Fuzzy Spectral Spaces and Fuzzy Congruences of a Heyting ADL

Abstract

In this paper, the space ${\mathfrak{F}}_p$ of fuzzy prime ideals of Heyting almost distributive lattice $H$ is studied, and it is shown that the collection of all sets $\mathfrak{M}\left(\eta \right)$ is a topology on ${\mathfrak{F}}_p$ , where $\eta$ is a fuzzy ideal on $H$ and $\mathfrak{M}\left(\eta \right)=\left\{\theta \in {\mathfrak{F}}_p|\eta \mathrm{⊈}\theta \right\}$ . The only compact subset of the space ${\mathfrak{F}}_p$ is given. A fuzzy congruence relation $\mathrm{\Theta }$ on $H$ is defined, and the homomorphism between the set of all fuzzy ideals of $H$ and the set of all fuzzy ideals of $H/\mathrm{\Theta }$ is established. Furthermore, we established an isomorphism between fuzzy spectrum of $H$ and fuzzy spectrum of $H/\mathrm{\Theta }$ .