Using Pictures in Combinatorial Group and Semigroup Theory

Pictures over group presentations are the duals of van Kampen diagrams, known widely in geometric group theory. They have proved to be an effective tool in obtaining results concerning groups. Pictures over semigroup and monoid presentations have recently been introduced and show promise in yielding algebraic information. In Chapter 1 we review existing theory concerning monoid and group presentations, and the concept of pictures over these. We remark that all the monoid presentations considered in this thesis have relations of the form A = B, where A and B are non-empty words. Therefore we refer to the monoid S or semigroup So defined by a monoid presentation. Related to any monoid presentation for a monoid S or semigroup S0, there is a group presentation defining a group G. It is of interest to ascertain exactly how the monoid or semigroup structures are related to the group structure. In Chapter 2 we prove a result which gives sufficient conditions on the group presentation for the embeddability of S in G. We prove that these conditions are implied by the conditions recently given by E.V. Kashintsev. Furthermore, we give an example of a group presentation which satisfies our embeddability conditions but has a corresponding monoid presentation which does not belong to the class of presentations defining embeddable semigroups, studied recently by Cuba. Chapter 3 is concerned with the relationship between conjugacy in S and conjugacy in G. We prove under Kashintsev's embeddablity conditions, and Goldstein and Teymouri's definition of conjugacy in S, that two elements of S which are conjugate in G, are conjugate in S in an 'elementary' way. Chapters 4 and 5 are concerned with relative monoid presentations. Generalising work by Adjan, we introduce the notions of left and right graphs for these presentations. We prove an asphericity result for mixed monoid presentations (for which there exists a natural notion of picture), as well as a cancellation result and an embeddability result for monoids given by relative monoid presentations