The rank of the endomorphism monoid of a uniform partition

Araújo, João; Schneider, Csaba

doi:10.1007/s00233-008-9122-0

Cited by 32 publications

(37 citation statements)

References 22 publications

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“…These monoids were introduced by Pei in [67], and have subsequently been studied by a number of different authors. In particular, the rank of T (X, P) was calculated in [5], while the rank and idempotent rank of the idempotent-generated subsemigroup of T (X, P) were calculated in [16]; for the corresponding studies of the non-uniform case, see [4,17]. As noted in [5], T (X, P) is isomorphic to the wreath product T m ≀ T n .…”

Section: An Idempotent-generated Presentation For M ≀ Sing N With M/lmentioning

confidence: 99%

Presentations for singular wreath products

Feng

Al-Aadhami

Dolinka

et al. 2019

Journal of Pure and Applied Algebra

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For a monoid M and a subsemigroup S of the full transformation semigroup T n , the wreath product M ≀ S is defined to be the semidirect product M n ⋊ S, with the coordinatewise action of S on M n . The full wreath product M ≀ T n is isomorphic to the endomorphism monoid of the free M -act on n generators. Here, we are particularly interested in the case that S = Sing n is the singular part of T n , consisting of all non-invertible transformations. Our main results are presentations for M ≀Sing n in terms of certain natural generating sets, and we prove these via general results on semidirect products and wreath products. We re-prove a classical result of Bulman-Fleming that M ≀ Sing n is idempotent-generated if and only if the set M/L of L -classes of M forms a chain under the usual ordering of L -classes, and we give a presentation for M ≀ Sing n in terms of idempotent generators for such a monoid M . Among other results, we also give estimates for the minimal size of a generating set for M ≀ Sing n , as well as exact values in some cases (including the case that M is finite and M/L is a chain, in which case we also calculate the minimal size of an idempotent generating set). As an application of our results, we obtain a presentation (with idempotent generators) for the idempotent-generated subsemigroup of the endomorphism monoid of a uniform partition of a finite set.

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Section: An Idempotent-generated Presentation For M ≀ Sing N With M/lmentioning

confidence: 99%

Presentations for singular wreath products

Feng

Al-Aadhami

Dolinka

et al. 2019

Journal of Pure and Applied Algebra

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“…Ranks of various finite monoids have been determined in the literature before (e.g. see [1,2,7,8,10]).…”

Section: Generating Sets Of Of Ca(g; A)mentioning

confidence: 99%

On Finite Monoids of Cellular Automata

Castillo-Ramirez

Gadouleau

2016

Cellular Automata and Discrete Complex Systems

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“…is the action of S α of permuting the coordinates of C α . For a more detailed description of the wreath product see [1]. The Rank function on monoids does not behave well when taking submonoids or subgroups: in other words, if N is a submonoid of M , there may be no relation between Rank(N ) and Rank(M ).…”

Section: Introductionmentioning

confidence: 99%

On the minimal number of generators of endomorphism monoids of full shifts

Castillo-Ramirez

2020

Nat Comput

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For a group G and a finite set A, denote by End(A G ) the monoid of all continuous shift commuting self-maps of A G and by Aut(A G ) its group of units. We study the minimal cardinality of a generating set, known as the rank, of End(A G ) and Aut(A G ). In the first part, when G is a finite group, we give upper and lower bounds for the rank of Aut(A G ) in terms of the number of conjugacy classes of subgroups of G. In the second part, we apply our bounds to show that if G has an infinite descending chain of normal subgroups of finite index, then End(A G ) is not finitely generated; such is the case for wide classes of infinite groups, such as infinite residually finite or infinite locally graded groups.

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The rank of the endomorphism monoid of a uniform partition

Cited by 32 publications

References 22 publications

Presentations for singular wreath products

Presentations for singular wreath products

On Finite Monoids of Cellular Automata

On the minimal number of generators of endomorphism monoids of full shifts

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