Homomorphisms

For a general description of homomorphisms, we refer to Chapter Mappings. This section describes some special aspects of homomorphisms whose domain is a rewrite monoid.

General Remarks

Monoids in the category MonRWS currently are accepted as codomains only for monoid homomorphisms, whose codomain is a rewrite monoid as well.

Construction of Homomorphisms

hom< M -> N | S >: Struct, Struct -> Map

Returns the homomorphism from the rewrite group \(M\) to the monoid \(N\) defined by the expression \(S\) which must be the one of the following:

(i)

A list, sequence or indexed set containing the images of the \(n\) generators \(M.1,\ldots,M.n\) of \(M\). Here, the \(i\)-th element of \(S\) is interpreted as the image of \(M.i\), 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 \(M\) and \(y_i\in N\) (\(i=1,\ldots,n\)) and the set \(\{x_1,\ldots,x_n\}\) is the full set of generators of \(M\). 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. Presently, \(N\) must be either a rewrite monoid or a group, and it is not possible to define a homomorphism by assigning images to the elements of an arbitrary generating set of \(M\).