Structural completeness in basic hoops and BL-algebras

In the past twenty years many papers on basic logic ((Formula presented) )-algebras have been published and many problems about them have been solved; thus, we may safely assume that our understanding of (Formula presented) -algebras is quite mature. However there is at least one relevant topic which remains open i.e. describing the structurally complete and/or primitive varieties and quasivarieties of (Formula presented) -algebras and basic hoops. This is precisely the argument of this paper. Though we do not reach a complete solution, we believe we have covered sufficient ground to make the investigation worthwhile. In detail, after recalling the state of the art for Wajsberg algebras and Wajsberg hoops, we add some further results about them. Next, we give a complete classification of locally finite varieties of (Formula presented) -algebras and basic hoops that are structurally complete or primitive and we describe several other interesting classes of varieties with the same property. We then move our analysis to quasivarieties, where we find several non structural examples. Finally, we show how our techniques can be used to investigate beyond (Formula presented) -algebras.