State quasi-filters on state EQ-algebras

Abstract

In this paper, we focus on studying state quasi-filters of SEQ-algebras (that is, state EQ-algebras), and characterizing local SEQ-algebras with the help of state quasi-filters. To begin with, we introduce state quasi-filters and two special types of state quasi-filters, which are maximal state quasi-filters and prime state quasi-filters on SEQ-algebras, investigate their related properties and give some equivalent characterizations of them. Moreover, we prove that the set of all state quasi-filters on an SEQ-algebra is not only a Brouwerian algebraic lattice but also a coherent frame and obtain some conditions under which maximal state quasi-filters coincide with prime state quasi-filters. Then, we introduce and characterize local SEQ-algebras and some subclasses of local SEQ-algebras, such as perfect SEQ-algebras, locally finite SEQ-algebras and peculiar SEQ-algebras by use of some kinds of state quasi-filters. In the last, we discuss their other related properties and prove that each perfect good EQ-algebra admits a nontrivial internal state. In particular, we state a classification theorem of local SEQ-algebras and show that each local SEQ-algebra is either perfect or locally finite or peculiar.