Projekt A15 - Homologische und homotopische Endlichkeitsbedingungen
(Homological and homotopical finiteness conditions)
(since 5/98, until 12/06)
Presentations of the form (\Sigma;R),
where \Sigma is an alphabet and
R is a string-rewriting system on \Sigma,
provide a means to define and describe monoids .
However, even if a monoid M is given through a finite
presentation, the word problem for M may be undecidable.
Therefore, one is interested in classes of finite presentations
which guarantee that the word problem is decidable.
A class of this form that has received a lot of attention in
recent years is the class of presentations that contain a
finite string-rewriting system that is convergent,
as such a system yields a complete set of unique
representatives for the monoid presented,
and given a string from \Sigma^*,
the corresponding normal form
can simply be computed by reduction .
Hence, the question arises for an algebraic or a combinatorial
characterization of the class of finitely presented monoids
that admit a presentation through a finite convergent string-rewriting system.
Meanwhile quite a few conditions have been identified that
a finitely presented monoid must necessarily satisfy,
if it is to admit such a presentation.
At first Squier proved that a monoid having a finite convergent
presentation satisfies the homological finiteness conditions
left FP_3 and right FP_3 [Squier 1987].
Next Kobayashi improved upon this result by showing that
these monoids even satisfy the conditions left and right FP_\infty
Next Squier introduced the homotopical finiteness condition FDT,
which is a combinatorial condition on the derivation graph
associated with a given monoid presentation [Squier 94].
This condition implies the conditions left and right FP_3,
but it does not imply left or right FP_\infty,
as for groups it is equivalent to the condition FP_3
[Cremanns, Otto 96].
Finally Wang and Pride introduced the finiteness
condition \FHT, which states that the homotopy bimodule of the given
presentation is finitely generated.
As the property \FDT\ this property is an invariant of finitely
presented monoids, and it is implied by FDT, while
it implies both left and right FP_3.
Here we are interested in the exact relationship between
these various finiteness conditions.