Homomorphisms

For a general description of homomorphisms, we refer to chapter MAPPINGS. This section describes some special aspects of homomorphisms whose domain or codomain is a rewrite group.

General Remarks

Groups in the category GrpRWS currently are accepted as codomains only in some special situations. The most important cases in which a rewrite group can be used as a codomain are group homomorphisms whose domain is in one of the categories FINITELY PRESENTED GROUPS, GrpGPC, GrpRWS or GrpAtc.

Construction of Homomorphisms

hom< R -> G | S > : Struct , Struct -> Map

Returns the homomorphism from the rewrite group R to the group G defined by the expression S which can be the one of the following:

(i)

A list, sequence or indexed set containing the images of the n generators R.1, ..., R.n of R. Here, the i-th element of S is interpreted as the image of R.i, i.e. the order of the elements in S is important.

(ii)

A list, sequence, enumerated set or indexed set, containing n tuples <x_i, y_i> or arrow pairs x_i -> y_i, where x_i is a generator of R and y_i∈G (i=1, ..., n) and the set {x₁, ..., x_n} is the full set of generators of R. In this case, y_i is assigned as the image of x_i, hence the order of the elements in S is not important.

It is the user's responsibility to ensure that the provided generator images actually give rise to a well-defined homomorphism. No checking is performed by the constructor.

Note that it is currently not possible to define a homomorphism by assigning images to the elements of an arbitrary generating set of R.