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Title: Finiteness conditions for monoids and small categories
Author: Pasku, Elton
ISNI:       0000 0001 3475 0246
Awarding Body: University of Glasgow
Current Institution: University of Glasgow
Date of Award: 2006
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Chapter 1 covers some basic notions and results from Algebraic Topology such as CW-complexes, homotopy and homology groups of a space in general and cellular homology for CW-complexes 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 Knuth-Bendix 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 CW-complexes such that ~n has dimension n, for every 2 ~ s ~ n the s-skeleton of ~n is ~s and F acts on ~n. This action is called translation. Also we show that, for 2 ~ s ~ n, the open s-cells of ~n are in a 1-1 correspondence with the s-tuples of positive edges of V with the same initial. For the critical s-tuples, the corresponding open s-cells are denoted by Ps-I and the set of their open translates by F.Ps-I.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, ... ,Ps-I) and the following is an exact sequence of (ZS, ZS)-bimodules where (D, Pl, ... , Ps-2) = V if s = 2. Using the above short exact sequences, we deduce that S is of type bi-FPn and that the free fi~ite resolution of'lS is S-graded. In Chapter 3 we generalize the notions left-(respectively right)-FPn and bi-FPn for small categories and show that bi-FPn implies left-(respectively right)-FPn . Also we show that another condition, which was introduced by Malbos and called FPn , implies bi-FPn . Since the name FPn is confusing, we call it here f-FPn 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 f-FPn . 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, E-unitary 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 O-simple semigroups, the local finiteness of its maximal subgroups implies the local finiteness of the semigroups.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: QA Mathematics