Beyond Regular Semigroups

The topic of this thesis is the class of weakly U-abundant semigroups. This class is very wide, containing inverse, orthodox, regular, ample, adequate, quasi-adequate, concordant, abundant, restriction, Ehresmann and weakly
abundant semigroups. A semigroup $S$ with subset of idempotents U is weakly U-abundant if every $\art_U$-class and every $\elt_U$-class contains an idempotent of U, where $\art_U$ and $\elt_U$ are relations extending the well known Green's relations $\ar$ and $\el$. We assume throughout that our semigroups satisfy a condition known as the Congruence Condition (C).

We take several approaches to weakly $U$-abundant semigroups. Our first results describe those that are analogous to completely simple semigroups. Together with an existing result of Ren this determines the structure of those weakly $U$-abundant semigroups that are analogues of completely regular semigroups, that is, they are superabundant. Our description is in terms of a semilattice of rectangular bands of monoids.

The second strand is to aim for an extension of the Hall-Yamada theorem for orthodox semigroups as spined products of inverse semigroups and fundamental orthodox semigroups. To this end we consider weakly $B$-orthodox semigroups, where $B$ is a band. We note that if
$B$ is a semilattice then a weakly $B$-orthodox semigroup is exactly an Ehresmann semigroup. We provide a description of a weakly $B$-orthodox semigroup $S$ as a spined product of a fundamental weakly $\overline{B}$-orthodox semigroup $S_B$ (depending only on $B$) and $S/\gamma_B$, where $\overline{B}$ is isomorphic to $B$ and $\gamma_B$ is the analogue of the least inverse congruence on an orthodox semigroup. This result is an analogue of the Hall-Yamada theorem for orthodox semigroups. In the case that $B$ is a normal band, or $S$ is weakly $B$-superabundant, we find a closed form $\delta_B$ for $\gamma_B$, which simplifies our result to a straightforward form.

For the above to work smoothly in the case $S$ is weakly $B$-superabundant, we need to find a canonical fundamental weakly $B$-superabundant subsemigroup of $S_B$. This we do, and give the corresponding answers in the case of the Hall semigroup $W_B$ and a number of intervening semigroups.

We then change our direction. A celebrated result of Nambooripad shows that regular semigroups are determined by ordered groupoids built over a regular biordered set. Our aim, achieved at the end of the thesis, is to extend Nambooripad's work to {\em weakly $U$-regular}
semigroups, that is, weakly $U$-abundant semigroups with (C) and $U$ generating a regular subsemigroup
whose set of idempotents is $U$.

As an intervening step we consider weakly $B$-orthodox semigroups in this light. We take two approaches. The first is via a new construction of an inductive generalised category over a band. In doing so we produce a new approach to characterising orthodox semigroups, by using inductive generalised groupoids.
We show that the category of weakly $B$-orthodox semigroups is isomorphic to the category of inductive generalised categories over bands. Our approach is influenced by that of Nambooripad, however, there are significant differences in strategy, the first
being the introduction of generalised categories and the second being that it is more convenient to consider (generalised) categories equipped with pre-orders, rather than with partial orders. Our work may be regarded as extending a result of Lawson for Ehresmann semigroups. We also examine the trace of a weakly $B$-orthodox semigroup, which is a primitive weakly $B$-orthodox semigroup.

We then take a more `traditional' approach to weakly $B$-orthodox semigroups via band categories and weakly orthodox categories over a band, equipped with two pre-orders. We show that the category of weakly $B$-orthodox semigroups is equivalent to the category of weakly orthodox categories over bands. To do so we must substantially adjust Armstrong's method for concordant semigroups.

Finally, we consider the most general case of weakly $U$-regular semigroups. Following Nambooripad's theorem, which establishes a correspondence between algebraic structures (inverse semigroups) and ordered structures (inductive group-oids), we build a correspondence between the category of weakly $U$-regular semigroups and the category of weakly regular categories over regular biordered sets, equipped with two pre-orders.