The largest subsemilattices of the endomorphism monoid of an independence algebra

Araújo, João; Bentz, Wolfram; Konieczny, Janusz

doi:10.1016/j.laa.2014.05.041

Cited by 6 publications

(11 citation statements)

References 51 publications

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“…Symmetrically, there exists t ∈ GL(i, F ) × GL(j, F ) such that At = A(s 3 , s 4 ) −1 (t 3 , t 4 ), for all A ∈ GL(i, F ) × GL(j, F ). Now, (5) shows that in the group GL(i, F ) × GL(j, F ), we have t = s −1 . Thus, in GL(i, F ) × GL(j, F ),…”

Section: Thenmentioning

confidence: 97%

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Congruences on direct products of transformation and matrix monoids

2018

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Malcev described the congruences of the monoid T n of all full transformations on a finite set X n = {1, . . . , n}. Since then, congruences have been characterized in various other monoids of (partial) transformations on X n , such as the symmetric inverse monoid I n of all injective partial transformations, or the monoid PT n of all partial transformations.The first aim of this paper is to describe the congruences of the direct products Q m × P n , where Q and P belong to {T , PT , I}.Malcev also provided a similar description of the congruences on the multiplicative monoid F n of all n × n matrices with entries in a field F ; our second aim is provide a description of the principal congruences of F m × F n .The paper finishes with some comments on the congruences of products of more than two transformation semigroups, and a fairly large number of open problems.

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Section: Thenmentioning

confidence: 97%

“…Possibly the best approach to attack Problem 8.10 is to rely on the classification theorem of these algebras ( [26,69,70]), as in [5].…”

Section: Problemsmentioning

confidence: 99%

Congruences on direct products of transformation and matrix monoids

2018

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“…Indeed, we have 4 ∼ p 3 (since 4 = 1 · 4 and 3 = 4 · 1) and 3 ∼ p 5 (since 3 = 1 · 5 and 5 = 5 · 1), but there are no x, y such that 4 = xy and 5 = yx. It is straightforward to check that ∼ p is the symmetric and reflexive closure of {(1, 2), (3,4), (3,5)}, that ∼ * p = ∼ tr , and that ∼ c = ∼ o has equivalence classes {0, 1, 2} and {3, 4, 5}. Thus we have the claimed strict inclusions.…”

Section: Conjugacy In Epigroups and Epigroup Elementsmentioning

confidence: 76%

Four notions of conjugacy for abstract semigroups

Araújo¹,

Kinyon²,

Konieczny³

et al. 2017

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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The action of any group on itself by conjugation and the corresponding conjugacy relation play an important role in group theory. There have been many attempts to find notions of conjugacy in semigroups that would be useful in special classes of semigroups occurring in various areas of mathematics, such as semigroups of matrices, operator and topological semigroups, free semigroups, transition monoids for automata, semigroups given by presentations with prescribed properties, monoids of graph endomorphisms, etc. In this paper we study four notions of conjugacy for semigroups, their interconnections, similarities and dissimilarities. They appeared originally in various different settings (automata, representation theory, presentations, and transformation semigroups). Here we study them in full generality. The paper ends with a large list of open problems.

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“…Now this graph can be found as a subgraph of Γ, as the union of two butterflies sharing a triangle; therefore the homomorphism can be realised as an endomorphism of Γ of rank 7, with kernel classes of sizes (10,10,5,5,5,5,5). The endomorphisms of ranks 5 and 3 can now be found by folding in one or both "wings" in the above figure.…”

Section: Rank 5 (And 7)mentioning

confidence: 99%

“…Problem 9.10 Solve the analogue of Problem 9.9 for independence algebras (for definitions and fundamental results see [3,9,10,11,5,23,27,28,30])…”

Section: Problem 94mentioning

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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The largest subsemilattices of the endomorphism monoid of an independence algebra

Cited by 6 publications

References 51 publications

Congruences on direct products of transformation and matrix monoids

Congruences on direct products of transformation and matrix monoids

Four notions of conjugacy for abstract semigroups

Primitive groups, graph endomorphisms and synchronization

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