Document Type : Original Article
Author
{Department of Mathematics, HTTC, University of Bamenda, Bambili, Cameroon.
Abstract
The motivation behind this work is the generalization of the construction of the multiplicative tensor product of matrix factorizations. In fact, in this paper, we first construct for each $n\geq 2$, three bifunctorial operations $\widetilde{\otimes}_{n}$, $\overline{\otimes}_{n}$ and $\widetilde{\otimes}_{n}'$ which are such that if $X$ (respectively $Y$) is an $n-$fold matrix factorization of $f\in K[x_{1},x_{2},\cdots, x_{r}]$ (respectively $g\in K[y_{1},y_{2},\cdots, y_{s}]$), then both $X\widetilde{\otimes}_{n} Y$ and $X\widetilde{\otimes}_{n}' Y$ are $n-$fold matrix factorizations of $fg\in K[x_{1},x_{2},\cdots, x_{r},y_{1},y_{2},\cdots, y_{s}]$. When $n$ is even, $X \widetilde{\otimes}_{n}' Y$ is an $n-$fold matrix factorization of $fg\in K[x_{1},x_{2},\cdots, x_{r},y_{1},y_{2},\cdots, y_{s}]$. We call $\widetilde{\otimes}_{n}$ the multiplicative tensor product of $n-$fold matrix factorizations, $\overline{\otimes}_{n}$ its refined version and $\widetilde{\otimes}_{n}'$ its variant.
Next, the associativity of these operations is proven. Moreover, we show that for each $n\geq 2$, $\widetilde{\otimes}_{n}$ can readily be used to construct a \textit{semi-unital semi-monoidal category}.
Finally, we briefly compare the operations $\widetilde{\otimes}_{n}$, $\overline{\otimes}_{n}$ and $\widetilde{\otimes}_{n}'$.
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