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In matematica e in particolare in algebra astratta , dato un magma
(
X
,
⋅
)
{\displaystyle (X,\cdot )}
elemento neutro
1
,
{\displaystyle 1,}
x
∈
X
{\displaystyle x\in X}
inverso destro di
x
{\displaystyle x}
d
x
∈
X
{\displaystyle d_{x}\in X}
x
⋅
d
x
=
1
{\displaystyle x\cdot d_{x}=1}
inverso sinistro di
x
{\displaystyle x}
s
x
∈
X
{\displaystyle s_{x}\in X}
s
x
⋅
x
=
1.
{\displaystyle s_{x}\cdot x=1.}
Se l'operazione binaria
⋅
{\displaystyle \cdot }
L'inverso destro e l'inverso sinistro di
x
,
{\displaystyle x,}
s
x
=
s
x
⋅
1
=
s
x
⋅
(
x
⋅
d
x
)
=
(
s
x
⋅
x
)
⋅
d
x
=
1
⋅
d
x
=
d
x
.
{\displaystyle s_{x}=s_{x}\cdot 1=s_{x}\cdot (x\cdot d_{x})=(s_{x}\cdot x)\cdot d_{x}=1\cdot d_{x}=d_{x}.}
L'inverso di
x
,
{\displaystyle x,}
a
{\displaystyle a}
b
{\displaystyle b}
x
,
{\displaystyle x,}
a
=
a
⋅
1
=
a
⋅
(
x
⋅
b
)
=
(
a
⋅
x
)
⋅
b
=
1
⋅
b
=
b
.
{\displaystyle a=a\cdot 1=a\cdot (x\cdot b)=(a\cdot x)\cdot b=1\cdot b=b.}
x
{\displaystyle x}
x
−
1
.
{\displaystyle x^{-1}.}
L'elemento
x
{\displaystyle x}
x
−
1
{\displaystyle x^{-1}}
Ogni elemento commuta con il suo inverso. Infatti,
x
⋅
x
−
1
=
1
=
x
−
1
⋅
x
.
{\displaystyle x\cdot x^{-1}=1=x^{-1}\cdot x.}
L'inverso dell'inverso è l'elemento stesso. Infatti, sia
x
{\displaystyle x}
(
x
−
1
)
−
1
{\displaystyle (x^{-1})^{-1}}
x
−
1
{\displaystyle x^{-1}}
x
=
(
x
−
1
)
−
1
.
{\displaystyle x=(x^{-1})^{-1}.}
Se
x
{\displaystyle x}
y
{\displaystyle y}
x
⋅
y
{\displaystyle x\cdot y}
y
−
1
⋅
x
−
1
.
{\displaystyle y^{-1}\cdot x^{-1}.}
(
x
⋅
y
)
⋅
(
y
−
1
⋅
x
−
1
)
=
x
⋅
(
y
⋅
y
−
1
)
⋅
x
−
1
=
x
⋅
1
⋅
x
−
1
=
x
⋅
x
−
1
=
1.
{\displaystyle (x\cdot y)\cdot (y^{-1}\cdot x^{-1})=x\cdot (y\cdot y^{-1})\cdot x^{-1}=x\cdot 1\cdot x^{-1}=x\cdot x^{-1}=1.}
