No abstract

No abstract

We exhibit a faithful representation of the stylic monoid of every finite rank as a monoid of upper unitriangular matrices over the tropical semiring. Thus, we show that the stylic monoid of finite rank n generates the pseudovariety $$\varvec{{\mathcal {J}}}_n$$ J n , which corresponds to the class of all piecewise testable languages of height n, in the framework of Eilenberg’s correspondence. From this, we obtain the equational theory of the stylic monoids of finite rank, show that they are finitely based if and only if $$n \le 3$$ n ≤ 3 , and that their identity checking problem is decidable in linearithmic time. We also establish connections between the stylic monoids and other plactic-like monoids, and solve the finite basis problem for the stylic monoid with involution.

Algorithm 179 (MDBETA) can be improved as shown.

We study semigroup varieties generated by full and upper triangular tropical matrix semigroups and the plactic monoid of rank 4. We prove that the upper triangular tropical matrix semigroup [Formula: see text] generates a different semigroup variety for each dimension [Formula: see text]. We show a weaker version of this fact for the full matrix semigroup: full tropical matrix semigroups of different prime dimensions generate different semigroup varieties. For the plactic monoid of rank 4, [Formula: see text], we find a new set of identities satisfied by [Formula: see text] shorter than those previously known, and show that the semigroup variety generated by [Formula: see text] is strictly contained in the variety generated by [Formula: see text].