  Stalactic, taiga, sylvester and Baxter monoids arise from the combinatorics
of tableaux by identifying words over a fixed ordered alphabet whenever they
produce the same tableau via some insertion algorithm. In this paper, three
sufficient conditions under which semigroups are finitely based are given. By
applying these sufficient conditions, it is shown that all stalactic and taiga
monoids of rank greater than or equal to $2$ are finitely based and satisfy the
same identities, that all sylvester monoids of rank greater than or equal to
$2$ are finitely based and satisfy the same identities and that all Baxter
monoids of rank greater than or equal to $2$ are finitely based and satisfy the
same identities.
