<Chapter Label="Ideals"><Heading>Ideals</Heading> 

  In this chapter we describe the various ways that an ideal of a semigroup
  can be created and manipulated in &Semigroups;.<P/>

  We write <E>ideal</E> to mean two-sided ideal everywhere in this chapter.<P/>

  The methods in the &Semigroups; package apply to any ideal of a
  transformation, partial permutation, or bipartition semigroup, or an ideal of
  a subsemigroup of a Rees 0-matrix semigroup or semigroup of matrices over a
  finite field, that is created by the function <Ref Func="SemigroupIdeal"/> or
  <C>SemigroupIdealByGenerators</C>. Anything that can be calculated for a
  semigroup defined by a generating set can also be found for an ideal. This
  works particularly well for regular ideals, since such an ideal can be
  represented using a similar data structure to that used to represent a
  semigroup defined by a generating set but without the necessity to find a
  generating set for the ideal. Many methods for non-regular ideals rely on
  first finding a generating set for the ideal, which can be costly (but not
  nearly as costly as an exhaustive enumeration of the elements of the ideal).
  We plan to improve the functionality of &Semigroups; for non-regular ideals
  in the future. 

  <Section><Heading>Creating ideals</Heading> 
    <#Include Label="SemigroupIdeal"> 
  </Section>
  
  <Section><Heading>Attributes of ideals</Heading> 
    <#Include Label="GeneratorsOfSemigroupIdeal">
    <#Include Label="MinimalIdealGeneratingSet">
    <#Include Label="SupersemigroupOfIdeal"> 
  </Section>

</Chapter>
