On Finite Complete Presentations and Exact Decompositions of Semigroups

Araújo, João; Malheiro, António

doi:10.1080/00927872.2010.514314

Cited by 4 publications

(3 citation statements)

References 22 publications

(26 reference statements)

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“…Consider the monoid M 0 obtained from the monoid M defined by Narendran and Otto in [25, page 35] which has undecidable o-conjugacy problem. Since M is defined by a finite complete presentation, the monoid M 0 is also defined by a finite complete presentation by [3,Proposition 3.1]. It can be seen that M does not have a zero.…”

Section: Proposition 42mentioning

confidence: 99%

Decidability and independence of conjugacy problems in finitely presented monoids

Araújo

Kinyon

Konieczny

et al. 2018

Theoretical Computer Science

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There have been several attempts to extend the notion of conjugacy from groups to monoids. The aim of this paper is study the decidability and independence of conjugacy problems for three of these notions (which we will denote by ∼p, ∼o, and ∼c) in certain classes of finitely presented monoids. We will show that in the class of polycyclic monoids, p-conjugacy is "almost" transitive, ∼c is strictly included in ∼p, and the p-and c-conjugacy problems are decidable with linear compexity. For other classes of monoids, the situation is more complicated. We show that there exists a monoid M defined by a finite complete presentation such that the c-conjugacy problem for M is undecidable, and that for finitely presented monoids, the c-conjugacy problem and the word problem are independent, as are the c-conjugacy and p-conjugacy problems.

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Section: Proposition 42mentioning

confidence: 99%

Decidability and independence of conjugacy problems in finitely presented monoids

Araújo

Kinyon

Konieczny

et al. 2018

Theoretical Computer Science

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“…The term 'Zappa-Szep Extension' appears in [2] for the category of semigroups, and the view of such internal products as extensions analogous to traditional split extensions appears in [3]. Early references to such internal products (of subgroups of a given group) appears, for example, in Szep's original paper [19] on the subject, but was studied by others as well, around the same time.…”

Section: Zappa-szep Extensionsmentioning

confidence: 99%

“…As such Q(D 3 ) = H(D 3 ), and that NHol(G) = QHol(G) ∼ = (S 3 × S 3 ) ⋊ C 2 where the C 2 component is that element conjugating λ(D 3 ) to ρ(D 3 ), so that π(Q(G)) is also this same subgroup of order 2. Indeed, if we embed D 3 as S 3 into S 6 then we have λ(S 3 ) = (1, 3)(2, 5) (4,6), (1,4,5)(2, 6, 3) ρ(S 3 ) = (1, 2)(3, 5) (4,6), (1,4,5)(2, 3, 6) QHol(S 3 ) = Hol(S 3 ) τ for any τ ∈ { (1,4), (1,5), (2,3), (2,6), (3,6), (4,5)} so that all π(Q(S 3 )) are isomorphic, which is unsurprising given that Q = H and QHol(S 3 ) = NHol(S 3 ) so that any π(Q(G)) would be isomorphic to T (S 3 ). For D n = x, t | x n = 1, t 2 = 1, xt = tx −1 in general, in [15], we have a complete enumeration of R(D n ) = R(D n , [D n ]).…”

Section: Dihedral Groupsmentioning

confidence: 99%

Mutually Normalizing Regular Permutation Groups and Zappa-Szep Extensions of the Holomorph

Kohl¹

2020

Preprint

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For a group G, embedded in its group of permutations B = P erm(G) via the left regular representation λ : G → P erm(G), the normalizer of λ(G) in B is Hol(G), the holomorph of G. It is known that Hol(G) is also the normalizer of ρ(G) as well and both λ(G) and ρ(G) are canonical examples of regular subgroups. The determination of those regular N ≤ P erm(G), where N ∼ = G with the same normalizer is keyed to the structure of the so-called multiple holomorph of G, NHol(G) = N orm B (N orm B (λ(G))). We wish to analyze the set of those subgroups of Hol(G) which mutually normalize each other, but which don't necessarily have the same normalizer. This analysis gives rise to a group that contains NHol(G), but is potentially larger. We introduce what we call the quasi-holomorph of G which allows one to enumerate this set of mutually normalizing subgroups of Hol(G). The quasi-holomorph will be a group properly containing Hol(G) and is frequently a Zappa-Szep product with the holomorph.

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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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On Finite Complete Presentations and Exact Decompositions of Semigroups

Cited by 4 publications

References 22 publications

Decidability and independence of conjugacy problems in finitely presented monoids

Decidability and independence of conjugacy problems in finitely presented monoids

Mutually Normalizing Regular Permutation Groups and Zappa-Szep Extensions of the Holomorph

Four notions of conjugacy for abstract semigroups

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