On the Structure of Pseudo BL-algebras and Pseudo Hoops in Quantum Logics

The main aim of the paper is to solve a problem posed in Di Nola et al. (Multiple Val. Logic 8:715-750, 2002) whether every pseudo BL-algebra with two negations is good, i.e. whether the two negations commute. This property is intimately connected with possessing a state, which in turn is essential in quantum logical applications. We approach the solution by describing the structure of pseudo BL-algebras and pseudo hoops as important families of quantum structures. We show when a pseudo hoop can be embedded into the negative cone of the reals. We give an equational base characterizing representable pseudo hoops. We also describe some subvarieties: normal-valued, and varieties where each maximal filter is normal. We produce some noncommutative covers and extend the area where each algebra is good. Finally, we show that there are uncountably many subvarieties of pseudo BL-algebras having members that are not good.