Semigroup and monoid presentations

The aim of this thesis is two-fold. First we investigate the class of right ideal Howson semigroups inspired by a question posed by Steinberg. Right ideal Howson semigroups are defined by the finitary property that the intersection of any two finitely generated right ideals is also finitely generated. We obtain semigroup presentations for right ideal Howson semigroups which are universal in a certain sense. In addition, we provide examples of right ideal Howson semigroups with a specific focus on coherent monoids, varieties of bands and other finiteness conditions. Dual results hold for left ideal Howson semigroups. The second part of this thesis concerns finding semigroup presentations for semigroups of the form ST, where S and T are subsemigroups of some common semigroup U, such that for every a in T we have aS contained in Sa and if xa = yb then x = y for every x,y in S and a,b in T. Significantly, we obtain a semigroup presentation for the singular part of the partial endomorphism monoid of a free G-act of finite rank. This builds on the work of Al-Aadhami, Dolinka, East, Feng and Gould. We also use our methods to give presentations for almost-factorisable inverse semigroups.