Abundant semigroups and constructions

This thesis studies two classes of semigroups, given by presentations, with regard to weak regularity properties. Since its introduction by Fountain in the late 1970s, the study of abundant and related semigroups has given upward thrust to this fruitful and deep research area. The class of abundant semigroups extends that of regular semigroups in a natural way and is itself contained in the class of weakly abundant semigroups. We are interested in the properties of abundance and weak abundance as not only do they arise from a number of different directions and there are many natural examples, but also (weakly) abundant semigroups have enough structure to allow for the development of a coherent theory. The study of the free idempotent generated semigroup IG(E) over a biordered set E began with the seminal work of Nambooripad in the 1970s. Given the universal nature of such semigroups, it is natural to investigate their structure. In 2016 Gould and Yang [16] showed that IG(B), where B is a band, is always a weakly abundant semigroup, but is not necessarily abundant. Moreover, they constructed a 10-element normal band B for which IG(B) is not abundant. Following these discoveries another interesting question comes out very naturally: what kind of normal bands are such that IG(B) is abundant? Our main result shows that if B is an iso-normal band, then IG(B) is an abundant semigroup. The above considerations of the structure of IG(B) led us to introduce the notion of graph product of semigroups. We first consider the special case of free product and show that the free product of (weakly) abundant semigroups is (weakly) abundant. To answer the questions of whether the graph product of (weakly) abundant semigroups is (weakly) abundant we introduce a special form for the elements of graph products, and use this to answer the foregoing questions in the positive.