  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 $\boldsymbol{\mathcal{J}}_n$, 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 $n
\leq 3$, 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.
