A Note on Free Idempotent Generated Semigroups over the Full Monoid of Partial Transformations

Dolinka, Igor

doi:10.1080/00927872.2011.607875

Cited by 15 publications

(17 citation statements)

References 18 publications

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“…Note that our result is independent of the exact nature of the group concerned, which is still unknown in general. As a corollary, we obtain the main result of Dolinka [7].…”

Section: Ig3) the Restriction Of φ To The Maximal Subgroup H E Of Ig(supporting

confidence: 48%

“…(2) follows from Theorem 4.8 and [5]. (3) has already been observed in [7] and is a special case of (2).…”

Section: Proof (I) Follows Immediately From Lemma 43 (I)mentioning

confidence: 82%

“…Dolinka [7] proved that the same holds when T n is replaced by PT n , where PT n is the finite partial transformation monoid on n elements, by reducing the problem for PT n to that for T n . Brittenham, Margolis and Meakin [3] studied the biordered set E of idempotents of the matrix monoid M n (D) of all n × n matrices over a division ring D, where n ≥ 3.…”

Section: Ig3) the Restriction Of φ To The Maximal Subgroup H E Of Ig(mentioning

confidence: 91%

“…Given an idempotent e ∈ E, we would like to explore conditions such that the maximal subgroup of IG(E) with identity e and the corresponding maximal subgroup of IG(F ) are isomorphic. This will enable us to use a reduction approach to determine maximal subgroups of IG(F ) in terms of those of IG(E), inspired by that mentioned above of Dolinka [7] for PT n . We then apply our main result to several special cases, and in particular, as exhibited in Theorem 3.3, to the study of the endomorphism monoid End A and the partial endomorphism monoid PEnd A of an independence algebra A of finite rank.…”

Section: Ig3) the Restriction Of φ To The Maximal Subgroup H E Of Ig(mentioning

confidence: 99%

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Free idempotent generated semigroups: subsemigroups, retracts and maximal subgroups

Yang

Gould

Quinn-Gregson

2017

Communications in Algebra

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Abstract. Let S be a subsemigroup of a semigroup T and let IG(E) and IG(F ) be the free idempotent generated semigroups over the biordered sets of idempotents of E of S and F of T , respectively. We examine the relationship between IG(E) and IG(F ), including the case where S is a retract of T . We give sufficient conditions satisfied by T and S such that for any e ∈ E, the maximal subgroup of IG(E) with identity e is isomorphic to the corresponding maximal subgroup of IG(F ). We then apply this result to some special cases and, in particular, to that of the partial endomorphism monoid PEnd A and the endomorphism monoid End A of an independence algebra A of finite rank. As a corollary, we obtain Dolinka's reduction result for the case where A is a finite set.

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“…Note that our result is independent of the exact nature of the group concerned, which is still unknown in general. As a corollary, we obtain the main result of Dolinka [7].…”

Section: Ig3) the Restriction Of φ To The Maximal Subgroup H E Of Ig(supporting

confidence: 48%

“…(2) follows from Theorem 4.8 and [5]. (3) has already been observed in [7] and is a special case of (2).…”

Section: Proof (I) Follows Immediately From Lemma 43 (I)mentioning

confidence: 82%

Section: Ig3) the Restriction Of φ To The Maximal Subgroup H E Of Ig(mentioning

confidence: 91%

Section: Ig3) the Restriction Of φ To The Maximal Subgroup H E Of Ig(mentioning

confidence: 99%

See 2 more Smart Citations

Free idempotent generated semigroups: subsemigroups, retracts and maximal subgroups

Yang

Gould

Quinn-Gregson

2017

Communications in Algebra

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“…We then check that the corresponding identities to determine G ≀ S r are satisfied by these generators, and it is then a short step to obtain our goal, namely, that H ∼ = H (Theorem 9.13). We note, however, that even at this stage more care is required than, for example, in the corresponding situation for T n [14] or PT n [3], since we cannot assume that G is finite. Indeed our particular choice of Schreier system will be seen to be a useful tool.…”

Section: Theorem 33 [13]mentioning

confidence: 99%

Free idempotent generated semigroups and endomorphism monoids of free G-acts

2015

Self Cite

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Abstract. 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 ≀ 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 End Fn(G) on n generators, and this provides us with a convenient approach.We say that the rank of an element of End Fn(G) is the minimal number of (free) generators in its image. Let ε = ε 2 ∈ End Fn(G). For rather straightforward reasons it is known that if rank ε = n − 1 (respectively, n), then the maximal subgroup of IG(E) containing ε is free (respectively, trivial). We show that if rank ε = r where 1 ≤ r ≤ n − 2, then the maximal subgroup of IG(E) containing ε is isomorphic to that in End Fn(G) and hence to G ≀ 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.

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