{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,2]],"date-time":"2025-11-02T10:32:10Z","timestamp":1762079530627},"reference-count":17,"publisher":"World Scientific Pub Co Pte Ltd","issue":"03","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Algebra Comput."],"published-print":{"date-parts":[[2016,5]]},"abstract":"<jats:p>For any biordered set of idempotents [Formula: see text] there is an initial object [Formula: see text], the free idempotent generated semigroup over[Formula: see text], in the category of semigroups generated by a set of idempotents biorder-isomorphic to [Formula: see text]. Recent research on [Formula: see text] has focused on the behavior 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 [Formula: see text], the latest being that of Dolinka and Ru\u0161kuc, who show that [Formula: see text] can be taken to be a band. From a result of Easdown, Sapir and Volkov, periodic elements of any [Formula: see text] lie in subgroups. However, little else is known of the \u201cglobal\u201d properties of [Formula: see text], other than that it need not be regular, even where [Formula: see text] is a semilattice. The aim of this paper is to deepen our understanding of the overall structure of [Formula: see text] in the case where [Formula: see text] is a biordered set with trivial products (for example, the biordered set of a poset) or where [Formula: see text] is the biordered set of a band [Formula: see text].<\/jats:p><jats:p>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 [Formula: see text] is a biordered set with trivial products then [Formula: see text] is abundant and (if [Formula: see text] is finite) has solvable word problem, and (2) for any band [Formula: see text], the semigroup [Formula: see text] is weakly abundant and moreover satisfies a natural condition called the congruence condition. Further, [Formula: see text] is abundant for a normal band [Formula: see text] for which [Formula: see text] satisfies a given technical condition, and we give examples of such [Formula: see text]. On the other hand, we give an example of a normal band [Formula: see text] such that [Formula: see text] is not abundant.<\/jats:p>","DOI":"10.1142\/s021819671650020x","type":"journal-article","created":{"date-parts":[[2016,4,7]],"date-time":"2016-04-07T08:41:59Z","timestamp":1460018519000},"page":"473-507","source":"Crossref","is-referenced-by-count":10,"title":["Free idempotent generated semigroups over bands and biordered sets with trivial products"],"prefix":"10.1142","volume":"26","author":[{"given":"Yang","family":"Dandan","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Xidian University, Xi\u2019an 710071, P. R. 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