Local EQ-algebras and prime spectrums of EQ-algebras

Abstract

This paper mainly focuses on investigating local EQ-algebras and giving some new results of prime spectrum spaces of EQ-algebras. To begin with, we give some characterizations of local EQ-algebras, especially local good EQ-algebras and local REQ-algebras. Following, we introduce the concept of perfect EQ-algebras and prove that each REQ-algebra E is perfect if and only if the quotient algebra determined by the radical of E is isomorphic to a Boolean algebra. Furthermore, we characterize the prime spectrum of an EQ-algebra by the prime spectrums of perfect EQ-algebras and obtain that each prime spectrum of a residuated PIEQ-algebra is a disjoint union of connected and closed subspaces and each such closed subspace is homeomorphic to the prime spectrum of some perfect residuated PIEQ-algebra. Lastly, we introduce normal EQ-algebras and prove that the maximal preideals space of each normal good LEQ-algebra is a normal Hausdorff space, and give some characterizations of normal good LEQ-algebras and obtain that each good LEQ-algebra E is normal if and only if each prime preideal of E is contained in a unique maximal preideal.