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THE FINITE BASIS PROBLEM FOR INVOLUTION SEMIGROUPS OF TRIANGULAR $2\times 2$ MATRICES

Part of:SemigroupsModel theory

Published online by Cambridge University Press:  02 October 2019

WEN TING ZHANG and
YAN FENG LUO
WEN TING ZHANG*
Affiliation:
School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, PR ChinaKey Laboratory of Applied Mathematics and Complex Systems, Lanzhou, Gansu 730000, PR China email zhangwt@lzu.edu.cn
YAN FENG LUO
Affiliation:
School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, PR ChinaKey Laboratory of Applied Mathematics and Complex Systems, Lanzhou, Gansu 730000, PR China email luoyf@lzu.edu.cn
*
✉zhangwt@lzu.edu.cn
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Abstract

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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 $\mathbb{F}$.

Keywords

semigroupmatrixidentity basisfinite basis problemvariety

MSC classification

Primary:20M05: Free semigroups, generators and relations, word problems
Secondary:03C05: Equational classes, universal algebra

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 101 , Issue 1 , February 2020, pp. 88 - 104
Copyright
© 2019 Australian Mathematical Publishing Association Inc. 

Footnotes

This research was partially supported by the National Natural Science Foundation of China (Nos. 11401275, 11771191 and 11371177) and the Fundamental Research Funds for the Central Universities (No. lzujbky-2016-96).

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