Abstract
The study of the free idempotent generated semigroup IG(E) over a biordered set E has recently received a deal of attention. Let G be a group, let n∈N with n≥3 and let E be the biordered set of idempotents of the wreath product G≀Tn. We show, in a transparent way, that for e∈E lying in the minimal ideal of G≀Tn, the maximal subgroup of e in IG(E) is isomorphic to G.
It is known that G≀Tn is the endomorphism monoid End F n (G) of the rank n free G-act F n (G). Our work is therefore analogous to that of Brittenham, Margolis and Meakin for rank 1 idempotents in full linear monoids. As a corollary we obtain the result of Gray and Ruškuc that any group can occur as a maximal subgroup of some free idempotent generated semigroup. Unlike their proof, ours involves a natural biordered set and very little machinery.
It is known that G≀Tn is the endomorphism monoid End F n (G) of the rank n free G-act F n (G). Our work is therefore analogous to that of Brittenham, Margolis and Meakin for rank 1 idempotents in full linear monoids. As a corollary we obtain the result of Gray and Ruškuc that any group can occur as a maximal subgroup of some free idempotent generated semigroup. Unlike their proof, ours involves a natural biordered set and very little machinery.
Original language | English |
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Pages (from-to) | 125-134 |
Number of pages | 10 |
Journal | Semigroup Forum |
Volume | 89 |
Issue number | 1 |
Early online date | 27 Nov 2013 |
DOIs | |
Publication status | Published - Aug 2014 |
Keywords
- G-act
- Idempotent
- Biordered set