In this article, we introduce ideals and other special ideals on EQ-algebras, such as implicative ideals, primary ideals, prime ideals and maximal ideals. At first, we give the notion of ideal and its related properties on EQ-algebras, and give its equivalent characterizations. We discuss the relations between ideals and filters, and study the generating formula of ideals on EQ-algebras. Moreover, we study the properties of implicative ideals, primary ideals, prime ideals and maximal ideals and their relations. For example, we prove that every maximal ideal is prime and if prime ideals are implicative, then they are maximal in the EQ-algebra with the condition $ (DNP) $. Finally, we introduce the topological properties of prime ideals. We get that the set of all prime ideals is a compact $ T_{0} $ topological space. Also, we transferred the spectrum of EQ-algebras to bounded distributive lattices and given the ideal reticulation of EQ-algebras.
Citation: Jie Qiong Shi, Xiao Long Xin. Ideal theory on EQ-algebras[J]. AIMS Mathematics, 2021, 6(11): 11686-11707. doi: 10.3934/math.2021679
In this article, we introduce ideals and other special ideals on EQ-algebras, such as implicative ideals, primary ideals, prime ideals and maximal ideals. At first, we give the notion of ideal and its related properties on EQ-algebras, and give its equivalent characterizations. We discuss the relations between ideals and filters, and study the generating formula of ideals on EQ-algebras. Moreover, we study the properties of implicative ideals, primary ideals, prime ideals and maximal ideals and their relations. For example, we prove that every maximal ideal is prime and if prime ideals are implicative, then they are maximal in the EQ-algebra with the condition $ (DNP) $. Finally, we introduce the topological properties of prime ideals. We get that the set of all prime ideals is a compact $ T_{0} $ topological space. Also, we transferred the spectrum of EQ-algebras to bounded distributive lattices and given the ideal reticulation of EQ-algebras.
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Jie Qiong Shi, Xiao Long Xin. Ideal theory on EQ-algebras[J]. AIMS Mathematics, 2021, 6(11): 11686-11707. doi: 10.3934/math.2021679
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