On the subsemigroup complex of an aperiodic Brandt semigroup

Margolis, Stuart W.; Rhodes, John

doi:10.1007/s00233-018-9927-4

Cited by 6 publications

(5 citation statements)

References 21 publications

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“…also a chain in L(S). It is easy to check that every X ∈ H is a partial transversal of the successive differences for some chain of type (8) or (9), hence S is boolean representable.…”

Section: Joinmentioning

confidence: 99%

“…and the maximum length of a chain in L(S) is d + 1, it follows easily that the maximal chains in L(S) must be of the form (8)…”

Section: Joinmentioning

confidence: 99%

“…(i) ⇒ (ii). Since P ≤d−1 (V ) ∪ {V } ⊆ L(S) and the maximum length of a chain in L(S) is d + 1, it follows easily that the maximal chains in L(S) must be of the form (8) or (9), with L ∈ L(S) \ (P ≤d−1 (V ) ∪ {V }). Thus (ii) holds.…”

Section: Joinmentioning

confidence: 99%

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On the topology of a boolean representable simplicial complex

Margolis

Rhodes

2017

Int. J. Algebra Comput.

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“…also a chain in L(S). It is easy to check that every X ∈ H is a partial transversal of the successive differences for some chain of type (8) or (9), hence S is boolean representable.…”

Section: Joinmentioning

confidence: 99%

“…and the maximum length of a chain in L(S) is d + 1, it follows easily that the maximal chains in L(S) must be of the form (8)…”

Section: Joinmentioning

confidence: 99%

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On the topology of a boolean representable simplicial complex

Margolis

Rhodes

2017

Int. J. Algebra Comput.

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“…The three authors of the present paper study the BRSC associated to the lattice R n (1) and more generally to the lattice of all subsemigroups of B(1, X) in [9].…”

Section: A Semigroup Theoretic Interpretation Of the Rhodes Latticementioning

confidence: 99%

On the Dowling and Rhodes lattices and wreath products

Margolis¹,

Rhodes²

2017

Preprint

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Dowling and Rhodes defined different lattices on the set of triples (Subset, Partition, Cross Section) over a fixed finite group G. Although the Rhodes lattice is not a geometric lattice, it defines a matroid in the sense of the theory of Boolean representable simplicial complexes. This turns out to be the direct sum of a complete matroid with a lift matroid of the complete biased graph over G. As is well known, the Dowling lattice defines the frame matroid over a similar biased graph. This gives a new perspective on both matroids and also an application of matroid theory to the theory of finite semigroups. We also make progress on an important question for these classical matroids: what are the minimal Boolean representations and the minimum degree of a Boolean matrix representation?

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“…Further, various aspects of affine near-semirings generated by affine maps on B n have been studied in [21]. The combinatorial study of B n have been related with theory of matroids and simplicial complexes in [23]. Various ranks of B n have been obtained in [14,15,24], where some of the ranks of B n were obtained by using graph theoretic properties of some graph associated on B n .…”

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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On the subsemigroup complex of an aperiodic Brandt semigroup

Cited by 6 publications

References 21 publications

On the topology of a boolean representable simplicial complex

On the topology of a boolean representable simplicial complex

On the Dowling and Rhodes lattices and wreath products

On the commuting graphs of Brandt semigroups

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