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monoid(...,SkewCommutative=>...) -- specify skew commuting variables in the ring

Description

The SkewCommutative option specifies which variables will skew-commute when the monoid is used to create a ring. The value true indicates that all of the variables skew-commute. Otherwise, the value of the option may be a list of symbols or indices corresponding to the skew-commuting variables.

i1 : ZZ/101[e_0..e_3,       SkewCommutative => true]

      ZZ
o1 = ---[e ..e ]
     101  0   3

o1 : PolynomialRing, 4 skew commutative variable(s)
 
i2 : (e_0+e_1+e_2+e_3)^2

o2 = 0

      ZZ
o2 : ---[e ..e ]
     101  0   3
 
i3 : ZZ/101[x,y,vars(0..4), SkewCommutative => vars(0..4)]

      ZZ
o3 = ---[x..y, a..e]
     101

o3 : PolynomialRing, 5 skew commutative variable(s)
 
i4 : c*b*a*d

o4 = -a*b*c*d

      ZZ
o4 : ---[x..y, a..e]
     101
 
i5 : ZZ/101[x,y,vars(0..4), SkewCommutative => {2..6}]

      ZZ
o5 = ---[x..y, a..e]
     101

o5 : PolynomialRing, 5 skew commutative variable(s)
 
i6 : c*b*a*d

o6 = -a*b*c*d

      ZZ
o6 : ---[x..y, a..e]
     101
 
i7 : R = ZZ[x,y,z, SkewCommutative => {x,y}]

o7 = R

o7 : PolynomialRing, 2 skew commutative variable(s)
 
i8 : x*y

o8 = x*y

o8 : R
 
i9 : y*x

o9 = -x*y

o9 : R
 
i10 : x*z-z*x

o10 = 0

o10 : R

This option may also be used when creating a new ring from an existing ring, creating a tensor product ring, or symmetric algebra.

See also

Functions with optional argument named SkewCommutative:

  • monoid(...,SkewCommutative=>...) -- specify skew commuting variables in the ring
  • newRing(...,SkewCommutative=>...) -- see newRing -- make a copy of a ring, with some features changed
  • symmetricAlgebra(...,SkewCommutative=>...) -- see symmetricAlgebra -- the symmetric algebra of a module
  • tensor(Monoid,Monoid,SkewCommutative=>...) -- see tensor(Monoid,Monoid) -- tensor product of monoids
  • tensor(Ring,Ring,SkewCommutative=>...) -- see tensor(Monoid,Monoid) -- tensor product of monoids

Further information

  • Default value: {}
  • Function: monoid -- make or retrieve a monoid
  • Option key: SkewCommutative -- an optional argument

The source of this document is in Macaulay2Doc/functions/monoid-doc.m2:563:0.