The finite basis problem for infinite involution semigroups of triangular 2  $$\times $$ ×  2 matrices

Zhang, Wen Ting; Ji, Ying; Luo, Yan

doi:10.1007/s00233-016-9832-7

Cited by 7 publications

(5 citation statements)

References 21 publications

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“…We establish two sufficient conditions under which an involution monoid is nonfinitely based. By applying one of the sufficient conditions to the involution semigroups (T 2 (F), * ), (UT 2 (F), * ) and (UT ±1 2 (F), * ), we show that (T 2 (F), * ) and (UT 2 (F), * ) are nonfinitely based when char(F) 0, and (UT ±1 2 (F), * ) is nonfinitely based for any field F. Together with the results of [24], this shows that (T 2 (F), * ), (UT 2 (F), * ) and (UT ±1 2 (F), * ) are nonfinitely based for any field F.…”

Section: Introductionsupporting

confidence: 80%

“…Note that (UT 2 (F), * ) is an involution submonoid of (T 2 (F), * ). If char(F) = p, then the result follows from Lemma 4.4 and Theorem 4.9; if char(F) = 0, then the result follows from [24,Corollary 14]. Therefore, Theorem 4.13 is still true if we substitute the field F by an arbitrary associative ring R with a unit element 1 satisfying either ( †) or ( ‡).…”

Section: Nonfinitely Based Involution Semigroups Of Triangular Matricesmentioning

confidence: 95%

“…Further, they proved that T n (F) with char(F) = 2 is hereditarily finitely based (that is, it generates a variety all semigroups of which are finitely based) if and only if n ≤ 2 in [26]. Chen et al [7] have shown that UT 2 (F) is finitely based for any field F. Zhang et al [24] have shown that the involution semigroups (T 2 (F), * ) and (UT 2 (F), * ) with char(F) = 0 are nonfinitely based. In [22], Volkov proved that UT 3 (F) as well as (UT 3 (F), * ) with char(F) = 0 are nonfinitely based.…”

Section: Introductionmentioning

confidence: 99%

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The Finite Basis Problem for Involution Semigroups of Triangular Matrices

Zhang

Luo

2019

Bull. Aust. Math. Soc.

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Let $T_{n}(\mathbb{F})$ be the semigroup of all upper triangular $n\times n$ matrices over a field $\mathbb{F}$. Let $UT_{n}(\mathbb{F})$ and $UT_{n}^{\pm 1}(\mathbb{F})$ be subsemigroups of $T_{n}(\mathbb{F})$, respectively, having $0$s and/or $1$s on the main diagonal and $0$s and/or $\pm 1$s on the main diagonal. We give some sufficient conditions under which an involution semigroup is nonfinitely based. As an application, we show that $UT_{2}(\mathbb{F}),UT_{2}^{\pm 1}(\mathbb{F})$ and $T_{2}(\mathbb{F})$ as involution semigroups under the skew transposition are nonfinitely based for any field $\mathbb{F}$.

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Section: Introductionsupporting

confidence: 80%

Section: Nonfinitely Based Involution Semigroups Of Triangular Matricesmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

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The Finite Basis Problem for Involution Semigroups of Triangular Matrices

Zhang

Luo

2019

Bull. Aust. Math. Soc.

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“…For example, if S is a commutative semiring into which the semiring of natural numbers can be embedded, then for n > 1 the monoid of all n × n (upper triangular) matrices over S satisfies no non-trivial identities, since the free monoid of rank 2 embeds into all such semigroups (see [19] for example). The finite basis problem is increasingly studied for families of infinite semigroups of combinatorial interest for which identities are known to exist, with complete results available for one-relator semigroups [14] and Kauffman monoids [2], and several recent partial results for various semigroups of upper triangular matrices with restrictions on the size of the matrices and the entries permitted on the diagonals [6,7,19,20].…”

Section: Introductionmentioning

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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“…For example, if S is a commutative semiring into which the semiring of natural numbers can be embedded, then for n > 1 the monoid of all n×n (upper triangular) matrices over S satisfies no non-trivial identities, since the free monoid of rank 2 embeds into all such semigroups (see [15] for example). The finite basis problem is increasingly studied for families of infinite semigroups of combinatorial interest for which identities are known to exist, with complete results available for one-relator semigroups [10] and Kauffman monoids [1], and several recent partial results for various semigroups of upper triangular matrices with restrictions on the size of the matrices and the entries permitted on the diagonals [3,4,15,16].…”

Section: Introductionmentioning

confidence: 99%

Identities in unitriangular and gossip monoids

Johnson¹,

Fenner²

2018

Preprint

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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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The finite basis problem for infinite involution semigroups of triangular 2 $$\times $$ × 2 matrices

Cited by 7 publications

References 21 publications

The Finite Basis Problem for Involution Semigroups of Triangular Matrices

The Finite Basis Problem for Involution Semigroups of Triangular Matrices

Identities in unitriangular and gossip monoids

Identities in unitriangular and gossip monoids

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