The plactic monoids can be obtained from the tensor product of crystals.
Similarly, the hypoplactic monoids can be obtained from the quasi-tensor
product of quasi-crystals. In this paper, we present a unified approach to
these constructions by expressing them in the context of quasi-crystals. We
provide a sufficient condition to obtain a quasi-crystal monoid for the
quasi-tensor product from a quasi-crystal monoid for the tensor product. We
also establish a sufficient condition for a hypoplactic monoid to be a quotient
of the plactic monoid associated to the same seminormal quasi-crystal.Comment: 23 pages. arXiv admin note: substantial text overlap with
arXiv:2301.0027