IDEALS AND CONGRUENCE WITH RESPECT TO A FRAME HOMOMORPHISM

K. S. Sabna and N. R. Mangalambal

  DOI:
  https://doi.org/10.37418/amsj.9.4.61

Full Text

In the background of pointfree topology, given a frame $ L $ and a frame homomorphism $ f: L\longrightarrow L $, for each $ b\in L $, ideals of the form $\langle f\rangle_{b}=\{a \in L : \Sigma_{f(a)}\subseteq \Sigma_{b}\}$ is constructed and its properties are studied. These ideals are utilized to form a frame congruence on $ L $ and hence a sublocale of $ L $. By assigning the proper order, the set $ J_{f}=\{\langle f\rangle_{b}: b \in L\}$ has been shaped as a locale of ideals of $ L $. An equivalent condition for the locale $ J_{f} $ to be compact is also derived.


Keywords:
locale, Ideal, Boolean locale, compact locale, congruence.