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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 $$n=2$$ .

Since the 1970s, the smallest known non-finitely based involution semigroups are of order six. This paper exhibits the first example of a non-finitely based involution semigroup of order five.

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 Auinger et al. Therefore involution monoid $$(BU_n, \,^*\,)$$ is finitely based if and only if $$n = 2$$.

Let n be a positive integer and $${\mathbf{S}_{n}}$$ the variety generated by all semigroups of order n. It was shown by Volkov that $${\mathbf{S}_{n}}$$ is non-finitely based for each $${n \geq 5}$$ . It was shown by Luo and Zhang that $${\mathbf{S}_{3}}$$ is finitely based. It is easy to show that the varieties $${\mathbf{S}_{1}, \mathbf{S}_{2}}$$ are finitely based. However, the finite basis problem for $${\mathbf{S}_{4}}$$ had been open for more than twenty years. In this paper, we solve this problem by showing that the variety $${\mathbf{S}_{4}}$$ is finitely based. Moreover, we give an explicit finite basis for $${\mathbf{S}_{4}}$$ .