NP-Completeness in the gossip monoid

Fenner, Peter; Johnson, Marianne; Kambites, Mark

doi:10.1142/s0218196718500297

Cited by 7 publications

(5 citation statements)

References 12 publications

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“…Another submonoid of M n (B), which has attracted recent attention, is the Gossip monoid G n ; see [7,19] for some recent results and [2,24] for the problem which inspired G n . This is typically defined as the monoid generated by the so-called phone-call matrices C[i, j], where There are a number of other submonoids of M n (B) that arise as generalisations of the Hall monoid.…”

Section: Boolean Matrix Monoidsmentioning

confidence: 99%

“…If n ∈ N + , then we denote by M n (B) the monoid consisting of all n × n matrices with entries in B. The semiring B is one of the simplest examples of a semiring, and the matrix monoids M n (B) for n ∈ N have been widely studied in the literature since the 1960s to the present day; see, for example, [5,6,8,9,10,11,19,35,38,40,51,50,53,54,55,59,63]. If α is a binary relation on the set {1, .…”

Section: Introductionmentioning

confidence: 99%

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Minimal generating sets for matrix monoids

Hivert,

Mitchell,

Smith

et al. 2020

Preprint

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In this paper, we determine minimal generating sets for several well-known monoids of matrices over certain semirings. In particular, we find minimal generating sets for the monoids consisting of: all n × n boolean matrices when n ≤ 8; the n × n boolean matrices containing the identity matrix (the reflexive boolean matrices) when n ≤ 7; the n × n boolean matrices containing a permutation (the Hall matrices) when n ≤ 8; the upper, and lower, triangular boolean matrices of every dimension; the 2 × 2 matrices over the semiring N ∪ {−∞} with addition ⊕ defined by x ⊕ y = max(x, y) and multiplication ⊗ given by x ⊗ y = x + y (the max-plus semiring); the 2 × 2 matrices over any quotient of the max-plus semiring by the congruence generated by t = t + 1 where t ∈ N; the 2 × 2 matrices over the min-plus semiring and its finite quotients by the congruences generated by t = t + 1 for all t ∈ N; and the n × n matrices over Z/nZ relative to their group of units.

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Section: Boolean Matrix Monoidsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Minimal generating sets for matrix monoids

Hivert,

Mitchell,

Smith

et al. 2020

Preprint

View full text Add to dashboard Cite

show abstract

“…Now let D(i, j) denote the n×n Boolean matrix with 1's on the diagonal and a single off-diagonal 1 in position (i, j), and let E(i, j) = D(i, j)D( j, i). The gossip monoid [5,9] is the submonoid of M n (B) generated by the set {E(i, j) : 1 ≤ i < j ≤ n}. The one directional gossip monoid G n is the submonoid of M n (B) generated by the set {D(i, j) : 1 ≤ i = j ≤ n}.…”

Section: Lemma 52mentioning

confidence: 99%

Identities in unitriangular and gossip monoids

Johnson

Fenner

2019

Semigroup Forum

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We establish a criterion for a semigroup identity to hold in the monoid of n × n upper unitriangular matrices with entries in a commutative semiring S. This criterion is combinatorial modulo the arithmetic of the multiplicative identity element of S. In the case where S is non-trivial and idempotent, the generated variety is the variety J n−1 , which by a result of Volkov is generated by any one of: the monoid of unitriangular Boolean matrices, the monoid R n of all reflexive relations on an n element set, or the Catalan monoid C n . We propose S-matrix analogues of these latter two monoids in the case where S is an idempotent semiring whose multiplicative identity element is the 'top' element with respect to the natural partial order on S, and show that each generates J n−1 . As a consequence we obtain a complete solution to the finite basis problem for Lossy gossip monoids.

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“…The gossip monoid [2,6] is the submonoid of M n (B) generated by the set {E(i, j) : 1 ≤ i < j ≤ n}. The one directional gossip monoid G n is the submonoid of M n (B) generated by the set {D(i, j) : 1 ≤ i = j ≤ n}.…”

Section: Catalan Monoids and Gossipmentioning

confidence: 99%

Identities in unitriangular and gossip monoids

Johnson¹,

Fenner²

2018

Preprint

Self Cite

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We establish a criterion for a semigroup identity to hold in the monoid of n × n upper unitriangular matrices with entries in a commutative semiring S. This criterion is combinatorial modulo the arithmetic of the multiplicative identity element of S. In the case where S is idempotent, the generated variety is the variety Jn−1, which by a result of Volkov is generated by any one of: the monoid of unitriangular Boolean matrices, the monoid Rn of all reflexive relations on an n element set, or the Catalan monoid Cn. We propose S-matrix analogues of these latter two monoids in the case where S is an idempotent semiring whose multiplicative identity element is the 'top' element with respect to the natural partial order on S, and show that each generates Jn−1. As a consequence we obtain a complete solution to the finite basis problem for lossy gossip monoids.

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NP-Completeness in the gossip monoid

Cited by 7 publications

References 12 publications

Minimal generating sets for matrix monoids

Minimal generating sets for matrix monoids

Identities in unitriangular and gossip monoids

Identities in unitriangular and gossip monoids

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