Maximal subsemigroups of finite semigroups

“…For any ∈ L we have e = , and hence p e, = λ(e) −1 λ(e )λ( ) −1 = λ(e e)λ( e) −1 = 1 H , using (1), (4). The multiplication between elements of S and M(L, H, R, P ) is defined by…”

Section: The Constructionmentioning

confidence: 99%

Finite groups are big as semigroups

Dolinka

Ruškuc

2011

Arch. Math.

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“…We can now derive some information on the facets of B(n). The next lemma is the graph theoretic interpretation of the construction of maximal subsemigroups for arbitrary finite 0-simple semigroups [5,6] in the special case of aperiodic Brandt semigroups. We introduce some more graph-theoretical tools.…”

Section: Proposition 515mentioning

confidence: 99%

“…It follows that S k−1 is a maximal subsemigroup of B(n) and that F ′ = F \{s k } is a facet of S k−1 . By [6], there is a non-trivial partition X, Y of {1, . .…”

Section: Proposition 516 Every Facet Ofmentioning

confidence: 99%

On the subsemigroup complex of an aperiodic Brandt semigroup

Margolis

Rhodes

2018

Semigroup Forum

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We introduce the subsemigroup complex of a finite semigroup S as a (boolean representable) simplicial complex defined through chains in the lattice of subsemigroups of S. We present a research program for such complexes, illustrated through the particular case of combinatorial Brandt semigroups. The results include alternative characterizations of faces and facets, asymptotical estimates on the number of facets, or establishing when the complex is pure or a matroid.

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“…This approach can be useful for investigating other notions of rank such as nilpotent and idempotent rank (see [Gra08] for example). In another related, and largely forgotten beautiful paper [GGR68] Graham, Graham and Rhodes use this approach to characterise maximal subsemigroups of finite semigroups in general.…”

Section: A Rees Matrix Semigroup and Let γ(S) Be Its Grahamhoughton Gmentioning

confidence: 99%

The minimal number of generators of a finite semigroup

Gray

2013

Semigroup Forum

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The rank of a finite semigroup is the smallest number of elements required to generate the semigroup. A formula is given for the rank of an arbitrary (non necessarily regular) Rees matrix semigroup over a group. The formula is expressed in terms of the dimensions of the structure matrix, and the relative rank of a certain subset of the structure group obtained from subgroups generated by entries in the structure matrix, which is assumed to be in Graham normal form. This formula is then applied to answer questions about minimal generating sets of certain natural families of transformation semigroups. In particular, the problem of determining the maximum rank of a subsemigroup of the full transformation monoid (and of the symmetric inverse semigroup) is considered.

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Maximal subsemigroups of finite semigroups

Cited by 21 publications

References 1 publication

Finite groups are big as semigroups

Finite groups are big as semigroups

On the subsemigroup complex of an aperiodic Brandt semigroup

The minimal number of generators of a finite semigroup

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