monoid -- make or retrieve a monoid

Synopsis

Usage:

monoid[a,b,c,...]

monoid{a,b,c,...}
Inputs:
- TT "[a,b,c,...]", an array, listing generators of the monoid, as well as optional arguments.
- TT "{a,b,c,...}", a list, same as above, but equivalent to also passing Local => true.
Optional inputs:
- Variables => a list, default value null, see monoid(...,Variables=>...);
- VariableBaseName => a symbol, default value "p", see monoid(...,VariableBaseName=>...);
- Global => a Boolean value, default value true, see monoid(...,Global=>...);
- Local => a Boolean value, default value false, see monoid(...,Local=>...);
- Inverses => a Boolean value, default value false, see monoid(...,Inverses=>...);
- Weights => a list, default value {}, see monoid(...,Weights=>...);
- Degrees => a list, default value null, see monoid(...,Degrees=>...);
- DegreeMap => a Boolean value, default value null, see monoid(...,DegreeMap=>...);
- DegreeLift => a Boolean value, default value null, see monoid(...,DegreeLift=>...);
- DegreeRank => an integer, default value null, see monoid(...,DegreeRank=>...);
- Heft => a list, default value null, see monoid(...,Heft=>...);
- Join => a Boolean value, default value null, see monoid(...,Join=>...);
- MonomialOrder => a list, default value {GRevLex, Position => Up}, see monoid(...,MonomialOrder=>...);
- MonomialSize => an integer, default value 32, see monoid(...,MonomialSize=>...);
- SkewCommutative => a Boolean value, default value {}, see monoid(...,SkewCommutative=>...);
- WeylAlgebra => a list, default value {}, see monoid(...,WeylAlgebra=>...);
- Constants (missing documentation) => ..., default value false,
- DegreeGroup (missing documentation) => ..., default value null,
Outputs:
- a monoid,

Description

The function monoid is called whenever a polynomial ring is created, see Ring Array, or when a local polynomial ring is made, see Ring List. Some of the options provided when making a monoid don't take effect until the monoid is made into a polynomial ring.

Let's make a free ordered commutative monoid on the variables a,b,c, with degrees 2, 3, and 4, respectively.

i1 : M = monoid[a,b,c, Degrees => {2,3,4}]

o1 = M

o1 : GeneralOrderedMonoid

i2 : degrees M

o2 = {{2}, {3}, {4}}

o2 : List

i3 : M_0 * M_1^6

        6
o3 = a*b

o3 : M

Call use to assign the variables their values in the monoid.

i4 : monoid[x,y,z]

o4 = monoid[x..z, Degrees => {3:1}, Heft => {1}]

o4 : GeneralOrderedMonoid

i5 : x

o5 = x

o5 : Symbol

i6 : use ooo

o6 = monoid[x..z, Degrees => {3:1}, Heft => {1}]

o6 : GeneralOrderedMonoid

i7 : x * y^6

        6
o7 = x*y

o7 : monoid[x..z, Degrees => {3:1}, Heft => {1}]

The options used when the monoid was created can be recovered with options.

i8 : options M

o8 = OptionTable{Constants => false                     }
                                  1
                 DegreeGroup => ZZ
                 DegreeLift => null
                 DegreeMap => null
                 DegreeRank => 1
                 Degrees => {{2}, {3}, {4}}
                 Global => true
                 Heft => {1}
                 Inverses => false
                 Join => null
                 Local => false
                 MonomialOrder => {MonomialSize => 32  }
                                  {GRevLex => {2, 3, 4}}
                                  {Position => Up      }
                 SkewCommutative => {}
                 Variables => {a, b, c}
                 WeylAlgebra => {}

o8 : OptionTable

i9 : describe M

o9 = monoid[a..c, Degrees => {2..4}, Heft => {1}, MonomialOrder => {MonomialSize => 32}]
                                                                   {GRevLex => {2..4} }
                                                                   {Position => Up    }

i10 : toExternalString M

o10 = monoid[a..c, Degrees => {2..4}, Heft => {1}, MonomialOrder =>
      VerticalList{MonomialSize => 32, GRevLex => {2..4}, Position => Up}]

The variables listed may be symbols or indexed variables. The values assigned to these variables are the corresponding monoid generators. The function baseName may be used to recover the original symbol or indexed variable.

The monoid(...,Heft=>...) option is used, for instance, by Ext(Module,Module).

i11 : R = ZZ[x,y, Degrees => {-1,-2}, Heft => {-1}]

o11 = R

o11 : PolynomialRing

i12 : degree \ gens R

o12 = {{-1}, {-2}}

o12 : List

i13 : transpose vars R

o13 = {1} | x |
      {2} | y |

              2       1
o13 : Matrix R  <--- R

By default, (multi)degrees are concatenated when forming polynomial rings over polynomial rings, as can be seen by examining the corresponding flattened monoid, which displays information about all of the variables.

i14 : QQ[x][y]

o14 = QQ[x][y]

o14 : PolynomialRing

i15 : oo.FlatMonoid

o15 = monoid[y, x, Degrees => {{1}, {0}}, Heft => {2:1}, MonomialOrder => {MonomialSize => 32}]
                               {0}  {1}                                   {GRevLex => {1}    }
                                                                          {Position => Up    }
                                                                          {GRevLex => {1}    }

o15 : GeneralOrderedMonoid

i16 : QQ[x][y][z]

o16 = QQ[x][y][z]

o16 : PolynomialRing

i17 : oo.FlatMonoid

o17 = monoid[z, y, x, Degrees => {{1}, {0}, {0}}, Heft => {3:1}, MonomialOrder => {MonomialSize => 32}]
                                  {0}  {1}  {0}                                   {GRevLex => {1}    }
                                  {0}  {0}  {1}                                   {Position => Up    }
                                                                                  {2:(GRevLex => {1})}

o17 : GeneralOrderedMonoid

That behavior can be overridden with the monoid(...,Join=>...) option.

i18 : QQ[x][y, Join => false]

o18 = QQ[x][y]

o18 : PolynomialRing

i19 : oo.FlatMonoid

o19 = monoid[y, x, Degrees => {2:1}, Heft => {1}, Join => false, MonomialOrder => {MonomialSize => 32}]
                                                                                  {GRevLex => {1}    }
                                                                                  {Position => Up    }
                                                                                  {GRevLex => {1}    }

o19 : GeneralOrderedMonoid

Ways to use monoid :

"monoid(Array)"
"monoid(List)"
monoid(Ring) -- make or retrieve a monoid

For the programmer

The object monoid is a method function with a single argument.

monoid(...,Variables=>...) -- specify the names of the indeterminates
monoid(...,Local=>...) -- specify local or global monomial order
monoid(...,Inverses=>...) -- allow negative exponents in monomials
monoid(...,Weights=>...) -- specify weights of the variables
monoid(...,Degrees=>...) -- specify the degrees of the variables
"monoid(...,DegreeMap=>...)" -- see monoid(...,DegreeLift=>...) -- specify maps between degree groups
monoid(...,Heft=>...) -- specify a heft vector
monoid(...,Join=>...) -- specify how to handle degrees in the coefficient ring
monoid(...,MonomialOrder=>...) -- specify the monomial ordering
monoid(...,MonomialSize=>...) -- specify the bit-length of monomial exponents in the ring
monoid(...,SkewCommutative=>...) -- specify Skew commuting variables in the ring
monoid(...,WeylAlgebra=>...) -- specify differential operators in the ring

monoid -- make or retrieve a monoid

Synopsis

Description

Ways to use monoid :

For the programmer

Menu

Optional arguments