y
−1
)
µ(xy) = µ(yx) µ ∈ N F(G)
µ ∈ N F(G) µ
∗
✁ G µ
∗
✁ G
µ ∈ F(G) x ∈ G µ(e)
{x}
◦µ
µ ◦ µ(e)
{x}
µ x
xµ µx µ ∈ N F(G) xµ = µx
xµ
(µ(e)
{x}

◦ µ)(z) = µ(x
−1
z)
µ ∈ F(G) x, y ∈ G
1) xµ = yµ ⇐⇒ xµ
∗
= yµ
∗
2) µx = µy ⇐⇒ µ
∗
x = µ
∗
y
µ ∈ N F(G) x, y ∈ G xµ = yµ µ(x) = µ(y)
µ ∈ N F(G) G/µ = {xµ|x ∈ G}
1) xµ ◦ yµ = (xy)µ, ∀x, y ∈ G
2) (G/µ, ◦)
3) G/µ
∼
=
G/µ
∗
4) µ
(∗)
∈ FP(G/µ) µ
(∗)
(xµ) = µ(x), ∀x ∈ G
µ
(∗)
∈ N F(G/µ)

1) x, y ∈ G
xµ ◦ yµ = (µ(e)
{x}
◦ µ) ◦ (µ(e)
{y}
◦ µ) = µ(e)
{x}
◦ (µ ◦ µ(e)
{y}
) ◦ µ
= µ(e)
{x}
◦ µ(e)
{y}
◦ (µ ◦ µ) = (µ(e)
{x}
◦ µ(e)
{y}
) ◦ µ = (xy)µ.
2) 1) ◦ G/µ (G/µ, ◦)
µ = eµ xµ ∈ G/µ x
−1
µ
3) xµ = yµ ⇐⇒ xµ
∗
= yµ
∗
f : G/µ → G/µ
∗
, xµ → xµ

∗
f(xµ ◦ yµ) = f((xy)µ) = xyµ
∗
= xµ
∗
yµ
∗
= f(xµ)f(yµ)
f G/µ
∼
=
G/µ
∗
4) ∀x, y ∈ G µ
(∗)
(y
−1
µ ◦ xµ) = µ
(∗)
(y
−1
xµ) = µ(y
−1
x)
≥ µ(y) ∧ µ(x) = µ
(∗)
(yµ) ∧ µ
(∗)
(xµ)
µ

(∗)
∈ F(G/µ)
µ
(∗)
(y
−1
µ ◦ xµ ◦ yµ) = µ
(∗)
(y
−1
xyµ) = µ(y
−1
xyµ) = µ(x) = µ
(∗)
(xµ)
µ
(∗)
∈ N F(G)
G/µ G
µ
ν ∈ F(G) N
G ξ ∈ FP(G)
ξ(xN) = ∨{ν(z)|z ∈ xN}, ∀x ∈ G
ξ ∈ F(G/N)
ξ
ν
N G ν/N
µ ∈ N F(G) ν ∈ N F(H) H
f G H
1) f(µ) ∈ N F(H)

2) f
−1
(ν) ∈ N F(G)
µ, ν ∈ F(G) µ ⊆ ν µ
ν µ ✁ ν
µ(xyx
−1
) ≥ µ(y) ∧ ν(x), ∀x, y ∈ G
1) G
1
G
2
G G
1
✁ G
2
1
G
1
✁ 1
G
2
1
G
✁ 1
G
1
e
✁ 1
G

2) µ ∈ N F(G), ν ∈ F(G) µ ⊆ ν µ ✁ ν
3)
4) µ ∈ FP(G) G µ✁1
G
µ ⊆ 1
G
x, y ∈ G µ(xyx
−1
) = µ(y) ≥ µ(y) ∧1
G
(x)
µ ✁ 1
G
µ ✁ 1
G
µ(xyx
−1
) ≥ µ(y) ∧ 1
G
(x) = µ(y)
∀x, y ∈ G µ ∈ N F(G)
µ, ν ∈ F(G) µ ⊆ ν
1) µ ✁ ν
2) µ(yx) ≥ µ(xy) ∧ ν(y), ∀x, y ∈ G
3) µ
a
✁ ν
a
, ∀a ∈ [0, µ(e)]
4) µ(e)

{x}
◦ µ ⊇ (µ ◦ µ(e)
{x}
) ∩ ν, ∀x ∈ G
1) =⇒ 2). ∀x, y ∈ G, µ(yx) = µ(y(xy)y
−1
) ≥ µ(xy) ∧ ν(y)
2) =⇒ 3). a ∈ [0, µ(e)] e ∈ µ
a
µ
a
= Ø x ∈ µ
a
µ(x) ≥ a ν(x) ≥ a x ∈ ν
a
µ
a
⊆ ν
a
∀x ∈ ν
a
, y ∈ µ
a
xy ∈ ν
a
µ((xy)x
−1
) ≥ µ(x
−1
xy) ∧ ν(xy) ≥ µ(y) ∧ ν(xy) ≥ a

xyx
−1
∈ µ
a
µ
a
✁ ν
a
3) =⇒ 1). x, y ∈ G µ(y) = a, ν(x) = b b ≥ a
x ∈ ν
a
xyx
−1
∈ µ
a
µ(xyx
−1
) ≥ a = a ∧ b = µ(y) ∧ ν(x) µ ✁ ν
2) =⇒ 4). ∀z ∈ G
(µ(e)
{x}
◦ µ)(z) = ∨{µ(e)
{x}
(u) ∧ µ(v)|u, v ∈ G, z = uv}
≥ ∨{µ(e)
{x}
(x) ∧ µ(v)|u, v ∈ G, z = xv} = ∨{µ(e)
{x}
(x) ∧ µ(x
−1

z)}
= ∨{µ(x
−1
z)} = ∨{µ(z
−1
x)}
≥ ∨{µ(xz
−1
)} ∧ ν(z
−1
) = ∨{µ(zx
−1
)} ∧ ν(z
−1
)
= (µ ◦ µ(e)
{x}
)(z) ∧ ν(z) = ((µ ◦ µ(e)
{x}
) ∩ ν)(z).
4) =⇒ 2).∀x, y ∈ G
µ(yx) = µ(x
−1
y
−1
) = (µ(e)
{x}
◦ µ)(y
−1
) ≥ ((µ ◦ µ(e)

{x}
) ∩ ν)(y
−1
)
= (µ◦µ(e)
{x}
)(y
−1
)∧ν(y
−1
) = µ(y
−1
x
−1
)∧ν(y
−1
) = µ(xy)∧ν(y).
µ, ν ∈ F(G) µ ✁ ν µ
∗
✁ ν
∗
µ
∗
✁ ν
∗
f : G −→ H
1) µ, ν ∈ F(G), µ ✁ ν f(µ) ✁ f(ν)
2) µ, ν ∈ F(H), µ ✁ ν f
−1
(µ) ✁ f

−1
(ν)
G H µ ∈ F(G) ν ∈ F(H)
1) f : G −→ H µ ν
f(µ) ⊆ ν µ ν µ
f
∼ ν µ ∼ ν
2) f : G −→ H µ ν
f(µ) ⊆ ν µ ν µ
f
 ν µ  ν
3) f : G −→ H µ ν
f(µ) = ν µ ν µ
f
≈ ν µ ≈ ν
4) f : G −→ H µ ν
f(µ) = ν µ ν µ
f
∼
=
ν µ
∼
=
ν
µ, ν ∈ F(G) µ ✁ ν µ
∗
✁ ν
∗
ν|
ν

∗
ν
∗
ν|
ν
∗
µ
∗
(ν|
ν
∗
)/µ
∗
:= ν/µ
ν µ
f : G → Y µ ∈ FP(G)
(f(µ))
∗
= f(µ
∗
)
µ ∈ N F(G) ν ∈ F(G) µ(e) = ν(e)
ν/(µ ∩ ν)  (µ ◦ ν)µ
µ
∗
✁ G
ν
∗
/(µ
∗

∩ ν
∗
)
f
∼
=
(µ
∗
ν
∗
)/µ
∗
µ
∗
∩ ν
∗
= (µ ∩ ν)
∗
µ
∗
ν
∗
= (µ ◦ ν)
∗
ν
∗
/(µ ∩ ν)
∗
f
∼

=
(µ ◦ ν)µ
∗
f(x(µ
∗
∩ ν
∗
)) = xµ
∗
∀x ∈ ν
∗
∀y ∈ ν
∗
f(ν/(µ ∩ ν))(yµ
∗
) = ∨{ν/(µ ∩ ν)(x(µ
∗
∩ ν
∗
))|x(µ
∗
∩ ν
∗
) ∈ ν
∗
/(µ
∗
∩ ν
∗
),

f(x(µ
∗
∩ ν
∗
)) = yµ
∗
}
f x(µ
∗
∩ ν
∗
) ∈ ν
∗
/(µ
∗
∩ ν
∗
)
f(x(µ
∗
∩ ν
∗
)) = yµ
∗
x(µ
∗
∩ ν
∗
) = y(µ
∗

∩ ν
∗
)
f(ν/(µ ∩ ν))(yµ
∗
) = ν/(µ ∩ ν)(y(µ
∗
∩ ν
∗
)) = ∨{ν(z)|z ∈ y(µ
∗
∩ ν
∗
)}
≤ ∨{(µ ◦ ν)(z)|z ∈ y(µ
∗
∩ ν
∗
)}
µ(e) = ν(e) ν(z) = ν(e) ∧ ν(z) = µ(e) ∧ ν(z) ≤ (µ ◦ ν)(z))
≤ ∨{(µ ◦ ν)(z)|z ∈ yµ
∗
}
= ((µ ◦ ν)/µ)(yµ
∗
).
f(ν/(µ ∩ ν)) ⊆ (µ ◦ ν)/µ ν/(µ ∩ ν)
f
 (µ ◦ ν)µ
µ, ν, ξ ∈ F(G) µ ν

ξ µ ⊆ ν (ξ/µ)/(ν/µ)
∼
=
ξ/ν
µ
∗
✁ ξ
∗
ν
∗
✁ ξ
∗
µ ⊆ ν
µ
∗
✁ ν
∗
(ξ
∗
/µ
∗
)/(ν
∗
/µ
∗
)
f
∼
=
ξ

∗
/ν
∗
f(xµ
∗
.(ν
∗
/µ
∗
)) = xν
∗
, ∀x ∈ ξ
∗
∀x ∈ ξ
∗
f((ξ/µ)/(ν/µ))(xν
∗
) = ∨{(ξ/µ)/(ν/µ)(yµ
∗
.(ν
∗
/µ
∗
))|f(yµ
∗
.(ν
∗
/µ
∗
)) = xν

∗
}
f yµ
∗
.(ν
∗
/µ
∗
) ∈ (ξ
∗
/µ
∗
)/(ν
∗
/µ
∗
)
f(yµ
∗
.(ν
∗
/µ
∗
)) = xν
∗
yµ
∗
.(ν
∗
/µ

∗
) = xµ
∗
.(ν
∗
/µ
∗
)
f((ξ/µ)/(ν/µ))(xν
∗
) = (ξ/µ)/(ν/µ)(xµ
∗
.(ν
∗
/µ
∗
))
= ∨{(ξ/µ)(yµ
∗
)|yµ
∗
∈ xµ
∗
.(ν
∗
/µ
∗
), y ∈ ξ
∗
}

= ∨{∨{ξ(z)|z ∈ yµ
∗
}|yµ
∗
∈ xµ
∗
.(ν
∗
/µ
∗
), y ∈ ξ
∗
}
= ∨{ξ(z)|z ∈ ξ
∗
, z ∈ xµ
∗
.(ν
∗
/µ
∗
)}
= ∨{ξ(z)|z ∈ ξ
∗
, f(z) ∈ f(xµ
∗
.(ν
∗
/µ
∗

)) = xν
∗
}
= (ξ/ν)(xν
∗
), ∀x ∈ ξ
∗
.
f((ξ/µ)/(ν/µ)) = ξ/ν (ξ/µ)/(ν/µ)
f
∼
=
ξ/ν
µ ∈ F(G) x ∈ G n
µ(x
n
) = µ(e)(∗) (∗)
x µ FO
µ
(x)
n (∗) x µ
µ ∈ F(G), x ∈ G FO
µ
(x) = n
1) m µ(x
m
) = µ(e) n|m
2) m F O
µ
(x

m
) =
n
(n, m)
3) x, y ∈ G xy = yx (F O
µ
(x), F O
µ
(y)) = 1 F O
µ
(xy) =
F O
µ
(x).F O
µ
(y)
1) m n m = nq + r 0 ≤ r < n
µ(x
r
) = µ(x
m−nq
) = µ(x
m
x
−nq
) ≥ µ(x
m
) ∧ µ((x
n
)

−q
)
≥ µ(x
m
) ∧ µ(x
n
) = µ(e) ∧ µ(e) = µ(e)
µ(x
r
) = µ(e) n µ(x
n
) = µ(e)
r = 0 n|m
2) F O
µ
(x
m
) = k d = (m, n) m = du, n = dv, (u, v) = 1
µ(x
m.
n
d
) = µ(x
nu
) ≥ µ(x
n
) = µ(e)
k|
n
d

d = (m, n) ∃i, j ∈ Z ni+ mj = d.
µ(x
kd
) = µ(x
k(ni+mj)
= µ(x
kni
.x
kmj
) ≥ µ(x
n
)∧µ(x
k
) = µ(e)∧µ(e) = µ(e)
µ(x
kd
) = µ(e) n|kd
n
d
|k k =
n
d
3) F O
µ
(xy) = k, F O
µ
(y) = l
µ((xy)
nl
) = µ(x

nl
.y
nl
) ≥ µ(x
nl
) ∧ µ(y
nl
) ≥ µ(x
n
) ∧ µ(y
l
) = µ(e)
k|nl
u = x
k
, v = y
k
n

= F O
µ
(u) =
n
(n, k)
, l

= F O
µ
(v) =
l

(l, k)
(n

, l

) = 1 (n, l) = 1
µ(e) = µ((xy)
k
) = µ(x
k
y
k
) = µ(uv) ≤ µ((uv)
l

) = µ(u
l

v
l

)
⇒ µ(u
l

v
l

) = µ(e)
µ(u

l

) = µ(u
l

v
l

v
−l

) ≥ µ(u
l

v
l

) ∧ µ(v
−l

) = µ(e) ∧ µ(e) = µ(e)
⇒ µ(u
l

) = µ(e) ⇒ n

|l

⇒ n


= 1 ⇒ µ(x
k
) = µ(e)
µ(y
k
) = µ(e) n|k l|k nl|k
k = nl (n, l) = 1
x ∈ G o(x) x G
o(x) F O
µ
(x) µ
G F O
µ
(x)|o(x) o(x)
n µ
n
G FO
µ
n
(x) = n
µ
n
G ∀y ∈ G,
µ
n
(y) =

t
0
y ∈ x

n

t
1
y /∈ x
n

0 ≤ t
1
< t
0
≤ 1 µ
n
∈ F(G) F O
µ
n
(x) = n
{G
i
|i ∈ I} e
i
G
i
, i ∈ I
G =

i∈I
G
i
(x

i
)
i∈I
(y
i
)
i∈I
= (x
i
y
i
)
i∈I
, ∀(x
i
)
i∈I
, (y
i
)
i∈I
∈ G
G (e
i
)
i∈I
G
i
, i ∈ I
∼


i∈I
G
i
I = {1, 2, . . . , n}, n ≥ 1
G
1
× G
2
× . . . × G
n
= {(x
1
, x
2
, . . . , x
n
)|x
i
∈ G
i
, i ∈ I}
G
1
∼
⊗ G
2
∼
⊗ . . .
∼

⊗ G
n
{G
i
|i ∈ I} e
i
G
i
, i ∈ I G =
∼

i∈I
G
i
µ
i
∈ F(G
i
), ∀i ∈ I
∼

i∈I
µ
i
∈ F(G)
∀(x
i
)
i∈I
(

∼

i∈I
µ
i
)((x
i
)
i∈I
) = ∧
i∈I
µ
i
(x
i
)
∼

i∈I
µ
i
G
µ
i
, i ∈ I
µ
i
∈ N F(G
i
), i ∈ I µ =

∼

i∈I
µ
i
∈ N F(G)
G e x
i
∈ G i ∈ I |I| > 1
x
i
e

i∈I
x
i

i∈I
x
i
:=

e x
i
= e, ∀i ∈ I

i∈I,x
i
=e
x

i
x
i
= e i ∈ I
x
i
e

i∈I
x
i

i∈i
x
i
|I| > 1

x
i

i∈I
x
i
G
i
G, ∀i ∈ I
x
i
∈ G
i

x
j
∈ G
j
, i, j ∈ I, i = j =⇒ x
i
x
j
= x
j
x
i
.
x =

x
i
y =

y
i
, x
i
, y
i
∈ G, ∀i ∈ I =⇒ xy =

x
i
y

j
.
A
i
, i ∈ I G {

x
i
|x
i
∈ A
i
, i ∈ I}
A
i
∗

i∈I
A
i
e ∈ A
i
, ∀i ∈ I, A
j
⊆
∗

i∈I
A
i

, ∀j ∈ I
G
i
G, i ∈ I G
G
i
G =
•

i∈I
G
i
1) G =
∗

i∈I
G
i
2) x
i
∈ G
i
x
j
∈ G
j
, i, j ∈ I, i = j =⇒ x
i
x
j

= x
j
x
i
3)

x
i
=

y
i
x
i
, y
i
∈ G
i
=⇒ x
i
= y
i
, ∀i ∈ I
2)
2

) G
i
G
3)

3

) e =

x
i
x
i
∈ G
i
=⇒ x
i
= e, ∀i ∈ I
3

) G
j
∩
∗

i∈I\{j}
G
i
= {e}, ∀j ∈ I
I
G =
•

i∈I
G

i
I = {1, 2, . . . , n} G
1
∼
⊗ G
2
∼
⊗ . . .
∼
⊗ G
n
G
1
⊗ G
2
⊗ . . . ⊗ G
n
G
i
G
i
i ∈ I µ
i
∈ FP(G)
µ G
µ(x) = ∨{∧
i∈I
µ
i
(x

i
)|x
i
∈ G, i ∈ I,

x
i
= x}, ∀x ∈ G
µ µ
i
µ =
∗

i∈I
µ
i
I = {1, 2, . . . , n} n ∈ N \ {1} µ
i
(e) ≥ µ
j
(x)
µ
i
⊆ µ ∀i, j ∈ I, ∀x ∈ G
∗

i∈I
µ
i
⊆ µ

1
◦ µ
2
◦ . . . ◦ µ
n
µ
i
x
i
∈ µ
∗
i
x
j
∈ µ
∗
j
, i, j ∈ I, i = j =⇒ x
i
x
j
= x
j
x
i
∗

i∈I
µ
i

= µ
1
◦ µ
2
◦ . . . ◦ µ
n
i ∈ I µ
i
∈ F(G) µ =
∗

i∈I
µ
i
1) µ
∗
⊇
∗

i∈I
(µ
i
)
∗
2) ∨{(∪
i∈I
µ
i
(G)) \ {µ(e)}} < µ(e) µ
∗

=
∗

i∈I
(µ
i
)
∗
3) µ
∗
=
∗

i∈I
(µ
i
)
∗
1) x ∈
∗

i∈I
(µ
i
)
∗
x =

x
i

x
i
∈ (µ
i
)
∗
µ
i
(x
i
) = µ
i
(e)
µ(x) = (
∗

i∈I
µ
i
)(x) = (
∗

i∈I
µ
i
)(

x
i
)

= ∨{∧
i∈I
µ
i
(x
i
)|x
i
∈ G, i ∈ I,

x
i
= x}
= ∨{∧
i∈I
µ
i
(e)|x
i
∈ G, i ∈ I,

x
i
= x} = µ(e)
x ∈ µ
∗
µ
∗
⊇
∗


i∈I
(µ
i
)
∗
2) 1) µ
∗
⊆
∗

i∈I
(µ
i
)
∗
∀x ∈ µ
∗
µ(x) = µ(e) ⇐⇒ (
∗

i∈I
µ
i
)(x) = (
∗

i∈I
µ
i

)(e)
⇐⇒ ∨{∧
i∈I
µ
i
(x
i
)|x
i
∈ G, i ∈ I,

x
i
= x} = {∨{∧
i∈I
µ
i
(y
i
)|y
i
∈ G,
i ∈ I,

y
i
= e}}
⇐⇒ ∨{∧
i∈I
µ

i
(x
i
)|x
i
∈ G, i ∈ I,

x
i
= x} = µ(e)
∨((∪
i∈I
µ
i
(G)) \ {µ(e)}) < µ(e) µ
i
(x
i
) = µ(e), ∀i ∈ I
µ
i
(e) = µ(e) µ
i
(x
i
) = µ
i
(e)
x =


x
i
, µ
i
(x
i
) = µ
i
(e), ∀i ∈ I x ∈
∗

i∈I
(µ
i
)
∗
3) x ∈ µ
∗
⇐⇒ µ(x) > 0 ⇐⇒
∗

i∈I
µ
i
)(x) > 0
⇐⇒ ∨{∧
i∈I
µ
i
(x

i
)|x
i
∈ G, i ∈ I,

x
i
= x} > 0
⇐⇒ x =

x
i
, µ
i
(x
i
) > 0, ∀i ∈ I
⇐⇒ x =

x
i
, x ∈ µ
∗
i
, i ∈ I
⇐⇒ x ∈
∗

i∈I
(µ

i
)
∗
µ
∗
=
∗

i∈I
(µ
i
)
∗
µ
i
∈ F(G) ∀i ∈ I µ
i
(e) = µ
j
(e) ∀i, j ∈ I
µ =
∗

i∈I
µ
i
H H
i
, i ∈ I G
µ

∗
i
⊆ H
i
, ∀i ∈ I H H
i
H =
•

i∈I
H
i
1) µ ∈ F(G)
2) µ
i
µ
3) µ
j
∩ (
∗

i∈I\{j}
µ
i
) = µ(e)
{e}
, ∀j ∈ I
4) µ
i
|

H
∈ N F(H), ∀i ∈ I µ|
H
∈ N F(H)
1) x, y ∈ G
x /∈ H y /∈ H x /∈ H x /∈
•

i∈I
H
i
x
i
∈ H
i
, i ∈ I x =

x
i
µ
∗
i
⊆ H
i
, ∀i ∈ I x
i
∈ µ
∗
i
, i ∈ I

x =

x
i
, i ∈ I x /∈
∗

i∈I
(µ
i
)
∗
= µ
∗
µ(x) = 0
y /∈ H µ(y) = 0 µ(xy) ≥ 0 = µ(x) ∧ µ(y)
x ∈ H y ∈ H xy ∈ H µ(xy) ≥ µ(x) ∧ µ(y) ∀z ∈ G,
µ(z
−1
) = ∨{∧
i∈I
µ
i
(y
i
)|y
i
∈ G, i ∈ I,

y

i
= z
−1
}
= ∨{∧
i∈I
µ
i
(y
−1
i
)|y
i
∈ G, i ∈ I,

y
−1
i
= z} = µ(z).
µ ∈ F(G)
2) j ∈ I µ
j
∈ F(G) µ
j
⊆ µ
µ
j
(yx) ≥ µ
j
(xy) ∧ µ(y), ∀x, y ∈ G

xy /∈ H
j
y /∈ H
µ
j
(yx) ≥ 0 = µ
j
(xy) ∧ 0 = µ(xy) ∧ µ(y), ∀x, y ∈ G.
xy ∈ H
j
y ∈ H H
j
⊆ H x = xyy
−1
∈ H
H
j
, j ∈ I H yx = x
−1
xyx ∈ H
j
µ(xy) = (
∗

i∈I
µ
i
)(xy) = ∨{∧
i∈I
µ

i
(x
i
)|x
i
∈ G, i ∈ I,

x
i
= xy} = µ
j
(xy)
xy ∈ H
j
∀i = j, µ
i
(x
i
) = µ(e) ≥ µ(z), ∀z ∈ G ⇒ ∧
i∈I
µ
i
(x
i
) = µ
j
(xy))
µ(yx) = µ
j
(yx)

µ
j
(yx) = µ(yx) = µ(yxyy
−1
) ≥ µ(xy) ∧ µ(y) = µ
j
(xy) ∧ µ(y).
3) x ∈ G
(µ
j
∩ (
∗

i∈I\{j}
µ
i
))(x) = µ
j
(x) ∧ (
∗

i∈I\{j}
µ
i
)(x)
= µ
j
(x) ∧ ∨{∧
i∈I\{j}
µ

i
(x
i
)|x
i
∈ G,

x
i
= x}
x = e µ
i
(e) = µ
j
(e), ∀i, j ∈ I
(µ
j
∩ (
∗

i∈I\{j}
µ
i
))(x) = µ
j
(e) = µ(e) = µ(e)
{e}
(e)
j ∈ I
x = e x /∈ H

j
µ
j
(x) = 0 (µ
j
∩ (
∗

i∈I\{j}
µ
i
))(x) = 0
x ∈ H
j
x /∈ H
i
, ∀i ∈ I \ {j}
0 = µ
i
(x) = µ
i
(

x
i
) ≥ ∧
i∈I\{j}
µ
i
(x

i
)
(µ
j
∩

i∈I\{j}
)(x) = 0 0 = (µ
j
∩

i∈I\{j}
)(x) = µ(e)
{e}
(x)
4) µ|
H
∈ F(H) x, y ∈ H x, y
x =

x
i
y =

y
i
, x
i
, y
i

∈ H
i
, ∀i ∈ I xy =

(x
i
y
i
)
yx =

(y
i
x
i
) µ(xy) = ∧
i∈I
µ
i
(x
i
y
i
) = ∧
i∈I
µ
i
(y
i
x

i
) = µ(yx)
µ|
H
∈ N F(H)
µ ∈ F(G) µ
i
∈ F(G), ∀i ∈ I µ
i
(e) = µ
j
(e)
∀i, j ∈ I µ µ
i
µ =
•

i∈I
µ
i
1) µ =
∗

i∈I
µ
i
2) µ
i
µ
3) µ

j
∩ (
∗

i∈I\{j}
µ
i
) = µ(e)
{e}
, ∀j ∈ I
I = {1, 2, . . . , n}, n ∈ N µ
1
•
⊗ µ
2
•
⊗ . . .
•
⊗ µ
n
µ
i
µ µ
µ
i
, i ∈ I
µ ∈ F(G) µ
i
∈ F(G) ∀i ∈ I µ
i

(e) = µ
j
(e)
∀i, j ∈ I
µ =
•

i∈I
µ
i
⇐⇒ µ =
∗

i∈I
µ
i
µ
∗
=
•

i∈I
µ
∗
i
H
i
= µ
∗
i

H = µ
∗
µ =
•

i∈I
µ
i
µ =
∗

i∈I
µ
i
µ
∗
=
∗

i∈I
µ
∗
i
µ
i
µ ∀i ∈ I µ
∗
i
µ
∗

2

)
x ∈ µ
∗
j
∩ (
∗

i∈I\{j}
µ
∗
i
) ⇐⇒ x ∈ µ
∗
j
∩ (
∗

i∈I\{j}
µ
i
)
∗
⇐⇒ µ
j
(x) > 0 (
∗

i∈I\{j}

µ
i
)
∗
(x) > 0 ⇐⇒ (µ
j
∩ (
∗

i∈I\{j}
µ
i
))(x) > 0
⇐⇒ x = e
3

)
µ
∗
=
•

i∈I
µ
∗
i
µ ∈ F(G) {µ
i
|i ∈ I}
G µ

i
⊆ µ, ∀i ∈ I
1) ∪
i∈I
|µ
i
(G)|
2) µ
∗
=
•

i∈I
µ
∗
i
µ =
∗

i∈I\{j}
µ
i
µ
a
=
∗

i∈I\{j}
(µ
i

)
a
, ∀a ∈ [0, µ(e)]
µ
i
(e) = µ(e), ∀i ∈ I
µ =
∗

i∈I
µ
i
µ
a
=
∗

i∈I
(µ
i
)
a
, ∀a ∈ [0, 1], a ≤ µ(e)
µ =
∗

i∈I
µ
i
µ(e) = ∨{∧

i∈I
µ
i
(x
i
)|e =

x
i
} ≤ ∨{∧
i∈I
µ
i
(e)} ≤ µ(e)( µ
i
⊆ µ)
µ
i
(e) = µ(e), ∀i ∈ I µ
a
=
•

i∈I
(µ
i
)
a
, ∀a ∈ [0, µ(e)] e ∈ µ
µ(e)

e ∈
∗

i∈I
(µ
i
)
µ(e)
(µ
i
)
µ(e)
= ∅ µ
i
(e) ≥ µ(e), ∀i ∈ I
µ
i
⊆ µ µ
i
(e) = µ(e), ∀i ∈ I
µ =
∗

i∈I
µ
i
x ∈ (
∗

i∈I

µ
i
)
a
⇐⇒ (
∗

i∈I
µ
i
)(x) ≥ a
⇐⇒ ∨{∧
i∈I
µ
i
(x
i
)|x =

x
i
} ≥ a
⇐= µ(x
i
) ≥ a x =

x
i
x, ∀i ∈ I
⇐⇒ x ∈

∗

i∈I
(µ
i
)
a
1) =⇒ ⇐⇒
2) x =⇒
⇐⇒
µ
a
=
∗

i∈I\{j}
(µ
i
)
a
, ∀a ∈ [0, µ(e)] x ∈ G
µ(x) = a x ∈ µ
a
y
i
∈ (µ
i
)
a
x =


y
i
(
∗

i∈I
µ
i
)(x) = ∨{∧
i∈I
|x =

x
i
} ≥ a = µ(x)
∗

i∈I
µ
i
⊇ µ
∗

i∈I
µ
i
= µ
µ ∈ F(G) {µ
i

|i ∈ I}
G µ
i
⊆ µ, ∀i ∈ I µ
i
(e) = µ
j
(e), ∀i, j ∈ I
µ =
•

i∈I
µ
i
⇐⇒ µ
a
=
•

i∈I
(µ
i
)
a
, ∀a ∈ (0, µ(e)]
µ =
•

i∈I
µ

i
µ
∗
=
•

i∈I
µ
∗
i
∀a ∈
(0, µ(e)] µ
a
=
∗

i∈I
(µ
i
)
a
(µ
i
)
a
∩ (µ
j
)
a
⊆ µ

∗
i
∩ µ
∗
j
= {e}
i = j µ
a
=
•

i∈I
(µ
i
)
a
, ∀a ∈ (0, µ(e)]
µ
a
=
•

i∈I
(µ
i
)
a
, ∀a ∈ (0, µ(e)] µ =
∗


i∈I
µ
i
x ∈ µ
∗
i
∩ µ
∗
j
a = µ
i
(x) ∧ µ
j
(x) a > 0
x ∈ (µ
i
)
a
∩ (µ
j
)
a
= {e} i = j x = e µ
∗
i
∩ µ
∗
j
= {e}, ∀i = j
µ

∗
=
∗

i∈I
µ
∗
i
µ =
•

i∈I
µ
i
µ ∈ F(G) {µ
i
|i ∈ I}
G µ
i
⊆ µ, ∀i ∈ I µ =
•

i∈I
µ
i
µ
∗
=
∗


i∈I
(µ
i
)
∗
µ
∗
=
•

i∈I
(µ
i
)
∗
µ
∗
G
µ
i
µ (µ
i
)
∗
µ
∗
j ∈ I x ∈ (µ
j
)
∗

∩
∗

i∈I\{j}
(µ
i
)
∗
x ∈ (µ
j
)
∗
∩
∗

i∈I\{j}
(µ
i
)
∗
(µ
j
∩
∗

i∈I\{j}
µ
i
)(x) > 0 x = e µ
∗

=
•

i∈I
(µ
i
)
∗
X x
t
X t ∈ [0, 1]
X
x
t
(y) =

t y = x
0 y = x.
x t x
t
G x
t
, y
s
G
x
t
y
s
= (xy)

t∧s
(x
t
)
−1
= (x
−1
)
t
.
I X = {x
i
|i ∈ I}
X
−1
= {x
−1
i
|i ∈ I} X Σ = X ∪ X
−1
Σ
∗
Σ
Σ Σ
∗
Σ Σ
∗
∼
∀w
1

, w
2
∈ Σ
∗
, w
1
∼ w
2
⇐⇒ w
1
w
2
x
i
x
−1
i
x
−1
i
x
i
i ∈ I ∼ Σ
∗
u ∈ Σ
∗
[u] Σ
∗
/ ∼= {[u]|u ∈ Σ
∗

} Σ
∗
/ ∼
∀[u], [v] ∈ Σ
∗
/ ∼, [u][v] = [uv] uv
u v
Σ
∗
/ ∼ X
F (X) F (X)
F (X) = X X F (X)
[u]
x
i
x
−1
i
x
−1
i
x
i
G B = {g
i
|i ∈ I} X = {x
i
|i ∈ I}
m : X −→ B m(x
i

) = g
i
∀x
i
∈ X ˆm : F (X) −→ G
ˆm([x]) = m(x), ∀x ∈ X
X
G G F (X)/Ker( ˆm)
G X
ˆm Ker( ˆm) S
S Ker( ˆm)
S R
G X, R G
G, H µ ν
G H ν µ
h : G −→ H h(µ) = ν h µ
ν
w ∈ Σ
∗
w = x
(1)
i
1
x
(2)
i
2
. . . x
(k)
i

k
(i) = ±1 x
1
= x {i
1
, i
2
, . . . , i
k
} I w
I(w) w w
−1
= x
−(k)
i
k
x
−(k−1)
i
k−1
. . . x
−(1)
i
1
T = {t
i
∈ [0, 1]|i ∈ I} t ∈ [0, 1] t ≥ ∨{s|s ∈ T}
f(X; T, t) F (X) ∀y ∈ F (X),
f(X; T, t)(y) = ∨{∧{t ∧ t
i

|i ∈ I(w)}|w ∈ y}
X T t
f(X; T, t) i ∈ I, t
i
x
i
x
−1
i
f(X; T, t) F (X)