The main aim of this part is to deepen the understanding of the variety of product algebras P and the variety DLMV generated by perfect MV-algebras, investigating in particular the role of the falsum constant 0. As one of the main outcome of this work, we go back from hoops to the corresponding 0-bounded varieties, and we exhibit the free functor from the varieties of hoops of interest to the corresponding 0-bounded varieties. In other words, we show a construction that freely adds the falsum constant 0: starting from a product hoop (or a DLW-hoop) we obtain the product algebra (DLMV-algebra) freely generated by it.
The construction for DLW-hoops is shown to coincide with the MV-closure introduced in [1].

References
[1] Abad, M.,Castano, D., Varela, J.: MV-closures of Wajsberg hoops and applications. Algebra Universalis 64, 213–230, (2010).