a
0
| b
0
| c
0
| and | d
0
|
and elements
a, b, c and d of
| a² = a | b² = b | c² = c | d² = d | a + b + c + d = 0 |
| a
0
| b
0
| c
0
| and | d
0
|
x
-
x
|
x
-
x
|
x
-
x
|
x
-
x
|
x
+
x
+
x
+
x.
|
be the algebra generated by
and let
a =
x
+ I, b =
x
+ I, c =
x
+ I, d =
x
+I.
When NCProcess is applied to the polynomials above, the
Gröbner Basis listed in the following spreadsheet is
obtained:
<
x
<
x
<
x
| YOUR SESSION HAS DIGESTED | ||
|---|---|---|
| THE FOLLOWING RELATIONS | ||
}
-x
-
x
-
x
,
x,
x}
x
x
x
x
x
x
x
- 2
x
- 2
x
- 2
x
-
x
x
-
x
x
-
x
x
-
x
x
-
x
x
x
x
x
- 8
x
- 6
x
- 6
x
- 6
x
x
-
x
x
(3 +
x)
-
(6 +
x)
x
x
-
x
x
(6 +
x)
-
(x
+
x)
x
x
-
x
x
(3 -
x)
== 0
| USER CREATIONS APPEAR BELOW | ||
|---|---|---|
| SOME RELATIONS WHICH APPEAR BELOW | ||||
|---|---|---|---|---|
| MAY BE UNDIGESTED | ||||
,
x,
x}
Therefore, a, b, c and d
are idempotents which add up to zero. Note that a = 0
if and only if
x
I.
The form of the above Gröbner Basis implies that
x
I
and so a
0.
Similarly, b
0,
c
0 and
d
0.
