Embedding finite semigroups in finite semibands of minimal depth

Giraldes, E.; Howie, James

doi:10.1007/bf02572480

Cited by 7 publications

(4 citation statements)

References 3 publications

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“…D Hence, as a consequence of this proposition and the discussion preceding it, we have the following result concerning linear transformations over finite dimensional vector spaces. In [5], a result of Laffey received in personal correspondance is mentioned. The result is that every nxn matrix of rank less than n/2 is a product of two idempotent matrices.…”

Section: Products Of Two Idempotents In £(F)mentioning

confidence: 98%

Products of two idempotent transformations over arbitrary sets and vector spaces

Thomas¹

1998

Bull. Austral. Math. Soc.

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In this paper we consider the characterisation of those elements of a transformation semigroup S which are a product of two proper idempotents. We give a characterisation where S is the endomorphism monoid of a strong independence algebra A, and apply this to the cases where A is an arbitrary set and where A is an arbitrary vector space. The results emphasise the analogy between the idempotent generated subsemigroups of the full transformation semigroup of a set and of the semigroup of linear transformations from a vector space to itself.

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Section: Products Of Two Idempotents In £(F)mentioning

confidence: 98%

Products of two idempotent transformations over arbitrary sets and vector spaces

Thomas¹

1998

Bull. Austral. Math. Soc.

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“…Our transformation A → H(A ) is a straightforward adaptation of a construction suggested by Higgins [11] in the realm of semigroup theory. Higgins used this construction to give a new simple proof of the following result independently obtained in [7] and [15]: an arbitrary (finite) semigroup may be embedded into another (finite) semigroup in which every element is the product of two idempotents. Our contribution consists in observing that, when restated in automata-theoretic terms, Higgins's construction leads to a transformation that preserves both the property of being synchronizing and the order of magnitude of reset threshold.…”

Section: Background and Overviewmentioning

confidence: 99%

Slowly Synchronizing Automata with Idempotent Letters of Low Rank

Волков¹

2018

Preprint

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We use a semigroup-theoretic construction by Peter Higgins in order to produce, for each even n, an n-state and 3-letter synchronizing automaton with the following two features: 1) all its input letters act as idempotent selfmaps of rank n 2 ; 2) its reset threshold is asymptotically equal to n 2 2 . * Supported by the Ministry of Science and Higher Education of the Russian Federation, project no. 1.580.2016, and the Competitiveness Enhancement Program of Ural Federal University.

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“…(2) C (n) denote the smallest integer k ≥ n such that every semigroup of C of order not greater than n can be embeddable into a semiband of depth 2 and order not greater than k. Let G be the class of all groups and let Reg be the class of all regular semigroups. Giraldes and Howie [3] showed that σ (2) G (n) ≤ 2n 2 and σ…”

Section: Proposition 56 a Regular Semigroup S Is Completely Regular mentioning

confidence: 99%

“…To answer to a question posed by Howie in [8], Laffey [12] and Hall presented each one a way to embed a finite semigroup into a finite semiband of depth 2 (the proof of the latter appears in [3]). Laffey's embedding is based on linear algebra techniques, while Hall's embedding makes use of the Rees matrix semigroup construction.…”

Section: Introductionmentioning

confidence: 99%

On the question of embedding a semigroup into a semiband

Oliveira

2014

Semigroup Forum

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In this paper we present a new embedding of a semigroup into a semiband (idempotent-generated semigroup) of depth 4 (every element is the product of 4 idempotents) using a semidirect product construction. Our embedding does not assume that S is a monoid (although it assumes a weaker condition), and works also for (nonmonoid) regular semigroups. In fact, this semidirect product is particularly useful for regular semigroups since we can defined another embedding for these semigroups into a smaller semiband of depth 2. In this paper we shall compare our construction with other known embeddings, and we shall see that some properties of S are preserved by it.

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Embedding finite semigroups in finite semibands of minimal depth

Cited by 7 publications

References 3 publications

Products of two idempotent transformations over arbitrary sets and vector spaces

Products of two idempotent transformations over arbitrary sets and vector spaces

Slowly Synchronizing Automata with Idempotent Letters of Low Rank

On the question of embedding a semigroup into a semiband

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