Given a semigroup and a congruence on the semigroup, one can construct a new semigroup: the quotient semigroup. The following funtions deal with quotient semigroups in GAP.
IsQuotientSemigroup( S ) C
is the category of semigroups constructed from another semigroup and a congruence on it
Elements of a quotient semigroup are equivalence classes of
elements of QuotientSemigroupPreimage(S)
under the congruence QuotientSemigroupCongruence(S).
It is probably most useful for calculating the elements of
the equivalence classes by using Elements or by looking at the
images of elements of the QuotientSemigroupPreimage(S) under
QuotientSemigroupHomomorphism(S):QuotientSemigroupPreimage(S)
® S.
For intensive computations in a quotient semigroup, it is probably worthwhile finding another representation as the equality test could involve enumeration of the elements of the congruence classes being compared.
HomomorphismQuotientSemigroup( cong ) F
for a congruence cong and a semigroup S. Returns the homomorphism from S to the quotient of S by cong.
QuotientSemigroupPreimage( S ) A
QuotientSemigroupCongruence( S ) A
QuotientSemigroupHomomorphism( S ) A
for a quotient semigroup S.
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GAP 4 manual