Equational theories of upper triangular tropical matrix semigroups

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 $n=2$.