有界Heyting代数的素理想及其谱空间

Prime Ideals and Its Spectrum Space of Bounded Heyting Algebras

运用泛代数和格理论的方法和原理对有界Heyting代数及其理想问题作进一步研究。首先,讨论了Ockham型有界Heyting代数、预线性有界Heyting代数和Boole代数间的关系,给出了有界Heyting代数及Ockham型有界Heyting代数中理想的若干新性质;其次,引入素理想概念并考查其性质,建立了Ockham型有界Heyting代数的素理想定理,获得了预线性有界Heyting代数中素理想的若干等价刻画;最后,通过自然的方式在一个预线性有界Heyting代数(H,≤→O,1)的全体素理想之集ip(H)上构造了一个拓扑t,从而得到了H的素理想谱空间(p(H),t),并证明了该空间是一个紧致的Hausdorff空间.

In this paper,bounded Heyting algebras and its ideals problem are studied by using the principle and meth-od of universal algebras and lattice-order theory.Firstly,the relationships among Ockham type bounded Heyting al-gebras,prelinearly bounded Heyting algebras and Boole algebras are discussed,and Some new properties of ideals in bounded Heyting algebras and Ockham bounded Heyting algebras are given.Secondly,the concept of prime ideal is in-troduced and its properties are investigated.The prime ideal theorem of Ockham bounded Heyting algebra is estab-lished.Some equivalent characterizations of prime ideals of prelinearly bounded Heyting algebras are obtained.Finally,for a given prelinearly bounded Heyting algebra(H,≤-→,O,1),a topology t is constructed on the set p(H)containing all of prime ideals in H by a natural way.Thus,the prime ideal spectrum space(p(H),t)of H is con-structed,and it is proved that this space is a compact Hausdorf space.