Equational theories of upper triangular tropical matrix semigroups

Han, Bin; Zhang, Wen Ting; Luo, Yan

doi:10.1007/s00012-021-00738-1

Cited by 7 publications

(3 citation statements)

References 35 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The plactic monoid of rank 3 satisfies exactly the same identities as the monoid of all 3 × 3 upper triangular tropical matrices [13,Corollary 4.5]. Thus the plactic monoid of rank 3 is non-finitely based by the result of Han et al [11]. The finite basis problems for the plactic monoids of rank greater than or equal to 4 are still open.…”

Section: Introductionmentioning

confidence: 91%

Finite basis problems for stalactic, taiga, sylvester and Baxter monoids

Han¹,

Zhang²

2021

Preprint

Self Cite

View full text Add to dashboard Cite

Stalactic, taiga, sylvester and Baxter monoids arise from the combinatorics of tableaux by identifying words over a fixed ordered alphabet whenever they produce the same tableau via some insertion algorithm. In this paper, three sufficient conditions under which semigroups are finitely based are given. By applying these sufficient conditions, it is shown that all stalactic and taiga monoids of rank greater than or equal to 2 are finitely based and satisfy the same identities, that all sylvester monoids of rank greater than or equal to 2 are finitely based and satisfy the same identities and that all Baxter monoids of rank greater than or equal to 2 are finitely based and satisfy the same identities.

show abstract

Section: Introductionmentioning

confidence: 91%

Finite basis problems for stalactic, taiga, sylvester and Baxter monoids

Han¹,

Zhang²

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…In [30,Section 5], it was shown that the involution monoid (U n+1 (T), ) is nonfinitely based, for n ≥ 3. It was also shown that (U 3 (T), ) satisfies, for each k ∈ N, the identity…”

Section: Proof It Suffices To Show Thatmentioning

confidence: 99%

Tropical representations and identities of the stylic monoid

Aird

Ribeiro

2022

Semigroup Forum

View full text Add to dashboard Cite

We exhibit a faithful representation of the stylic monoid of every finite rank as a monoid of upper unitriangular matrices over the tropical semiring. Thus, we show that the stylic monoid of finite rank n generates the pseudovariety $$\varvec{{\mathcal {J}}}_n$$ J n , which corresponds to the class of all piecewise testable languages of height n, in the framework of Eilenberg’s correspondence. From this, we obtain the equational theory of the stylic monoids of finite rank, show that they are finitely based if and only if $$n \le 3$$ n ≤ 3 , and that their identity checking problem is decidable in linearithmic time. We also establish connections between the stylic monoids and other plactic-like monoids, and solve the finite basis problem for the stylic monoid with involution.

show abstract

“…An important consequence of these representations, which are specifically representations using upper triangular tropical matrices, is that each finite-rank plactic monoid satisfies a non-trivial semigroup identity [27,Theorem 3.1]. In particular, the plactic monoids of rank equal to 2 and 3 are non-finitely based by [9,16] and [24,27] respectively, and their involution cases are also non-finitely based by using the result of [37,Theorem 4]. The finite basis problem for the plactic monoids of rank greater than 3 and their involution cases are still open.…”

Section: Introductionmentioning

confidence: 99%

Representations and identities of Baxter monoids with involution

Han¹,

Zhang²,

Luo³

et al. 2023

Preprint

View full text Add to dashboard Cite

Let (baxtn, ♯ ) be the Baxter monoid of finite rank n with Schützenberger's involution ♯ . In this paper, it is shown that (baxtn, ♯ ) admits a faithful representation by an involution monoid of upper triangular matrices over any semiring from a large class including the tropical semiring under the skew transposition. Then a transparent combinatorial characterization of the word identities satisfied by (baxtn, ♯ ) is given. Further, it is proved that (baxtn, ♯ ) is finitely based if and only if n = 3, and shown that the identity checking problem for (baxtn, ♯ ) can be done in polynomial time.

show abstract

Equational theories of upper triangular tropical matrix semigroups

Cited by 7 publications

References 35 publications

Finite basis problems for stalactic, taiga, sylvester and Baxter monoids

Finite basis problems for stalactic, taiga, sylvester and Baxter monoids

Tropical representations and identities of the stylic monoid

Representations and identities of Baxter monoids with involution

Contact Info

Product

Resources

About