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ON THE VARIETY GENERATED BY THE MONOID OF TRIANGULAR 2×2 MATRICES OVER A TWO-ELEMENT FIELD

Part of: Semigroups

Published online by Cambridge University Press:  28 May 2012

WEN TING ZHANG ,
JIAN RONG LI  and
YAN FENG LUO
WEN TING ZHANG
Affiliation:
Department of Mathematics, Lanzhou University, Lanzhou, Gansu 730000, PR China
JIAN RONG LI
Affiliation:
Department of Mathematics, Lanzhou University, Lanzhou, Gansu 730000, PR China
YAN FENG LUO*
Affiliation:
Department of Mathematics, Lanzhou University, Lanzhou, Gansu 730000, PR China (email: luoyf@lzu.edu.cn)
*
✉For correspondence; e-mail: luoyf@lzu.edu.cn
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Abstract

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Let 𝒯n(F) denote the monoid of all upper triangular n×n matrices over a finite field F. It has been shown by Volkov and Goldberg that 𝒯n(F) is nonfinitely based if ∣F∣>2 and n≥4, but the cases when ∣F∣>2 and n=2,3 or when ∣F∣=2 have remained open. In this paper, it is shown that the monoid 𝒯2 (F) is finitely based when ∣F∣=2 , and a finite identity basis for it is given. Moreover, all maximal subvarieties of the variety generated by 𝒯2 (F) with ∣F∣=2 are determined.

Keywords

finite basis problemsemigroup of triangular matricesfinite fieldsemigroup varietysubvarieties

MSC classification

Secondary: 20M07: Varieties and pseudovarieties of semigroups
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 86 , Issue 1 , August 2012 , pp. 64 - 77
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

Footnotes

This research was partially supported by the National Natural Science Foundation of China (No. 10971086), the Mathematical Tianyuan Foundation of China (No. 11126186), the Natural Science Foundation of Gansu Province (No. 1107RJZA218), and the Fundamental Research Funds for the Central Universities (No. lzujbky-2012-12).

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