Groups that together with any transformation generate regular semigroups or idempotent generated semigroups

Araújo, João; Mitchell, James D.; Schneider, Csaba

doi:10.1016/j.jalgebra.2011.07.002

Cited by 16 publications

(37 citation statements)

References 20 publications

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“…• t = 2, G ≤ AΣL(1, q), H = {1}: λ = (n − 2, 2); • t = 3, G ≤ PΣL(2, q), H = A 3 : λ = (n − 3, 3) or (n − 3, 2, 1); • t = 3, G = AGL (1,8) or AΓL (1,32), H = {1}: λ = (n − 3, 3);…”

Section: Introductionmentioning

confidence: 99%

“…• t = 3, G = AΓL (2,8), H = A 3 : λ = (n − 3, 3) or (n − 3, 2, 1); • t = 4, G = PGL (2,8) or PΓL (2,32), H = V 4 : λ = (n − 4, 4) or (n − 4, 3, 1); • t = 4, G = PΓL (2,8), H = A 4 : λ is any partition with largest part n − 4 except (n − 4, 1, 1, 1, 1). The classification of λ-homogeneous groups is a very natural problem in group theory, and it is quite a mystery why it took so long for this idea to appear in the literature, after the introduction of λ-transitive groups by Martin and Sagan in [34].…”

Section: Introductionmentioning

confidence: 99%

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The classification of partition homogeneous groups with applications to semigroup theory

2016

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Let λ = (λ 1 , λ 2 , . . .) be a partition of n, a sequence of positive integers in non-increasing order with sum n. Let Ω := {1, . . . , n}. An ordered partition P = (A 1 , A 2 , . . .) of Ω has type λ if |A i | = λ i .Following Martin and Sagan, we say that G is λ-transitive if, for any two ordered partitions P = (A 1 , A 2 , . . .) and Q = (B 1 , B 2 , . . .) of Ω of type λ, there exists g ∈ G with A i g = B i for all i. A group G is said to be λhomogeneous if, given two ordered partitions P and Q as above, inducing the setsThe first goal of this paper is to classify the λ-homogeneous groups (Theorems 1.1 and 1.2). The second goal is to apply this classification to a problem in semigroup theory.Let Tn and Sn denote the transformation monoid and the symmetric group on Ω, respectively. Fix a group H Sn. Given a non-invertible transformation a ∈ Tn \ Sn and a group G Sn, we say that (a, G) is an H-pair if the semigroups generated by {a} ∪ H and {a} ∪ G contain the same non-units, that is, a, G \ G = a, H \ H. Using the classification of the λ-homogeneous groups we classify all the Sn-pairs (Theorem 1.7). For a multitude of transformation semigroups this theorem immediately implies a description of their automorphisms, congruences, generators and other relevant properties (Theorem 8.5).This topic involves both group theory and semigroup theory; we have attempted to include enough exposition to make the paper self-contained for researchers in both areas.The paper finishes with a number of open problems on permutation and linear groups.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The classification of partition homogeneous groups with applications to semigroup theory

2016

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“…Given the enormous progress made in the last three or four decades, permutation groups now has the tools to answer questions coming from the real world through transformation semigroups; these questions translate into beautiful statements in the language of permutation groups and combinatorial structures, as shown in many recent investigations (as a small sample, please see [2,4,6,7,8,12,13,20,29,39,42]).…”

Section: Theorem 15 a Transformation Semigroup Does Not Contain A Comentioning

confidence: 99%

Primitive groups, graph endomorphisms and synchronization

Araújo

Bentz

Cameron

et al. 2016

Proc. London Math. Soc.

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Let Ω be a set of cardinality n, G be a permutation group on Ω and f : Ω → Ω be a map that is not a permutation. We say that G synchronizes f if the transformation semigroup G, f contains a constant map, and that G is a synchronizing group if G synchronizes every non-permutation.A synchronizing group is necessarily primitive, but there are primitive groups that are not synchronizing. Every non-synchronizing primitive group fails to synchronize at least one uniform transformation (that is, transformation whose kernel has parts of equal size), and it had previously been conjectured that this was essentially the only way in which a primitive group could fail to be synchronizing, in other words, that a primitive group synchronizes every non-uniform transformation.The first goal of this paper is to prove that this conjecture is false, by exhibiting primitive groups that fail to synchronize specific non-uniform transformations of ranks 5 and 6. As it has previously been shown that primitive groups synchronize every non-uniform transformation of rank at most 4, these examples are of the lowest possible rank. In addition, we produce graphs with primitive automorphism groups that have approximately √ n non-synchronizing ranks, thus refuting another conjecture on the number of non-synchronizing ranks of a primitive group.The second goal of this paper is to extend the spectrum of ranks for which it is known that primitive groups synchronize every non-uniform transformation of that rank. It has previously been shown that a primitive group of degree n synchronizes every non-uniform transformation of rank n − 1 and n − 2, and here this is extended to n − 3 and n − 4.In the process, we will obtain a purely graph-theoretical result showing that, with limited exceptions, in a vertex-primitive graph the union of neighbourhoods of a set of vertices A is bounded below by a function that is asymptotically |A|.Determining the exact spectrum of ranks for which there exist non-uniform transformations not synchronized by some primitive group is just one of several natural, but possibly difficult, problems on automata, primitive groups, graphs and computational algebra arising from this work; these are outlined in the final section.

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“…It is clear that if a, G \G is idempotent generated, for all rank k transformation a ∈ T n \ S n , then G has the k-ut property (see [8]). Problem 3.4 Classify the groups G ≤ S n such that a, G \ G is idempotent generated, for all rank k maps, where k ≤ n/2.…”

Section: Problemsmentioning

confidence: 99%

“…The following portmanteau theorem lists some previously known results on this problem. The first part is due to Levi [26], the other two to Araújo, Mitchell and Schneider [8]. Theorem 1.3 (a) For any a ∈ T n \ S n the semigroups g −1 ag : g ∈ S n and g −1 ag : g ∈ A n are equal.…”

Section: Introductionmentioning

confidence: 99%

Permutation groups and transformation semigroups: results and problems

Araújo

Cameron

2015

Groups St Andrews 2013

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J.M. Howie, the influential St Andrews semigroupist, claimed that we value an area of pure mathematics to the extent that (a) it gives rise to arguments that are deep and elegant, and (b) it has interesting interconnections with other parts of pure mathematics.This paper surveys some recent results on the transformation semigroup generated by a permutation group G and a single non-permutation a. Our particular concern is the influence that properties of G (related to homogeneity, transitivity and primitivity) have on the structure of the semigroup. In the first part of the paper, we consider properties of S = G, a such as regularity and generation. The second is a brief report on the synchronization project, which aims to decide in what circumstances S contains an element of rank 1. The paper closes with a list of open problems on permutation groups and linear groups, and some comments about the impact on semigroups are provided.These two research directions outlined above lead to very interesting and challenging problems on primitive permutation groups whose solutions require combining results from several different areas of mathematics, certainly fulfilling both of Howie's elegance and value tests in a new and fascinating way.

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Groups that together with any transformation generate regular semigroups or idempotent generated semigroups

Cited by 16 publications

References 20 publications

The classification of partition homogeneous groups with applications to semigroup theory

The classification of partition homogeneous groups with applications to semigroup theory

Primitive groups, graph endomorphisms and synchronization

Permutation groups and transformation semigroups: results and problems

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