Identities in unitriangular and gossip monoids

Johnson, Marianne; Fenner, Peter

doi:10.1007/s00233-019-09996-x

Cited by 12 publications

(27 citation statements)

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“…Hence it suffices to show that each of (i)-(iv) implies (v). It is clear from the definitions that (iii) implies (iv), which by [23,Lemma 4.1] implies (v). Since (ii) implies condition (i), it remains to show that (i) implies (v).…”

Section: 2mentioning

confidence: 91%

“…6. If E is tight in only 1 → 2 → 4, 1 → 3 → 4: In this case (23) is satisfied, and for (24) to hold requires α 2,3 = 1. But then, since E is tight in 1 → 2 → 4, we see that (22) also holds, giving 7.…”

Section: Suppose Now Thatmentioning

confidence: 99%

“…We show that the idempotent generated subsemigroup of U T n (S * ) is precisely U n (S * ) (Corollary 4. 23).…”

Section: Introductionmentioning

confidence: 99%

“…Def E (1 → 2 → 3)α 1,3 and either: E is tight in the path 1 → 2 → 4; or α 1,2 = 1; or α 2,3 = Def E (1 → 2 → 3)α 1,3 . Equation(23) holds if and only if either E is tight in the path 1 → 3 → 4 or α 1,3 = 1; whilst(24) holds if and only if either E is tight in the path 2 → 3 → 4 or α 2,3 = 1.Similarly, if A L E, then by (16) we may writeA • = [β 2,3 ] 2,3 • [β 2,4 ] 2,4 • [β 3,4 ] 3,4 • E, where β 2,3 , β 2,4 , β 3,4 ≥ 1. Since A L E if and only if (A • ) ( * ) = E, the L class of E can be note that 1 < Def E (2 → 3 → 4)β 2,4.…”

mentioning

confidence: 99%

“…5. If E is tight only in 1 → 2 → 3, 2 → 3 → 4:In this case (24) is satisfied, and for(23) to hold requires α 1,3 = 1. Since E is tight in 1 → 2 → 3, we find that the only way that (22) can hold is if α 2,3 = 1.…”

mentioning

confidence: 99%

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Matrix semigroups over semirings

Gould

Johnson

Naz

2019

Int. J. Algebra Comput.

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The multiplicative semigroup Mn(F ) of n × n matrices over a field F is well understood, in particular, it is a regular semigroup. This paper considers semigroups of the form Mn(S), where S is a semiring, and the subsemigroups U Tn(S) and Un(S) of Mn(S) consisting of upper triangular and unitriangular matrices. Our main interest is in the case where S is an idempotent semifield, where we also consider the subsemigroups U Tn(S * ) and Un(S * ) consisting of those matrices of U Tn(S) and Un(S) having all elements on and above the leading diagonal non-zero. Our guiding examples of such S are the 2-element Boolean semiring B and the tropical semiring T. In the first case, Mn(B) is isomorphic to the semigroup of binary relations on an n-element set, and in the second, Mn(T) is the semigroup of n × n tropical matrices.It is well known that the subsemigroup of Mn(F ) consisting of the singular matrices is idempotent generated. We begin consideration of the analogous questions for Mn(S) and its subsemigroups. In particular we show that the idempotent generated subsemigroup of U Tn(T * ) is precisely the semigroup Un(T * ).Il'in has proved that for any semiring R and n > 2, the semigroup Mn(R) is regular if and only if R is a regular ring. We therefore base our investigations for Mn(S) and its subsemigroups on the analogous but weaker concept of being Fountain (formerly, weakly abundant). These notions are determined by the existence and behaviour of idempotent left and right identities for elements, lying in particular equivalence classes. We show that certain subsemigroups of Mn(S), including several generalisations of well-studied monoids of binary relations (Hall relations, reflexive relations, unitriangular Boolean matrices), are Fountain. We give a detailed study of a family of Fountain semigroups arising in this way that has particularly interesting and unusual properties.

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Section: 2mentioning

confidence: 91%

Section: Suppose Now Thatmentioning

confidence: 99%

“…We show that the idempotent generated subsemigroup of U T n (S * ) is precisely U n (S * ) (Corollary 4. 23).…”

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Matrix semigroups over semirings

Gould

Johnson

Naz

2019

Int. J. Algebra Comput.

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Equational theories of upper triangular tropical matrix semigroups

2021

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Identities of the stylic monoid

Volkov

2022

Semigroup Forum

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We observe that for each n ≥ 2, the identities of the stylic monoid with n generators coincide with the identities of n-generated monoids from other distinguished series of J -trivial monoids studied in the literature, e.g., Catalan monoids and Kiselman monoids. This solves the Finite Basis Problem for stylic monoids.A monoid identity is a pair of words, i.e., elements of the free monoid X * over an alphabet X, written as a formal equality. An identity w = w ′ with w, w ′ ∈ X * is said to hold in a monoid M if wϕ = w ′ ϕ for each homomorphism ϕ : X * → M; alternatively, we say that the monoid satisfies w = w ′ . Clearly, if M satisfies w = w ′ , then so does every homomorphic image of M.Given any set Σ of monoid identities, we say that an identity w = w ′ follows from Σ if every monoid satisfying all identities of Σ satisfies the identity w = w ′ as well. A subset ∆ ⊆ Σ is called a basis for Σ if each identity in Σ follows from ∆. The Finite Basis Problem for a monoid M is the question of whether or not the set of all identities that hold in M admits a finite basis.A monoid M is said to be J -trivial if every principal ideal of M has a unique generator, that is, MaM = MbM implies a = b for all a, b ∈ M. Finite J -trivial monoids attract much attention because of their distinguished role in algebraic theory of regular languages [14,15] and representation theory [7]. Several series of finite J -trivial monoids parameterized by positive integers appear in the literature, including Straubing monoids [17,18], Catalan monoids [16], double Catalan monoids [13], Kiselman monoids [9,11,12], and gossip monoids [5,8,10]. These monoids arise due to completely unrelated reasons and consist of elements of a very different nature. Surprisingly, studying identities of the listed monoids has revealed that the n-th monoids in each series satisfy exactly the same identities! Recently, a new family of finite J -trivial monoids, coined stylic monoids, have been introduced by Abram and Reutenauer [1], with motivation coming from combinatorics of Young tableaux. It is quite natural to ask whether the above phenomenon extends to this new family, i.e., whether the n-th stylic monoid again satisfies the same identities as do the n-th monoids in each aforementioned series. The present note aims to answers this question in the affirmative.The combinatorial definition of stylic monoids can be found in [1], but here we only need their presentation via generators and relations established in [1, Theorem 8.1(ii)]. Thus, let the stylic monoid Styl n be the monoid generated by a 1 , a 2 , . . . , a n subject to the

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Identities in unitriangular and gossip monoids

Cited by 12 publications

References 18 publications

Matrix semigroups over semirings

Matrix semigroups over semirings

Equational theories of upper triangular tropical matrix semigroups

Identities of the stylic monoid

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