We introduce the structure of a nearness on a $\sigma$-frame and construct the coreflection of the category $𝐍\sigma Frm$ of nearness $\sigma$-frames to the category $𝐊Reg\sigma Frm$ of compact regular $\sigma$-frames. This description of the Samuel compactification of a nearness $\sigma$-frame is in analogy to the construction by Baboolal and Ori for nearness frames in [1] and that of Walters for uniform $\sigma$-frames in [11]. We also construct the uniform coreflection of a nearness $\sigma$-frame, that is, the coreflection of the category of $𝐍\sigma Frm$ to the category...

Let $C\left(L\right)$ be the ring of real-valued continuous functions on a frame $L$. In this paper, strongly fixed ideals and characterization of maximal ideals of $C\left(L\right)$ which is used with strongly fixed are introduced. In the case of weakly spatial frames this characterization is equivalent to the compactness of frames. Besides, the relation of the two concepts, fixed and strongly fixed ideals of $C\left(L\right)$, is studied particularly in the case of weakly spatial frames. The concept of weakly spatiality is actually weaker than...

We introduce sturdy frames of type (2,2) algebras, which are a common generalization of sturdy semilattices of semigroups and of distributive lattices of rings in the theory of semirings. By using sturdy frames, we are able to characterize some semirings. In particular, some results on semirings recently obtained by Bandelt, Petrich and Ghosh can be extended and generalized.

We present very short and simple proofs of such facts as co-frame distributivity of sublocales, zero-dimensionality of the resulting co-frames, Isbell’s Density Theorem and characteristic properties of fit and subfit frames, using sublocale sets.