We prove a sufficient condition under which a semigroup admits no finite identity basis. As an application, it is shown that the identities of the Kauffman monoid $mathcal{K}_n$ are nonfinitely based for each $nge 3$. This result holds also for the c
ase when $mathcal{K}_n$ is considered as an involution semigroup under either of its natural involutions.
We exhibit a simple condition under which a finite involutary semigroup whose semigroup reduct is inherently nonfinitely based is also inherently nonfinitely based as a unary semigroup. As applications, we get already known as well as new examples of
inherently nonfinitely based involutory semigroups. We also show that for finite regular semigroups, our condition is not only sufficient but also necessary for the property of being inherently nonfinitely based to persist. This leads to an algorithmic description of regular inherently nonfinitely based involutory semigroups.
The Krohn--Rhodes complexity of the Brauer semigroup $mathfrak{B}_n$ and of the annular semigroup $mathfrak{A}_n$ is computed.
We study matrix identities involving multiplication and unary operations such as transposition or Moore-Penrose inversion. We prove that in many cases such identities admit no finite basis.
