finite basis problem
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2021 ◽  
Vol 29 (2) ◽  
pp. 291-324
Author(s):  
Vesselin Drensky

Abstract Let R be an associative algebra over a field K generated by a vector subspace V. The polynomial f(x 1, . . . , xn ) of the free associative algebra K〈x 1, x 2, . . .〉 is a weak polynomial identity for the pair (R, V) if it vanishes in R when evaluated on V. We survey results on weak polynomial identities and on their applications to polynomial identities and central polynomials of associative and close to them nonassociative algebras and on the finite basis problem. We also present results on weak polynomial identities of degree three.



2021 ◽  
pp. 1-23
Author(s):  
Meng Gao ◽  
Wen-Ting Zhang ◽  
Yan-Feng Luo


2020 ◽  
Vol 6 (2) ◽  
pp. 117
Author(s):  
Xianzhong Zhao ◽  
Miaomiao Ren ◽  
Siniša Crvenković ◽  
Yong Shao ◽  
Petar Dapić

Up to isomorphism, there are 61 ai-semirings of order three. The finite basis problem for these semirings is investigated. This problem for 45 semirings of them is answered by some results in the literature. The remaining semirings are studied using equational logic. It is shown that with the possible exception of the semiring \(S_7\), all ai-semirings of order three are finitely based.



2020 ◽  
Vol 81 (1) ◽  
Author(s):  
Wen Ting Zhang ◽  
Yan Feng Luo ◽  
Nan Wang


2019 ◽  
Vol 101 (1) ◽  
pp. 88-104 ◽  
Author(s):  
WEN TING ZHANG ◽  
YAN FENG LUO

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 $\mathbb{F}$.



2016 ◽  
Vol 94 (2) ◽  
pp. 426-441 ◽  
Author(s):  
Wen Ting Zhang ◽  
Ying Dan Ji ◽  
Yan Feng Luo




2016 ◽  
Vol 94 (1) ◽  
pp. 54-64 ◽  
Author(s):  
YUZHU CHEN ◽  
XUN HU ◽  
YANFENG LUO ◽  
OLGA SAPIR

For each positive $n$, let $\mathbf{u}_{n}\approx \boldsymbol{v}_{n}$ denote the identity obtained from the Adjan identity $(xy)(yx)(xy)(xy)(yx)\approx (xy)(yx)(yx)(xy)(yx)$ by substituting $(xy)\rightarrow (x_{1}x_{2}\ldots x_{n})$ and $(yx)\rightarrow (x_{n}\ldots x_{2}x_{1})$. We show that every monoid which satisfies $\mathbf{u}_{n}\approx \boldsymbol{v}_{n}$ for each positive $n$ and generates a variety containing the bicyclic monoid is nonfinitely based. This implies that the monoid $U_{2}(\mathbb{T})$ (respectively, $U_{2}(\overline{\mathbb{Z}})$) of two-by-two upper triangular tropical matrices over the tropical semiring $\mathbb{T}=\mathbb{R}\cup \{-\infty \}$ (respectively, $\overline{\mathbb{Z}}=\mathbb{Z}\cup \{-\infty \}$) is nonfinitely based.



2015 ◽  
Vol 48 (4) ◽  
Author(s):  
D. N. Ashikhmin ◽  
M. V. Volkov ◽  
Wen Ting Zhang

AbstractIn an earlier paper, the second-named author has described the identities holding in the so-called Catalan monoids. Here we extend this description to a certain family of Hecke-Kiselman monoids including the Kiselman monoids K



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