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Equational theories of upper triangular tropical matrix semigroups


Let \(\mathbb {S}\) be the commutative and idempotent semiring with additive identity \(\mathbf {0}\) and multiplicative identity \(\mathbf {1}\). The tropical semiring \(\mathbb {T}\) and the Boolean semiring \(\mathbb {B}\) are common important examples of such semirings. Let \(UT_{n}(\mathbb {S})\) be the semigroup of all \(n\times n\) upper triangular matrices over \(\mathbb {S}\), both \(UT^{\pm }_n(\mathbb {S})\) and \(UT^{+}_n(\mathbb {S})\) be subsemigroups of \(UT_n(\mathbb {S})\) with \(\mathbf {0}\) and/or \(\mathbf {1}\) on the main diagonal, and \(\mathbf {1}\) on the main diagonal respectively. It is known that \(UT_{2}(\mathbb {T})\) is non-finitely based and \(UT^{\pm }_{2}(\mathbb {S})\) is finitely based. Combining these results, the finite basis problems for \(UT_{n}(\mathbb {T})\) and \(UT^{\pm }_{n}(\mathbb {S})\) with \(n=2, 3\) both as semigroups and involution semigroups under the skew transposition are solved. It is well known that the semigroups \(UT^{+}_n(\mathbb {S})\) and \(UT^{+}_n(\mathbb {B})\) are equationally equivalent. In this paper, we show that the involution semigroups \(UT^{+}_n(\mathbb {S})\) and \(UT^{+}_n(\mathbb {B})\) under the skew transposition are not equationally equivalent. Nevertheless, the finite basis problems for involution semigroups \(UT_n^{+}(\mathbb {S})\) and \(UT_n^{+}(\mathbb {B})\) share the same solution, that is, the involution semigroup \(UT_n^{+}(\mathbb {S})\) is finitely based if and only if \(n=2\).

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The authors are very grateful to the anonymous referee whose meticulous reading and insightful suggestions led to improvements in readability and simplification and generalization of some arguments in the paper.

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Correspondence to Wen Ting Zhang.

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This research was partially supported by the National Natural Science Foundation of China (Nos. 11771191, 11401275, 11371177) and the Natural Science Foundation of Gansu Province (No. 20JR5RA275)

Presented by E. W. H. Lee.

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Han, B.B., Zhang, W.T. & Luo, Y.F. Equational theories of upper triangular tropical matrix semigroups. Algebra Univers. 82, 44 (2021).

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Mathematics Subject Classification

  • 20M07


  • Tropical semiring
  • Matrix
  • Identity
  • Involution
  • Finite basis problem