Knuth-Bendix on Monoids and Automatic Groups

Version 1.5.9

Released 2019-07-07

This project is maintained by The GAP Team

This package uses external binaries and currently works only under UNIX/LINUX systems.

KBMAG (pronounced `Kay-bee-mag`

) stands for ```
Knuth-Bendix on Monoids, and
Automatic Groups
```

. It is a stand-alone package written in C, for use under
UNIX, with an interface to GAP. There are interfaces for the use of KBMAG with
finitely presented groups, monoids and semigroups defined within GAP. The
package also contains a collection of routines for manipulating finite state
automata, which can be accessed via the GAP interface.

The overall objective of KBMAG is to construct a normal form for the elements of a finitely presented group G in terms of the given generators together with a word reduction algorithm for calculating the normal form representation of an element in G, given as a word in the generators. If this can be achieved, then it is also possible to enumerate the words in normal form up to a given length, and to determine the order of the group, by counting the number of words in normal form. In most serious applications, this will be infinite, since finite groups are (with some exceptions) usually handled better by Todd-Coxeter related methods. In fact a finite state automaton W is calculated that accepts precisely the language of words in the group generators that are in normal form, and W is used for the enumeration and counting functions. It is possible to inspect W directly if required; for example, it is often possible to use W to determine whether an element in G has finite or infinite order.

The normal form for an element g in G is defined to be the least word in the group generators (and their inverses) that represents G, with respect to a specified ordering on the set of all words in the group generators.

KBMAG offers two possible means of achieving these objectives. The first is to apply the Knuth-Bendix algorithm to the group presentation, with one of the available orderings on words, and hope that the algorithm will complete with a finite confluent presentation. (If the group is finite, then it is guaranteed to complete eventually but, like the Todd-Coxeter procedure, it may take a long time, or require more space than is available.) The second is to use the automatic group program. This also uses the Knuth-Bendix procedure as one component of the algorithm, but it aims to compute certain finite state automata rather than to obtain a finite confluent rewriting system, and it completes successfully on many examples for which such a finite system does not exist. In the current implementation, its use is restricted to the shortlex ordering on words. That is, words are ordered first by increasing length, and then words of equal length are ordered lexicographically, using the specified ordering of the generators.

The GAP4 version of KBMAG also offers extensive facilities for finding confluent presentations and finding automatic structures relative to a specified finitely generated subgroup of the group G. Finally, there is a collection of functions for manipulating finite state automata that may be of independent interest.

This package uses external binaries and currently works only under UNIX/LINUX systems.

It will work only on GAP versions >= 4.7.

To complete the installation of the `kbmag`

package go to the
directory `kbmag`

created (the directory contains a copy of this
README file) and call

```
./configure <PATH>
```

where `PATH`

is a path to the main GAP root directory; so normally you
would call

```
./configure ../..
```

and then call

```
make
```

to compile the binary.

If you installed GAP on several architectures, you must execute this
configure/make step for the `kbmag`

package on each of the architectures
immediately after configuring GAP itself on this architecture.

Full information and documentation can be found in the manual, available
as PDF `doc/manual.pdf`

or as HTML `doc/chap0.html`

, or on the package
homepage at

https://gap-packages.github.io/kbmag/

Please submit bug reports and feature requests via our GitHub issue tracker:

https://github.com/gap-packages/kbmag/issues

kbmag is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version.

For details see the file LICENSE.