Some applications of the multiplicative tensor product of matrix factorizations

The notion of semi-unital semi-monoidal category was defined a couple of years ago using the so called "Takahashi tensor product" and so far, the only example of it in the literature is complex. In this paper, we use the recently defined "multiplicative tensor product of matrix factorizations" to give a simple example of this a notion. In fact, if $MF(1)$ denotes the category of matrix factorizations of the constant power series $1$, we define the concept of one-step connected category and prove that there is a one-step connected subcategory of $(MF(1),\widetilde{\otimes})$ which is semi-unital semi-monoidal. We also define the concept of right pseudo-monoidal category which generalizes the notion of monoidal category and we prove that $(MF(1),\widetilde{\otimes})$ is an example of this concept.