scholarly journals Weak polynomial identities and their applications

2021 ◽  
Vol 29 (2) ◽  
pp. 291-324
Author(s):  
Vesselin Drensky
Keyword(s):  
Associative Algebra ◽  
Finite Basis ◽  
Vector Subspace ◽  
Free Associative Algebra ◽  
Polynomial Identities ◽  
Finite Basis Problem ◽  
Basis Problem ◽  
Nonassociative Algebras ◽  
Survey Results ◽  
Algebra Over A Field

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.

On the finite basis problem for certain 2-limited words

2012 ◽  
Vol 29 (3) ◽  
pp. 571-590 ◽  
Author(s):  
Jian Rong Li ◽  
Wen Ting Zhang ◽  
Yan Feng Luo
Keyword(s):  
Finite Basis ◽  
Finite Basis Problem ◽  
Basis Problem

On the finite basis problem for the monoids of partial extensive injective transformations

2015 ◽  
Vol 91 (2) ◽  
pp. 524-537
Author(s):  
Xun Hu ◽  
Yuzhu Chen ◽  
Yanfeng Luo
Keyword(s):  
Finite Basis ◽  
Finite Basis Problem ◽  
Basis Problem

Varieties generated by 2-testable monoids

2012 ◽  
Vol 49 (3) ◽  
pp. 366-389 ◽  
Author(s):  
Edmond Lee
Keyword(s):  
Present Article ◽  
Chain Condition ◽  
Finite Basis ◽  
Finitely Based ◽  
Finitely Generated ◽  
Finite Basis Problem ◽  
Basis Problem ◽  
Descending Chain Condition ◽  
Ascending Chain Condition

The smallest monoid containing a 2-testable semigroup is defined to be a 2-testable monoid. The well-known Brandt monoid B21 of order six is an example of a 2-testable monoid. The finite basis problem for 2-testable monoids was recently addressed and solved. The present article continues with the investigation by describing all monoid varieties generated by 2-testable monoids. It is shown that there are 28 such varieties, all of which are finitely generated and precisely 19 of which are finitely based. As a comparison, the sub-variety lattice of the monoid variety generated by the monoid B21 is examined. This lattice has infinite width, satisfies neither the ascending chain condition nor the descending chain condition, and contains non-finitely generated varieties.

scholarly journals The finite basis problem for Kauffman monoids

2015 ◽  
Vol 74 (3-4) ◽  
pp. 333-350 ◽  
Author(s):  
K. Auinger ◽  
Yuzhu Chen ◽  
Xun Hu ◽  
Yanfeng Luo ◽  
M. V. Volkov
Keyword(s):  
Finite Basis ◽  
Finite Basis Problem ◽  
Basis Problem

REFLEXIVE RELATIONS, EXTENSIVE TRANSFORMATIONS AND PIECEWISE TESTABLE LANGUAGES OF A GIVEN HEIGHT

2004 ◽  
Vol 14 (05n06) ◽  
pp. 817-827 ◽  
Author(s):  
M. V. VOLKOV
Keyword(s):  
Complete Solution ◽  
Finite Basis ◽  
Finite Basis Problem ◽  
Basis Problem ◽  
Transformation Monoid ◽  
Single Relation

Straubing deduced from Simon's theorem that monoids of reflexive relations as well as monoids of order preserving extensive transformations generate the pseudovariety of [Formula: see text]-trivial monoids. We refine these results by showing that each pseudovariety in Simon's hierarchy of [Formula: see text]-trivial monoids is generated by a single relation/transformation monoid. From this and from some results by Blanchet–Sadri we obtain a complete solution of the finite basis problem for Straubing's monoids.

scholarly journals THE FINITE BASIS PROBLEM FOR THE MONOID OF TWO-BY-TWO UPPER TRIANGULAR TROPICAL MATRICES

2016 ◽  
Vol 94 (1) ◽  
pp. 54-64 ◽  
Author(s):  
YUZHU CHEN ◽  
XUN HU ◽  
YANFENG LUO ◽  
OLGA SAPIR
Keyword(s):  
Finite Basis ◽  
Finite Basis Problem ◽  
Basis Problem ◽  
Tropical Semiring ◽  
Bicyclic Monoid ◽  
Nonfinitely Based

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.

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