62.24 SmallestIdeal

SmallestIdeal( sg )

A subset I of a semigroup (S,cdot) is defined as an ideal of S if forall i in I, s in S: i cdot s in I & s cdot i in I. An ideal I is called minimal, if for any ideal J, J subseteq I implies J = I. If a minimal ideal exists, then it is unique and therefore the smallest ideal of S.

The function SmallestIdeal returns the smallest ideal of the transformation semigroup sg. Note that for a finite semigroup the smallest ideal always exists. (Which is not necessarily true for an arbitrary semigroup).

  gap> SmallestIdeal( s );
  [ Transformation( [ 1, 2, 3 ], [ 1, 1, 1 ] ), 
    Transformation( [ 1, 2, 3 ], [ 2, 2, 2 ] ), 
    Transformation( [ 1, 2, 3 ], [ 3, 3, 3 ] ) ]

Previous Up Top Next
Index