Minimality of Group and Monoid Presentations
İÇİNDEKİLER.Contents Statement Acknowledgements Abstract Notation 1 Preliminaries 1.1 Words . . 1.2 Group presentations..................... .1.2.1 Tietze transformations................ 1.2.2 Pictures over group presentations.......... 1.2.3 Aspherical and Cockcroft presentations....... 1.2.4 Efficiency of group presentations.......... . 1.3 Monoid presentations..................... 1.3.1 Fox derivations.................... 1.3.2 Pictures over monoid presentations......... 1.3.3 Aspherical and Cockcroft monoid presentations . . 1.3.4 Efficiency of monoid presentations.......... 2 The Cockcroft property of central extensions of groups 2.1 Introduction.......................... 2.2 Central extensions ...................... 2.3 The p-Cockcroft property for the central extensions........... 2.3.1 The general theorem........................ 2.3.2 The generating pictures of tt^P)................. 2.3.3 The proof of Theorem 2.3.1............. 2.4 Some examples........................ 3 The efficiency of standard wreath products of groups 3.1 Some background.............................. 3.2 The main theorem............................. 3.2.1 Calculation of d{H2{BA)) and 8{B A)............. ■ 3.2.2 To obtain an efficient presentation for G = B A........ . 3.3 Examples and applications......................... 4 The p-Cockcroft property of the semi-direct products of monoids 4.1 Introduction................................. 4.2 Monoid presentations............................ 4.2.1 Homomorphisms of monoids defined by presentations...... 4.2.2 Presentations of given monoids .................. 4.2.3 Endomorphisms of monoids.................... 4.3 Semi-clirect products of monoids...................... 4.3.1 The definition............................ 4.3.2 A generating set for D....................... 4.3.3 A presentation for D........................ 4.3.4 Trivializer of the Squier complex V(Vd) ............. 4.3.5 Defining a homomorphism 6 : A —V End(K)........... 4.4 The p-Cockcroft property for semi-direct products............ 4.4.1 The general theorem........................ 4.4.2 Direct products........................... 4.4.3 Semi-clirect products of finite cyclic monoids........... 5 Minimal presentations of semi-direct products of some monoids 5.1 Introduction............................, . . . 5.2 Semi-direct products of one-relator monoids by infinite cyclic monoids 5.2.1 The p-Cockcroft property.................... 5.3 Some minimal but inefficient presentations...............  |
