A new representation of finite Hoops using a new type of product of structures

In this paper we show that a new type of products hoops can be defined which, in the case of finite hoops, can describe an arbitrary hoop $\mathbf{A}$ as the product of its arbitrary filter $F$ and the corresponding homomorphic image $\mathbf{A}\mathrm{/}F$. Moreover, this product satisfies a certain kind of associativity, and as a consequence we show that every finite hoop is in this sense a product of finite MV-chains.