randomMonomialAlgebra -- Generate random monomial algebra.

Synopsis

Usage:

randomMonomialAlgebra(a,d,c)
Inputs:
- a, an integer, positive
- d, an integer, positive
- c, an integer, positive
Optional inputs:
- Num => ..., default value 1, Option of randomSemigroup and randomMonomialAlgebra to return a list of several semigroups.
- SetSeed => ..., default value false, Option to set the random seed for randomSemigroup and randomMonomialAlgebra.
- Simplicial => ..., default value false, Option of randomSemigroup and randomMonomialAlgebra to return a simplicial semigroup.
Outputs:
- a MonomialAlgebra, or a list of those if the option Num is bigger or equal to 2.

Description

Generate a random monomial algebra such that the semigroup generated by the degrees is in \mathbb{N}^d of full dimension with coordinate sum a and codimension c.

Note that the random number generator can be controlled by the command setRandomSeed. Alternatively you can also use the option SetSeed.

The option Num can be used to return a list of semigroups.

The option Simplicial can be used to return a simplicial semigroup such that the standard vectors a*e_i are among the Hilbert basis.

i1 : randomMonomialAlgebra(5,3,7)

      ZZ
o1 = ---[x ..x ]
     101  0   9

o1 : MonomialAlgebra generated by {{1, 3, 1}, {0, 1, 4}, {2, 1, 2}, {0, 4, 1}, {2, 2, 1}, {3, 2, 0}, {3, 1, 1}, {5, 0, 0}, {2, 3, 0}, {0, 3, 2}}

i2 : setRandomSeed()

i3 : randomMonomialAlgebra(5,3,7)

      ZZ
o3 = ---[x ..x ]
     101  0   9

o3 : MonomialAlgebra generated by {{5, 0, 0}, {1, 3, 1}, {1, 4, 0}, {0, 2, 3}, {3, 0, 2}, {0, 1, 4}, {0, 5, 0}, {4, 1, 0}, {2, 0, 3}, {4, 0, 1}}

i4 : setRandomSeed()

i5 : randomMonomialAlgebra(5,3,7)

      ZZ
o5 = ---[x ..x ]
     101  0   9

o5 : MonomialAlgebra generated by {{5, 0, 0}, {1, 3, 1}, {1, 4, 0}, {0, 2, 3}, {3, 0, 2}, {0, 1, 4}, {0, 5, 0}, {4, 1, 0}, {2, 0, 3}, {4, 0, 1}}

i6 : setRandomSeed()

i7 : randomMonomialAlgebra(5,3,7,Num=>2)

       ZZ           ZZ
o7 = {---[x ..x ], ---[x ..x ]}
      101  0   9   101  0   9

o7 : List

i8 : setRandomSeed()

i9 : randomMonomialAlgebra(5,3,7,Simplicial=>true)

      ZZ
o9 = ---[x ..x ]
     101  0   9

o9 : MonomialAlgebra generated by {{5, 0, 0}, {0, 5, 0}, {0, 0, 5}, {0, 1, 4}, {4, 0, 1}, {1, 1, 3}, {1, 3, 1}, {2, 3, 0}, {2, 0, 3}, {1, 4, 0}}

Ways to use randomMonomialAlgebra :

randomMonomialAlgebra(ZZ,ZZ,ZZ)

For the programmer

The object randomMonomialAlgebra is a method function with options.