randomSemigroup -- Generate random semigroups.

Synopsis

Usage:

randomSemigroup(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:
- L, a list,

Description

Generate a random generating set of a semigroups B 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 : randomSemigroup(5,3,7)

o1 = {{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}}

o1 : List

i2 : setRandomSeed()

i3 : randomSemigroup(5,3,7)

o3 = {{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}}

o3 : List

i4 : setRandomSeed()

i5 : randomSemigroup(5,3,7)

o5 = {{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}}

o5 : List

i6 : setRandomSeed()

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

o7 = {{{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}}, {{1, 3, 1}, {1, 4, 0}, {2, 3,
     ------------------------------------------------------------------------
     0}, {0, 5, 0}, {1, 2, 2}, {3, 1, 1}, {0, 0, 5}, {1, 1, 3}, {4, 0, 1},
     ------------------------------------------------------------------------
     {0, 2, 3}}}

o7 : List

i8 : setRandomSeed()

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

o9 = {{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}}

o9 : List

Ways to use randomSemigroup :

randomSemigroup(ZZ,ZZ,ZZ)

For the programmer

The object randomSemigroup is a method function with options.