The Ideal Structure of Semigroups of Transformations With Restricted Range

Mendes-Gonçalves, Suzana; Sullivan, R. P.

doi:10.1017/s0004972710001814

Cited by 14 publications

(6 citation statements)

References 6 publications

(14 reference statements)

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“…Once the previous problem is solved, the following is also natural given the results in this paper. Our original goal was the description of the congruences on the direct product of some classic transformation monoids; of course, there are many more such classes whose congruences have also been classified (see for example [27,36,37,38,39,66]). Therefore the following is a natural question.…”

Section: Problemsmentioning

confidence: 99%

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: Problemsmentioning

confidence: 99%

Congruences on direct products of transformation and matrix monoids

2018

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“…These subsemigroups have been studied extensively in the literature [20,24,44,46,47,53,54], and will play an important role in the current work. Note that for any a ∈ T X with im(a) = A, the subsemigroup T X (A) is precisely the principal left ideal…”

Section: Full Transformation Semigroupsmentioning

confidence: 99%

Minimum degrees of finite rectangular bands, null semigroups, and variants of full transformation semigroups

Cameron¹,

East²,

Fitzgerald³

et al. 2021

Preprint

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For a positive integer n, the full transformation semigroup T n consists of all self maps of the set {1, . . . , n} under composition. Any finite semigroup S embeds in some T n , and the least such n is called the (minimum transformation) degree of S and denoted µ(S). We find degrees for various classes of finite semigroups, including rectangular bands, rectangular groups and null semigroups. The formulae we give involve natural parameters associated to integer compositions. Our results on rectangular bands answer a question of Easdown from 1992, and our approach utilises some results of independent interest concerning partitions/colourings of hypergraphs.As an application, we prove some results on the degree of a variant T a n . (The variant S a = (S, ) of a semigroup S, with respect to a fixed element a ∈ S, has underlying set S and operation x y = xay.) It has been previously shown that n ≤ µ(T a n ) ≤ 2n − r if the sandwich element a has rank r, and the upper bound of 2n − r is known to be sharp if r ≥ n − 1. Here we show that µ(T a n ) = 2n − r for r ≥ n − 6. In stark contrast to this, when r = 1, and the above inequality says nAmong other results, we also classify the 3-nilpotent subsemigroups of T n , and calculate the maximum size of such a subsemigroup.

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“…The semigroups T 1 and T 2 are special cases of semigroups of transformations with restricted range or kernel. Such semigroups have been studied extensively by many authors, particularly from the Thai school of semigroup theory; see for example [12][13][14]28,29,[32][33][34][35][39][40][41][42][43][44][45][46][47][48][49][50]52].…”

Section: Introductionmentioning

confidence: 99%