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, ..., 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∈N (i=1, ..., n) and the set {x₁, ..., 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.