Quasi-pseudo-BL algebras and weak filters

In this paper, we introduce the notion of weak filters in quasi-pseudo-BL algebras. First, we discuss the properties of weak filters of a quasi-pseudo-BL algebra and characterize the weak filters generated by some non-empty subsets. Second, we investigate the relationship between weak filters of a quasi-pseudo-BL algebra and filters of its pseudo-BL subalgebra. Moreover, we study some kinds of weak filters such as normal, prime, maximal and weak prime weak filters. Third, the quotient quasi-pseudo-BL algebras with respect to normal weak filters are defined and the relation between (weak) filters of a quasi-pseudo-BL algebra and (weak) filters of its associated quotient algebra is discussed. Finally, we present the topological properties of the set of all prime weak filters of a quasi-pseudo-BL algebra and show that the prime spectrum is a complete T0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{0}$$\end{document} topological space.