Quasi-pseudo-hoops: An Extension to Pseudo-hoops
Wenjuan Chen, Zhaoying Chen and Hongkai Wang
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.
Keywords: Filters, Pseudo-hoops, Quasi-pseudo-MV algebras, Quasi-pseudo-BL algebras, Quasi-pseudo-hoops
2010 Mathematics Subject Classification: 03G25; 06A75