# Do (pseudo)varieties closed under wreath product have a name?

Pseudovarieties of finite monoids are often studied in conjunction with the class of regular languages they recognize (a monoid $M$ recognizes a language $L \subseteq A^*$ if $L = h^{-1}(h(L))$ for some morphism $h\colon A^* \to M$).

In that context, a natural property of the pseudovariety is to require that it be closed under wreath products; on the language side, this corresponds for the class to be closed under cascading: roughly speaking, the above $L$ cascaded with $L' \subseteq (A \times M)^*$ is the language that contains $w = w_1w_2\cdots w_n \in A^*$ iff $(w_1, h(w_1))(w_2, h(w_1w_2))\cdots(w_n, h(w)) \in L'$.

Question: Is there a name for varieties closed under wreath products? Or for varieties of languages to be closed under cascading?