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 \(n=2\).

令\(\mathbb {S}\)是具有加法恒等式\(\mathbf {0}\)和乘法恒等式\(\mathbf {1}\)的交换和幂等半环。热带半环\(\mathbb {T}\)和布尔半环\(\mathbb {B}\)是此类半环的常见重要示例。让\（UT_ {N}（\ mathbb {S}）\）是所有的半群\（N \ n次\）上三角矩阵超过\（\ mathbb {S} \） ，两者\（UT ^ {\ pm }_n(\mathbb {S})\)和\(UT^{+}_n(\mathbb {S})\)是\(UT_n(\mathbb {S})\)与\(\mathbf {0}\)和/或\(\mathbf {1}\)在主对角线上，\(\mathbf {1}\)在主对角线上。众所周知，\(UT_{2}(\mathbb {T})\)是非有限基的，而\(UT^{\pm }_{2}(\mathbb {S})\)是有限基的。结合这些结果，\(UT_{n}(\mathbb {T})\)和\(UT^{\pm }_{n}(\mathbb {S})\)的有限基问题与\(n =2, 3\)作为偏转转位下的半群和对合半群都求解。众所周知，半群\(UT^{+}_n(\mathbb {S})\)和\(UT^{+}_n(\mathbb {B})\)是等价的。在本文中，我们证明了对合半群\(UT^{+}_n(\mathbb {S})\)和\(UT^{+}_n(\mathbb {B})\)在偏斜转置下不是等价的。然而，对合半群\(UT_n^{+}(\mathbb {S})\)和\(UT_n^{+}(\mathbb {B})\)的有限基问题共享相同的解决方案，即，对合半群\(UT_n^{+}(\mathbb {S})\)是有限基的当且仅当\(n=2\)。