Abstract
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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Presented by E. W. H. Lee.
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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)
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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). https://doi.org/10.1007/s00012-021-00738-1
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DOI: https://doi.org/10.1007/s00012-021-00738-1