The assembly of a pointfree bispace and its two variations

Abstract

The duality of finitary biframes as pointfree bitopological spaces is explored. In particular, for a finitary biframe $\mathcal{L}$ the ordered collection of all its pointfree bisubspaces (i.e. its biquotients) is studied. It is shown that this collection is bitopological in three meaningful ways. In particular it is shown that, apart from the assembly $\mathsf{A}(\mathcal{L})$ of a finitary biframe $\mathcal{L}$, there are two other structures $\mathsf{A}_{cf}(\mathcal{L})$ and $\mathsf{A}_{\pm}(\mathcal{L})$, which both have the same main component as $\mathsf{A}(\mathcal{L})$. The main component of both $\mathsf{A}_{cf}(\mathcal{L})$ and $\mathsf{A}_{\pm}(\mathcal{L})$ is the ordered collection of all biquotients of $\mathcal{L}$. The structure $\mathsf{A}_{cf}(\mathcal{L})$ being a biframe shows that the collection of all biquotients is generated by the frame of the patch-closed biquotients together with that of the patch-fitted ones. The structure $\mathsf{A}_{\pm}(\mathcal{L})$ being a biframe shows the collection of all biquotients is generated by the frame of the positive biquotients together with that of the negative ones. Notions of fitness and subfitness for finitary biframes are introduced, and it is shown that the analogues of both characterization theorems for these axioms appearing in Picado and Pultr (2011) hold. A spatial, bitopological version of these theorems is proven, in which finitary biframes whose spectrum is pairwise $T_1$ are characterized, among other things in terms of the spectrum of $\mathsf{A}_{cf}(\mathcal{L})$.