ON THE VARIETY GENERATED BY THE MONOID OF TRIANGULAR $2\times 2$ MATRICES OVER TWO-ELEMENT FIELD

Authors

  • Yan Feng Luo

Keywords:

finite basis problem, semigroup of triangular matrices, finite field, semigroup variety, subvarieties

Abstract

Let $\mathscr{T}_n(F)$ denote the submonoid of all upper triangular $n\times n$ matrices over a finite field $F$. It is shown by Volkov and Goldberg that $\mathscr{T}_n(F)$ is nonfinitely based if $|F|>2$ and $n\geq 4$, but the cases when $|F|>2$ and $n=2,3$ or when $|F|=2$ remained open. In this paper, it is shown that the monoid $\mathscr{T}_2(F)$ is finitely based when $|F|=2$, and a finite identity basis for it is given. Moreover, all maximal subvarieties of the variety generated by $\mathscr{T}_2(F)$ with $|F|=2$ are determined. DOI: 10.1017/S0004972711002905

Published

2012-07-01

Issue

Vol. 86 No. 1 (2012)

Section

Articles
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