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# cellularBinomialIsPrimary -- test for primaryness of a binomial ideal

- Usage:
    `cellularBinomialIsPrimary I`
- Inputs:
  - I, a binomial ideal
- Optional inputs:
  - CellVariables => ..., default value null, cellular variables
  - ReturnPChars => ..., default value false, return two partial characters
  - ReturnPrimes => ..., default value false, return two associated primes
- Outputs:
  - true or false, depending on whether I is a binomial primary ideal

## Description

A binomial ideal is primary only if it is cellular. If the cellular variables are known they can be given via the CellVariables option. If the ideal is not primary, either 'false' or two distinct associated primes can be returned. The behaviour can be changed using the options ReturnPrimes and ReturnPChars.

    i1 : R = QQ[x,y]

    o1 = R

    o1 : PolynomialRing

    i2 : I = ideal(x^2-1)

                2
    o2 = ideal(x  - 1)

    o2 : Ideal of R

    i3 : cellularBinomialIsPrimary (I,ReturnPrimes=>true)
    The radical is not prime, as the character is not saturated

    o3 = {ideal (x - 1, x - 1), ideal(x + 1)}

    o3 : List

## See also

- cellularBinomialIsPrimary -- test for primaryness of a binomial ideal
- CellVariables -- cellular variables
- ReturnPrimes -- return two associated primes
- ReturnPChars -- return two partial characters

## Ways to use cellularBinomialIsPrimary:

- `cellularBinomialIsPrimary(Ideal)`

## For the programmer

The object cellularBinomialIsPrimary is a method function with options.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Binomials.m2:1615:0.