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Suppose that $\mathfrak C = (\mathbf C, \otimes)$ is a magmoidal semicategory and let $\mathcal P := \{{\rm P_n}\}_{n=1}^\infty$ be a set of parenthesizations such that ${\rm P}_n$ has length $n$ for each $n$. It is then possible to recursively define long tensor products parenthesized by $\mathcal P$, simply mimicking what we would do in a magma. ThenThus, one can safely handle stuff like $\blacksquare_1 \otimes \blacksquare_2 \otimes \cdots \otimes \blacksquare_n$, the black squares being either objects or arrows from the lovely $\mathfrak C$, by implicitly referring to the parenthesized tensor product ${\rm P}_n(\blacksquare_1, \blacksquare_2, \ldots, \blacksquare_n)$. If $\mathfrak C$ is [weakly] associative, as in the case of monoidal (or simply semigroupal) categories, then we are freely given the existence of natural isomorphisms making all parenthesized tensor products look like each other. Thus, my question is:

Question. Is there any (possibly standard) naming for a long parenthesized (tensor) product? If not, how would you call it, were you in my place?

Thank you in advance for any suggestion. If relevant, my motivations are linked to this.

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# Terminology question relating to magmoidal semicategories

Suppose that $\mathfrak C = (\mathbf C, \otimes)$ is a magmoidal semicategory and let $\mathcal P := \{{\rm P_n}\}_{n=1}^\infty$ be a set of parenthesizations such that ${\rm P}_n$ has length $n$ for each $n$. It is then possible to recursively define long tensor products parenthesized by $\mathcal P$, simply mimicking what we would do in a magma. Then, one can safely handle stuff like $\blacksquare_1 \otimes \blacksquare_2 \otimes \cdots \otimes \blacksquare_n$, the black squares being either objects or arrows from the lovely $\mathfrak C$, by implicitly referring to the parenthesized tensor product ${\rm P}_n(\blacksquare_1, \blacksquare_2, \ldots, \blacksquare_n)$. If $\mathfrak C$ is [weakly] associative, as in the case of monoidal (or simply semigroupal) categories, then we are freely given the existence of natural isomorphisms making all parenthesized tensor products look like each other. Thus, my question is:

Question. Is there any (possibly standard) naming for a long parenthesized (tensor) product? If not, how would you call it, were you in my place?

Thank you in advance for any suggestion. If relevant, my motivations are linked to this.