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7.5.7.0. SuperCommutative
Procedure from library nctools.lib (see nctools_lib).

Usage:
SuperCommutative([b,[e, [Q]]]);

Return:
qring

Purpose:
create the super-commutative algebra (as a GR-algebra) 'over' a basering,

Note:
activate this qring with the "setring" command.

Note:
as a side effect the basering will be changed (if not in a commutative case) to bo the ground G-algebra (without factor).

Note:
if b==e then the resulting ring is commutative.

Theory:
given a basering, this procedure introduces the anticommutative relations x(j)x(i)=-x(i)x(j) for all e>=j>i>=b,
moreover, creates a factor algebra modulo the two-sided ideal, generated by x(b)^2, ..., x(e)^2[ + Q]

Example:
 
LIB "nctools.lib";
ring R = 0,(x(1..4)),dp; // global!
def ER = SuperCommutative(); // the same as Exterior (b = 1, e = N)
setring ER; ER;
==> //   characteristic : 0
==> //   number of vars : 4
==> //        block   1 : ordering dp
==> //                  : names    x(1) x(2) x(3) x(4) 
==> //        block   2 : ordering C
==> //   noncommutative relations:
==> //    x(2)x(1)=-x(1)*x(2)
==> //    x(3)x(1)=-x(1)*x(3)
==> //    x(4)x(1)=-x(1)*x(4)
==> //    x(3)x(2)=-x(2)*x(3)
==> //    x(4)x(2)=-x(2)*x(4)
==> //    x(4)x(3)=-x(3)*x(4)
==> // quotient ring from ideal
==> _[1]=x(4)^2
==> _[2]=x(3)^2
==> _[3]=x(2)^2
==> _[4]=x(1)^2
"Alternating variables: [", AltVarStart(), ",", AltVarEnd(), "].";
==> Alternating variables: [ 1 , 4 ].
kill R; kill ER;
ring R = 0,(x(1..4)),(lp(1), dp(3)); // global!
def ER = SuperCommutative(2); // b = 2, e = N
setring ER; ER;
==> //   characteristic : 0
==> //   number of vars : 4
==> //        block   1 : ordering lp
==> //                  : names    x(1) 
==> //        block   2 : ordering dp
==> //                  : names    x(2) x(3) x(4) 
==> //        block   3 : ordering C
==> //   noncommutative relations:
==> //    x(3)x(2)=-x(2)*x(3)
==> //    x(4)x(2)=-x(2)*x(4)
==> //    x(4)x(3)=-x(3)*x(4)
==> // quotient ring from ideal
==> _[1]=x(4)^2
==> _[2]=x(3)^2
==> _[3]=x(2)^2
"Alternating variables: [", AltVarStart(), ",", AltVarEnd(), "].";
==> Alternating variables: [ 2 , 4 ].
kill R; kill ER;
ring R = 0,(x(1..6)),(ls(2), dp(2), lp(2)); // local!
def ER = SuperCommutative(3,4); // b = 3, e = 4
setring ER; ER;
==> //   characteristic : 0
==> //   number of vars : 6
==> //        block   1 : ordering ls
==> //                  : names    x(1) x(2) 
==> //        block   2 : ordering dp
==> //                  : names    x(3) x(4) 
==> //        block   3 : ordering lp
==> //                  : names    x(5) x(6) 
==> //        block   4 : ordering C
==> //   noncommutative relations:
==> //    x(4)x(3)=-x(3)*x(4)
==> // quotient ring from ideal
==> _[1]=x(3)^2
==> _[2]=x(4)^2
"Alternating variables: [", AltVarStart(), ",", AltVarEnd(), "].";
==> Alternating variables: [ 3 , 4 ].
kill R; kill ER;


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