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 Kn are nonfinitely based for each n ≥ 3. This result holds also for the case when Kn is considered as an involution semigroup under either of its natural involutions.(M. V. Volkov

For each positive n, let un ≈ vn denote the identity obtained from the Adjan identity (xy)(yx)(xy)(xy)(yx) ≈ (xy)(yx)(yx)(xy)(yx) by substituting (xy) → (x1x2 . . . xn) and (yx) → (xn . . . x2x1). We show that every monoid which satisfies un ≈ vn for each positive n and generates a variety containing the bicyclic monoid is nonfinitely based.This implies that the monoid U2(T) (resp., U2(Z)) of 2 × 2 upper triangular tropical matrices over the tropical semiring T = R∪{−∞} (resp., Z = Z∪{−∞}) is nonfinitely based.2010 Mathematics subject classification: 20M07, 03C05

We give a transparent combinatorial characterization of the identities satisfied by the Kauffman monoid K3. Our characterization leads to a polynomial time algorithm to check whether a given identity holds in K3.