Free idempotent generated semigroups over bands and biordered sets with trivial products

For any biordered set of idempotents E there is an initial object IG(E), the free idempotent generated semigroup over E, in the category of semigroups generated by a set of idempotents biorder-isomorphic to E. Recent research on IG(E) has focussed on the behaviour of the maximal subgroups. Inspired by an example of Brittenham, Margolis and Meakin, several proofs have been offered that any group occurs as a maximal subgroup of some IG(E), the latest being that of Dolinka and Ruškuc, who show that E can be taken to be a band. From a result of Easdown, Sapir and Volkov, periodic elements of any IG(E) lie in subgroups. However, little else is known of the ‘global’ properties of IG(E), other than that it need not be regular, even where E is a semilattice. The aim of this article is to deepen our understanding of the overall structure of IG(E) in the case where E is a biordered set with trivial products (for example, the biordered set of a poset) or where E is the biordered set of a band B. Since its introduction by Fountain in the late 1970s, the study of abundant and related semigroups has given rise to a deep and fruitful research area. The class of abundant semigroups extends that of regular semigroups in a natural way and itself is contained in the class of weakly abundant semigroups. Our main results show that (1) if E is a biordered set with trivial products then IG(E) is abundant and (if E is finite) has solvable word problem, and (2) for any band B, the semigroup IG(B) is weakly abundant and moreover satisfies a natural condition called the congruence condition. Further, IG(B) is abundant for a normal band B for which IG(B) satisfies a given technical condition, and we give examples of such B. On the other hand, we give an example of a normal band B such that IG(B) is not abundant.