Title:

Finiteness conditions for monoids and small categories

Chapter 1 covers some basic notions and results from Algebraic Topology such as CWcomplexes,
homotopy and homology groups of a space in general and cellular homology for CWcomplexes
in particular. Also we give some basic ideas from abstract reduction systems and some supporting
material such as several order relations on a set and the KnuthBendix completion
procedure. There are only two original results of the author in this chapter, Theorem 1.4.5
and Theorem 1.7.3. The material related to Topology and Homological Algebra can be found
in [12], [21], [40], [62], [82], [91] and [92]. The material related to reduction systems can be
found in [5] and [11].
The original work of the author is included in Chapters 2, 3 and 4 apart from Section 3.2
which contains general notions from Category Theory, Section 3.5.2 which contains an account
of the work in [67] and Section 4.1 which contains some basics from Combinatorial Semigroup
Theory. The results of Section 4.2 are part of [83] which is accepted for publication in the
International Journal of Algebra and Computation. The material related to Category Theory
can be found in [59], [64], [66], [67], [74], [75], [76], [82] and [93]. The material related to
Semigroup Theory is in [24] and [34].In Chapter 2 we show that for every monoid S which is given by a finite and complete presentation
P = P[x, r], and for every n ~ 2, there is a chain of CWcomplexes
such that ~n has dimension n, for every 2 ~ s ~ n the sskeleton of ~n is ~s and F acts on
~n. This action is called translation. Also we show that, for 2 ~ s ~ n, the open scells of ~n
are in a 11 correspondence with the stuples of positive edges of V with the same initial. For
the critical stuples, the corresponding open scells are denoted by PsI and the set of their
open translates by F.PsI.F. The following holds true.
if s ~ 3
if s = 2,
where U stands for the disjoint union. Also, for every 2 ~ s ~ n  1, there exists a cellular
equivalence "'s on Ks = (~s X ~8)(s+1) such that Ks/ "'s= (V, PI, ... ,PsI) and the following
is an exact sequence of (ZS, ZS)bimodules
where (D, Pl, ... , Ps2) = V if s = 2. Using the above short exact sequences, we deduce that S
is of type biFPn and that the free fi~ite resolution of'lS is Sgraded.
In Chapter 3 we generalize the notions left(respectively right)FPn and biFPn for small
categories and show that biFPn implies left(respectively right)FPn . Also we show that another
condition, which was introduced by Malbos and called FPn , implies biFPn . Since the
name FPn is confusing, we call it here fFPn for a reason which will be made clear in Section
3.1. Restricting to monoids, we show that, if a monoid is given by a finite and complete presentation,
then it is of type fFPn . Lastly, for every small category C, we show how to construct
free resolutions of ZC, at lea..'lt up to dimension 3, using some geometrical ideas which can be
generalized to construct free resolutions of ZC of any length.
vi
In Chapter 4 we study finiteness conditions of ~onoids of a combinatorial nature. We show
that there are semigroups S in which min'R., is independent of other conditions which S may
satisfy such as being finitely generated, periodic, inverse, Eunitary and even from the finiteness
of the maximal subgroups of S. We also relate the congruences of a monoid with the finiteness
condition minQ, and show that, if S is a monoid which satisfies minQ, then every congruence
JC on S which contains Q is of finite index in S. If a semigroup satisfies minQ and has all its
maximal subgroups locally finite, then we show that it is finite. Lastly, we show that, for trees
of completely Osimple semigroups, the local finiteness of its maximal subgroups implies the
local finiteness of the semigroups.
