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Title: | Free idempotent generated semigroups and endomorphism monoids of free G-acts | Authors: | Dolinka Igor Gould Victoria Yang Dandan |
Issue Date: | 2015 | Journal: | Journal of Algebra | Abstract: | © 2015 Elsevier Inc. 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 and has seen a recent revival with a number of new approaches, both geometric and combinatorial. Here we study IG(E) in the case E is the biordered set of a wreath product G{wreath product}Tn, where G is a group and Tn is the full transformation monoid on n elements. This wreath product is isomorphic to the endomorphism monoid of the free G-act EndFn(G) on n generators, and this provides us with a convenient approach. We say that the rank of an element of EndFn(G) is the minimal number of (free) generators in its image. Let ε=ε2∈EndFn(G). For rather straightforward reasons it is known that if rankε=n-1 (respectively, n), then the maximal subgroup of IG(E) containing ε is free (respectively, trivial). We show that if rankε=r where 1≤r≤n-2, then the maximal subgroup of IG(E) containing ε is isomorphic to that in EndFn(G) and hence to G{wreath product}Sr, where Sr is the symmetric group on r elements. We have previously shown this result in the case r=1; however, for higher rank, a more sophisticated approach is needed. Our current proof subsumes the case r=1 and thus provides another approach to showing that any group occurs as the maximal subgroup of some IG(E). On the other hand, varying r again and taking G to be trivial, we obtain an alternative proof of the recent result of Gray and Ruškuc for the biordered set of idempotents of Tn. | URI: | https://open.uns.ac.rs/handle/123456789/30507 | ISSN: | 0021-8693 | DOI: | 10.1016/j.jalgebra.2014.12.041 |
Appears in Collections: | PMF Publikacije/Publications |
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