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 Jn, 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≤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.Comment: 22 pages. Added results on the finite basis problem for the stylic
monoid with involution and updated reference