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Finite basis problem for involution monoids of unitriangular Boolean matrices. (English) Zbl 1477.20104

Summary: Let \((BU_n, \,^*\,)\) be the involution monoid of all Boolean upper triangular \(n\times n\) matrices with 1s on the main diagonal under the ordinary matrix multiplication and the skew transposition. The involution monoid \((BU_2, \,^*\,)\) is easily seen to be finitely based. In this paper, we shown that \((BU_n, \,^*\,)\) is non-finitely based for each \(n \ge 3\), which answers an open question posed by K. Auinger et al. [J. Eur. Math. Soc. (JEMS) 14, No. 3, 937–969 (2012; Zbl 1261.20068)]. Therefore involution monoid \((BU_n, \,^*\,)\) is finitely based if and only if \(n = 2\).

MSC:

20M07 Varieties and pseudovarieties of semigroups
20M05 Free semigroups, generators and relations, word problems
08B05 Equational logic, Mal’tsev conditions
20M20 Semigroups of transformations, relations, partitions, etc.

Citations:

Zbl 1261.20068
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References:

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