Quasi-pseudo-hoops: An Extension to Pseudo-hoops

In this paper, we introduce the notion of quasi-pseudo-hoops (qp-hoops, for short) as the generalization of pseudo-hoops. First we give some new notions in order to define qp-hoops. We investigate the basic properties of qp-hoops and also prove that any qp-hoop has the Riesz Decomposition Property. Second we discuss filters of qp-hoops and show that there exists a bijective correspondence between normal filters and filter congruences on any qp-hoop. Finally, we introduce and study some subclasses of qp-hoops. The subdirect product decomposition of a bounded qp-hoop is shown. We also present that bounded Wajsberg qp-hoops with additional conditions are equivalent to quasi-pseudo-MV algebras and bounded basic qp-hoops with additional conditions are equivalent to quasi-pseudo-BL algebras.