Maximality properties of some subsemigroups of Baer-Levi semigroups

Hotzel, Eckehart

doi:10.1007/bf02573628

Cited by 11 publications

(4 citation statements)

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“…Some work has been done on determining all maximal subsemigroups of certain transformation semigroups defined on a finite set X : see [1] for T (X ), [11] for the ideals of T (X ), and [12] for E(X ). There are partial results in the same direction for Baer-Levi semigroups defined on an infinite set: see [4,8]. Here we describe all maximal subsemigroups of A(X ) when X is uncountable.…”

Section: Maximal Subsemigroupsmentioning

confidence: 96%

Injective Transformations With Equal Gap and Defect

Sanwong

Sullivan²

2009

Bull. Aust. Math. Soc.

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Suppose that X is an infinite set and I (X ) is the symmetric inverse semigroup defined on X . If α ∈ I (X ), we let dom α and ran α denote the domain and range of α, respectively, and we say that g(α) = |X \ dom α| and d(α) = |X \ ran α| is the 'gap' and the 'defect' of α, respectively. In this paper, we study algebraic properties of the semigroup A(X ) = {α ∈ I (X ) | g(α) = d(α)}. For example, we describe Green's relations and ideals in A(X ), and determine all maximal subsemigroups of A(X ) when X is uncountable.2000 Mathematics subject classification: primary 20M20.

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Section: Maximal Subsemigroupsmentioning

confidence: 96%

Injective Transformations With Equal Gap and Defect

Sanwong

Sullivan²

2009

Bull. Aust. Math. Soc.

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“…Nichols [6] and Reilly [7] have studied the maximal inverse subsemigroups of T X . Levi and Wood [4] and Hotzel [1] have studied the maximal subsemigroups of Baer-Levi semigroups. Todorov and Kracolova [9] have determined the maximal subsemigroups of the ideals of finite symmetric semigroups.…”

Section: Introductionmentioning

confidence: 99%

Maximal regular subsemigroups of certain semigroups of transformations

You

2002

Semigroup Forum

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“…There are numerous papers in the literature about finding maximal subsemigroups of particular classes of semigroups; for example [5,6,7,8,9,10,12,20,21,22,26,28,34,35,36,37]. Perhaps the most important paper on this topic for finite semigroups is that by Graham, Graham, and Rhodes [18].…”

Section: Introductionmentioning

confidence: 99%

Computing maximal subsemigroups of a finite semigroup

Donoven¹,

Mitchell²,

Wilson³

2018

Journal of Algebra

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A proper subsemigroup of a semigroup is maximal if it is not contained in any other proper subsemigroup. A maximal subsemigroup of a finite semigroup has one of a small number of forms, as described in a paper of Graham, Graham, and Rhodes. Determining which of these forms arise in a given finite semigroup is difficult, and no practical mechanism for doing so appears in the literature. We present an algorithm for computing the maximal subsemigroups of a finite semigroup S given knowledge of the Green's structure of S, and the ability to determine maximal subgroups of certain subgroups of S, namely its group H -classes.In the case of a finite semigroup S represented by a generating set X, in many examples, if it is practical to compute the Green's structure of S from X, then it is also practical to find the maximal subsemigroups of S using the algorithm we present. In such examples, the time taken to determine the Green's structure of S is comparable to that taken to find the maximal subsemigroups. The generating set X for S may consist, for example, of transformations, or partial permutations, of a finite set, or of matrices over a semiring. Algorithms for computing the Green's structure of S from X include the Froidure-Pin Algorithm, and an algorithm of the second author based on the Schreier-Sims algorithm for permutation groups. The worst case complexity of these algorithms is polynomial in |S|, which for, say, transformation semigroups is exponential in the number of points on which they act.Certain aspects of the problem of finding maximal subsemigroups reduce to other well-known computational problems, such as finding all maximal cliques in a graph and computing the maximal subgroups in a group.The algorithm presented comprises two parts. One part relates to computing the maximal subsemigroups of a special class of semigroups, known as Rees 0-matrix semigroups. The other part involves a careful analysis of certain graphs associated to the semigroup S, which, roughly speaking, capture the essential information about the action of S on its J -classes. arXiv:1606.05583v4 [math.CO] 6 Jul 2018 are L -related. Green's R-relation is defined dually to Green's L -relation; Green's H -relation is the meet, in the lattice of equivalence relations on S, of L and R. In any semigroup, xJ y if and only if the (2-sided) principal ideals generated by x and y are equal. However, in a finite semigroup J is the join of L and R. We will refer to the equivalence classes as K -classes where K is any of R, L , H , or J , and the K -class of x ∈ S will be denoted by K x . We write K S if it is necessary to explicitly refer to the semigroup on which the relation is defined. We denote the set of K -classes of a semigroup S by S/K .An idempotent is an element x ∈ S such that x 2 = x. We denote the set of idempotents in a semigroup S by E(S). A J -class of a finite semigroup is regular if it contains an idempotent, and a finite semigroup is called regular if each of its J -classes is regular. Containment of principal ideals induces a p...

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Maximality properties of some subsemigroups of Baer-Levi semigroups

Cited by 11 publications

References 11 publications

Injective Transformations With Equal Gap and Defect

Injective Transformations With Equal Gap and Defect

Maximal regular subsemigroups of certain semigroups of transformations

Computing maximal subsemigroups of a finite semigroup

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