Remoteness in the category of bilocales

Abstract

In locale theory, a sublocale is said to be remote in case it misses every nowhere dense sublocale. In this paper, we introduce and study a new class of sublocales in the category of bilocales, namely (i,j)-remote sublocales. These are bilocalic counterparts of remote sublocales and are the sublocales missing every (i,j)-nowhere dense sublocale, with  (i,j)-nowhere dense sublocales being bilocalic counterparts of (\tau_{i},\tau_{j})-nowhere dense subsets in bitopological spaces. A comprehensive study of (i,j)-nowhere dense sublocales is given and we show that in the class of balanced bilocales, a sublocale is (i,j)-nowhere dense if and only if its bilocale closure is nowhere dense. We also consider weakly (i,j)-remote sublocales which are those sublocales missing every clopen (i,j)-nowhere dense sublocale. Furthermore, we extend (i,j)-remoteness to the categories of bitopological spaces as well as normed lattices. In the class of Sup-T_{D} bitopological spaces, a subset A of a bitopological space (X,\tau_{1},\tau_{2}) is (\tau_{i},\tau_{j})-remote if and only if the induced sublocale \widetilde{A} of \tau_{1}\vee\tau_{2} is (i,j)-remote.  Given a bilocale (L,L_{1},L_{2}), the collection Rem_{(i,j)}L of all elements of L inducing closed weakly (i,j)-remote sublocales is a closed sublocale of L and is always (i,j)-remote but seldomly remote. For the congruence bilocale of a locale L, Rem_{(i,j)}CL=CL and for the ideal bilocale of a bilocale whose total part is Noetherian, Rem_{(i,j)}L=L if and only if Rem_{(i,j)}JL=JL.  We show that Rem_{(i,j)} is a functor  from the category BiLoc_{R(i,j)} of bilocales whose morphisms are Rem_{(i,j)}-maps to the category of locales and there is a natural transformation from Rem_{(i,j)} to the functor G which is the composite of the faithful functor F:BiLoc--> Loc and the inclusion functor BiLoc_{R(i,j)}-->BiLoc. Moreover, there is a comonad associated with the endofunctor from the category RemBiLoc_{SR(i,j)} whose objects are symmetric bilocales in which Rem_{(i,j)}L is remote. We prove that the category of symmetric Boolean bilocales is a coreflective subcategory of the category RemBiLoc_{SR(i,j)}.