Maximal subgroups of free idempotent generated semigroups over the full linear monoid

Dolinka, Igor; Gray, Robert D.

doi:10.1090/s0002-9947-2013-05864-3

Cited by 29 publications

(23 citation statements)

References 39 publications

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“…In Section 3 we show how to use the generic presentation for maximal subgroups given in [13] (restated here as Theorem 3.3) to obtain a presentation of H ε ; once these technicalities are in place we sketch the strategy employed in the rest of the paper, and work our way through this in subsequent sections. By the end of Section 6 we are able to show that for 1 ≤ r ≤ n/3, H ε ∼ = H ε (Theorem 6.3), a result corresponding to that in [4] for full linear monoids. To proceed further, we need more sophisticated analysis of the generators of H ε .…”

Section: Introductionmentioning

confidence: 75%

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

2015

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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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Section: Introductionmentioning

confidence: 75%

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

2015

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“…Firstly, these graphs are important in the study of free idempotent generated semigroups. There has been a recent resurgence of interest in the study of these semigroups [9,17,18,21,31,33,34,61,68,69,119], with a particular focus on describing their maximal subgroups. The theory developed in [9] shows that maximal subgroups of free idempotent generated semigroups are precisely the fundamental groups of Graham-Houghton complexes.…”

Section: Introductionmentioning

confidence: 99%

Diagram monoids and Graham–Houghton graphs: Idempotents and generating sets of ideals

East

Gray

2017

Journal of Combinatorial Theory, Series A

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Abstract. We study the ideals of the partition, Brauer, and Jones monoid, establishing various combinatorial results on generating sets and idempotent generating sets via an analysis of their Graham-Houghton graphs. We show that each proper ideal of the partition monoid Pn is an idempotent generated semigroup, and obtain a formula for the minimal number of elements (and the minimal number of idempotent elements) needed to generate these semigroups. In particular, we show that these two numbers, which are called the rank and idempotent rank (respectively) of the semigroup, are equal to each other, and we characterize the generating sets of this minimal cardinality. We also characterize and enumerate the minimal idempotent generating sets for the largest proper ideal of Pn, which coincides with the singular part of Pn. Analogous results are proved for the ideals of the Brauer and Jones monoids; in each case, the rank and idempotent rank turn out to be equal, and all the minimal generating sets are described. We also show how the rank and idempotent rank results obtained, when applied to the corresponding twisted semigroup algebras (the partition, Brauer, and Temperley-Lieb algebras), allow one to recover formulae for the dimensions of their cell modules (viewed as cellular algebras) which, in the semisimple case, are formulae for the dimensions of the irreducible representations of the algebras. As well as being of algebraic interest, our results relate to several well-studied topics in graph theory including the problem of counting perfect matchings (which relates to the problem of computing permanents of {0, 1}-matrices and the theory of Pfaffian orientations), and the problem of finding factorizations of Johnson graphs. Our results also bring together several well-known number sequences such as Stirling, Bell, Catalan and Fibonacci numbers.

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“…Gray and Ruškuc [22] investigated the maximal subgroups of IG(E), where E is the biordered set of idempotents of a full transformation monoid T n , showing that for any e ∈ E with rank r , where 1 ≤ r ≤ n − 2, the maximal subgroup of IG(E) containing e is isomorphic to the maximal subgroup of T n containing e, and hence to the symmetric group S r . Another strand of this popular theme is to consider the biordered set E of idempotents of the matrix monoid M n (D) of all n × n matrices over a division ring D. By using similar topological methods to those of [3], Brittenham, Margolis and Meakin [2] proved that if e ∈ E is a rank 1 idempotent, then the maximal subgroup of IG(E) containing e is isomorphic to that of M n (D), that is, to the multiplicative group D * of D. Dolinka and Gray [10] went on to generalise the result of [2] to e ∈ E with higher rank r , where r < n/3, showing that the maximal subgroup of IG(E) containing e is isomorphic to the maximal subgroup of M n (D) containing e, and hence to the r dimensional general linear group G L r (D). So far, the structure of maximal subgroups of IG(E) containing e ∈ E, where rank e = r and r ≥ n/3 remains unknown.…”

Section: Proposition 11 Let S S E = E(s) Ig(e) and φ Be As Abovementioning

confidence: 98%

Free idempotent generated semigroups and endomorphism monoids of independence algebras

Yang

Gould

2016

Semigroup Forum

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We study maximal subgroups of the free idempotent generated semigroup IG(E), where E is the biordered set of idempotents of the endomorphism monoid End A of an independence algebra A, in the case where A has no constants and has finite rank n. It is shown that when n ≥ 3 the maximal subgroup of IG(E) containing a rank 1 idempotent ε is isomorphic to the corresponding maximal subgroup of End A containing ε. The latter is the group of all unary term operations of A. Note that the class of independence algebras with no constants includes sets, free group acts and affine algebras.

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Maximal subgroups of free idempotent generated semigroups over the full linear monoid

Cited by 29 publications

References 39 publications

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

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

Diagram monoids and Graham–Houghton graphs: Idempotents and generating sets of ideals

Free idempotent generated semigroups and endomorphism monoids of independence algebras

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