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 GrpFP, 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 \(R.1,\ldots,R.n\) 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 \(x_i \rightarrow y_i\), where \(x_i\) is a generator of \(R\) and \(y_i\in G\) (\(i=1,\ldots,n\)) and the set \(\{x_1,\ldots,x_n\}\) 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 \(R\).