<!-- %W minimal.xml Images documentation Christopher Jefferson --> <!-- %H --> <!-- %Y Copyright (C) 2016, School of Comp. Sci., St Andrews, Scotland -->
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Chapter Label="MinimalChapter">
<Heading>Minimal and Canonical Images</Heading>
Given a group G and action A, the minimal image of an object O is the smallest image
of O under any element of G, under the action A.
<P/>
As a more concrete example, let us consider the minimal image of the set [2,3,5,7] under a group G.
<P/>
We can calculate all the images of our set under G, then choose the smallest one.
This is very inefficient, as it requires enumerating all members of G.
The images package produces a function MinimalImage, which performs this same operation more efficiently.
The most common use of MinimalImage is to categorise objects into equivalence classes. This next example shows [2,3,5,7] and [1,6,7,8] are in the same orbit, while [3,5,7,8] is in a different orbit.
In this situation, we do not really need the minimal image, just a method of telling if two sets are in the same equivalence class.
<P/>
Motivated by this, this package provides CanonicalImage. CanonicalImage(G,O,A) returns some image of O by an element of G under the action A, guaranteeing that if two objects O1 and O2 are in the same orbit of G then CanonicalImage(G,O1,A) = CanonicalImage(G,O2,A). However, the canonical image is not "minimal" under any sensible ordering. The advantage of CanonicalImage is that it is much faster than MinimalImage, often by orders of magnitude.
<P/>
<B>WARNING:</B> The value of MinimalImage will remain identical between versions of GAP and the Images package, unless bugs are discovered. This is NOT true for CanonicalImage.
Die Informationen auf dieser Webseite wurden
nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit,
noch Qualität der bereit gestellten Informationen zugesichert.
Bemerkung:
Die farbliche Syntaxdarstellung ist noch experimentell.
