Erratum to: Semigroup identities in the monoid of triangular tropical matrices

1 Erratum to: Semigroup Forum (2014) 88:145–161 DOI 10.1007/s00233-013-9507-6

One of the conditions in the main result of the paper [1] is too strict as it stands: Theorem 4.8 should be replaced as follows.

Theorem 4.8

Any two triangular tropical matrices $X,Y \in U_n(\mathbb {T})$ having the same diagonal satisfy the (nontrivial) identities:

$$\begin{aligned} \widetilde{w}_{(C,P,n-1)} \ {X} \ \widetilde{w}_{(C,P,n-1)} = \widetilde{w}_{(C,P,n-1)} \ {Y} \ \widetilde{w}_{(C,P,n-1)}, \end{aligned}$$

where $\widetilde{w}_{(C,P,n-1)}$ is any word having as its factors all the words of length $n-1$ generated by $C = \{ X, Y\}$ of powers $P = \{ 1,2,\dots , n-1\}$, such that $\widetilde{w}_{(C,P,n-1)} \ {X} \ \widetilde{w}_{(C,P,n-1)}$ and $\widetilde{w}_{(C,P,n-1)} \ {Y} \ \widetilde{w}_{(C,P,n-1)}$ are generated by C and powers P.

The proof of the theorem appears in [2, Theorem 4.5].