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# isSymmetric -- test whether the semigroup generated by L is symmetric

- Usage:
    `t = isSymmetric L`
- Inputs:
  - L, a list,
- Outputs:
  - t, a Boolean value,

## Description

Suppose that c = conductor S, so that c-1 is the last gap. If x in S, and x<c then c-1-x must be a gap, so the number g of gaps <c is at least the number of semigroup elements e < c. The semigroup is called \emph{symmetric} if g = e, or equivalently if the semigroup ring is Gorenstein. Thus for example any semigroup generated by just 2 elements is symmetric.

    i1 : isSymmetric{3,4,5}

    o1 = false

    i2 : isSymmetric{3, 5}

    o2 = true

    i3 : gaps {3,5}

    o3 = {1, 2, 4, 7}

    o3 : List

    i4 : mu {3,5}

    o4 = {3, 1}

    o4 : List

## See also

- apery -- Compute the apery set, multiplicity and conductor
- semigroup -- Compute the semigroup generated by a list of positive integers
- gaps -- The gap sequence of a semigroup
- mu -- Compute the point representing a semigroup in the Kunz cone

## Ways to use isSymmetric:

- `isSymmetric(List)`

## For the programmer

The object isSymmetric is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/NumericalSemigroups.m2:1743:0.