In an earlier paper, the second-named author has described the identities holding in the so-called Catalan monoids. Here we extend this description to a certain family of Hecke--Kiselman monoids including the Kiselman monoids $mathcal{K}_n$. As a con
sequence, we conclude that the identities of $mathcal{K}_n$ are nonfinitely based for every $nge 4$ and exhibit a finite identity basis for the identities of each of the monoids $mathcal{K}_2$ and $mathcal{K}_3$. In the third version a question left open in the initial submission has beed answered.
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.
We exhibit a 6-element semigroup that has no finite identity basis but nevertheless generates a variety whose finite membership problem admits a polynomial algorithm.
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.
