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We study the equational theories and bases of meets and joins of severalvarieties of plactic-like monoids. Using those results, we constructsublattices of the lattice of varieties of monoids, generated by saidvarieties. We calculate the axiomatic ranks of their elements, obtainplactic-like congruences whose corresponding factor monoids generate varietiesin the lattice, and determine which varieties are joins of the variety ofcommutative monoids and a finitely generated variety. We also show that thehyposylvester and metasylvester monoids generate the same variety as thesylvester monoid.

Lattices of varieties of plactic-like monoids