Endomorphism Seminear-Rings of Brandt Semigroups

Gilbert, N. D.; Samman, Mohammad

doi:10.1080/00927870903286892

Cited by 4 publications

(7 citation statements)

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“…End(B n ) and Aff(B n ). We find the size of End(B n ) by generalizing the corresponding result for End ϑ (B n ) in (Gilbert and Samman, 2010b). They have shown that the monoid…”

Section: The Elements Ofmentioning

confidence: 68%

“…In this section, we provide a necessary background material and fix our notation. For more details one may refer to (Gilbert and Samman, 2010b;Krishna, 2005).…”

Section: Preliminariesmentioning

confidence: 99%

“…The endomorphism near-semiring, denoted by E + (Γ), is the subnear-semiring generated by End(Γ) in M (Γ). Indeed, the subsemigroup of (M (Γ), +) generated by End(Γ) equals (E Gilbert and Samman (2010b)…”

Section: Preliminariesmentioning

confidence: 99%

“…Note that the notion of full support maps used in (Gilbert and Samman, 2010b) is too restrictive and is limited to the elements of E + (B n ) whose support size is at most n. Whereas, the above definition can be adopted to any map.…”

Section: Preliminariesmentioning

confidence: 99%

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Affine Near-Semirings Over Brandt Semigroups

Kumar

Krishna

2014

Communications in Algebra

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Abstract. This work investigates the rank properties of A + (Bn), the additive semigroup reduct of affine near-semiring over Brandt semigroup Bn. In this connection, this work reports the ranks r 1 , r 2 , r 3 and r 5 of A + (Bn) and identifies a lower bound for the upper rank r 4 (A + (Bn)). While this lower bound is found to be the r 4 (A + (Bn)) for n ≥ 6, in other cases where 2 ≤ n ≤ 5, the upper rank of A + (Bn) is still open for investigation.

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“…End(B n ) and Aff(B n ). We find the size of End(B n ) by generalizing the corresponding result for End ϑ (B n ) in (Gilbert and Samman, 2010b). They have shown that the monoid…”

Section: The Elements Ofmentioning

confidence: 68%

“…In this section, we provide a necessary background material and fix our notation. For more details one may refer to (Gilbert and Samman, 2010b;Krishna, 2005).…”

Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

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Affine Near-Semirings Over Brandt Semigroups

Kumar

Krishna

2014

Communications in Algebra

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“…Brandt semigroup B n play an important role in inverse semigroup theory and arises in number of different ways, see [9,19] and the references therein. Endomorphism seminear-rings on B n have been classified by Gilbert and Samman [10]. Further, various aspects of affine near-semirings generated by affine maps on B n have been studied in [21].…”

Section: Introductionmentioning

confidence: 99%

On the commuting graphs of Brandt semigroups

Kumar¹,

Dalal²,

Pandey³

2020

Preprint

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The commuting graph of a finite non-commutative semigroup S, denoted by ∆(S), is the simple graph whose vertices are the non-central elements of S and two distinct vertices x, y are adjacent if xy = yx. In the present paper, we study various graph theoretic properties of the commuting graph ∆(B n ) of Brandt semigroup B n including its diameter, clique number, chromatic number, independence number, strong metric dimension and dominance number. Moreover, we obtain the automorphism group Aut(∆(B n )) and the endomorphism monoid End(∆(B n )) of ∆(B n ). We show that Aut(∆(B n )) ∼ = S n × Z 2 , where S n is the symmetric group of degree n and Z 2 is the additive group of integers modulo 2. Further, for n ≥ 4, we prove that End(∆(B n )) =Aut(∆(B n )). In order to provide an answer to the question posed in [2], we ascertained a class of inverse semigroups whose commuting graph is Hamiltonian.

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