# The monoid of \(2 \times 2\) triangular boolean matrices under skew transposition is non-finitely based

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## Abstract

Let \(\mathscr {T\!B}_n\) be the involution semigroup of all upper triangular boolean \(n\times n\) matrices under the ordinary matrix multiplication and the skew transposition. It is shown by Auinger et al. that the involution semigroup \(\mathscr {T\!B}_n\) is non-finitely based if \(n > 2\), but the case when \(n=2\) still remains open. In this paper, we give a sufficient condition under which an involution semigroup is non-finitely based. As an application, we show that the involution semigroup \(\mathscr {T\!B}_2\) is non-finitely based. Hence \(\mathscr {T\!B}_n\) is non-finitely based for all \(n \ge 2\).

## Keywords

Semigroup Boolean matrix Identity basis Finite basis problem Variety## Notes

### Acknowledgements

The authors are very grateful to the anonymous referees for valuable comments and pertinent suggestions, especially for pointing out the gap described in Remark 16, and also to Professor Mikhail Volkov for his valuable remarks and suggestions.

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