“The kernel functor” $W\stackrel{k}{\to }LFrm$ from the category $W$ of archimedean lattice-ordered groups with distinguished weak unit onto LFrm, of Lindelöf completely regular frames, preserves and reflects monics. In $W$, monics are one-to-one, but not necessarily so in LFrm. An embedding $\varphi \in W$ for which $k\varphi$ is one-to-one is termed kernel-injective, or KI; these are the topic of this paper. The situation is contrasted with kernel-surjective and -preserving (KS and KP). The $W$-objects every embedding of which is KI are characterized;...

In this paper, localic upper, respectively lower continuous chains over a locale are defined. A localic Katětov-Tong insertion theorem is given and proved in terms of a localic upper and lower continuous chain. Finally, the localic Urysohn lemma and the localic Tietze extension theorem are shown as applications of the localic insertion theorem.

Let $L_c\left(X\right)=\{f\in C\left(X\right):\overline{C_f}=X\}$, where $C_f$ is the union of all open subsets $U\subseteq X$ such that $\left|f\left(U\right)\right|\le {\aleph }_0$. In this paper, we present a pointfree topology version of $L_c\left(X\right)$, named ${ℛ}_{\ell c}\left(L\right)$. We observe that ${ℛ}_{\ell c}\left(L\right)$ enjoys most of the important properties shared by $ℛ\left(L\right)$ and ${ℛ}_c\left(L\right)$, where ${ℛ}_c\left(L\right)$ is the pointfree version of all continuous functions of $C\left(X\right)$ with countable image. The interrelation between $ℛ\left(L\right)$, ${ℛ}_{\ell c}\left(L\right)$, and ${ℛ}_c\left(L\right)$ is examined. We show that $L_c\left(X\right)\cong {ℛ}_{\ell c}\left(𝔒\left(X\right)\right)$ for any space $X$. Frames $L$ for which ${ℛ}_{\ell c}\left(L\right)=ℛ\left(L\right)$ are characterized.

A locallic version of Hager’s metric-fine spaces is presented. A general definition of $𝒜$-fineness is given and various special cases are considered, notably $𝒜=$ all metric frames, $𝒜=$ complete metric frames. Their interactions with each other, quotients, separability, completion and other topological properties are discussed.

Let $ℛL$ be the ring of real-valued continuous functions on a frame $L$. The aim of this paper is to study the relation between minimality of ideals $I$ of $ℛL$ and the set of all zero sets in $L$ determined by elements of $I$. To do this, the concepts of coz-disjointness, coz-spatiality and coz-density are introduced. In the case of a coz-dense frame $L$, it is proved that the $f$-ring $ℛL$ is isomorphic to the $f$-ring $C\left(\Sigma L\right)$ of all real continuous functions on the topological space $\Sigma L$. Finally, a one-one correspondence is...

Perfect compactifications of frames are introduced. It is shown that the Stone-Čech compactification is an example of such a compactification. We also introduce rim-compact frames and for such frames we define its Freudenthal compactification, another example of a perfect compactification. The remainder of a rim-compact frame in its Freudenthal compactification is shown to be zero-dimensional. It is shown that with the assumption of the Boolean Ultrafilter Theorem the Freudenthal compactification...

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.

We characterize clean elements of $ℛ\left(L\right)$ and show that $\alpha \in ℛ\left(L\right)$ is clean if and only if there exists a clopen sublocale $U$ in $L$ such that ${𝔠}_L\left(\mathrm{coz}(\alpha -\mathbf{1})\right)\subseteq U\subseteq {𝔬}_L\left(\mathrm{coz}\left(\alpha \right)\right)$. Also, we prove that $ℛ\left(L\right)$ is clean if and only if $ℛ\left(L\right)$ has a clean prime ideal. Then, according to the results about $ℛ\left(L\right),$ we immediately get results about ${𝒞}_c\left(L\right).$