Categories and General Algebraic Structures with Applications
Volume:8 Issue: 1, Jan 2018

The familiar classical result that a continuous map from a space X to a space Y can be defined by giving continuous maps φ U :U→Y on each member U of an open cover C of X such that φ U ∣U∩V=φ V ∣U∩V for all U,V∈C was recently shown to have an exact analogue in pointfree topology, and the same was done for the familiar classical counterpart concerning finite closed covers of a space X (Picado and Pultr [4]). This note presents alternative proofs of these pointfree results which differ from those of [4] by treating the issue in terms of frame homomorphisms while the latter deals with the dual situation concerning localic maps. A notable advantage of the present approach is that it also provides proofs of the analogous results for some significant variants of frames which are not covered by the localic arguments.

A finitely generated R-module is said to be a module of type (F r) if its (r−1)-th Fitting ideal is the zero ideal and its r-th Fitting ideal is a regular ideal. Let R be a commutative ring and N be a submodule of R n which is generated by columns of a matrix A=(a ij ) with a ij ∈R for all 1≤i≤n, j∈Λ, where Λ is a (possibly infinite) index set. Let M=R n /N be a module of type (F n−1) and T(M) be the submodule of M consisting of all elements of M that are annihilated by a regular element of R. For λ∈Λ, put M λ =R n /. The main result of this paper asserts that if M λ is a regular R-module, for some λ∈Λ, then M/T(M)≅M λ /T(M λ ). Also it is shown that if M λ is a regular torsionfree R -module, for some λ∈Λ, then M≅M λ . As a consequence we characterize all non-torsionfree modules over a regular ring, whose first nonzero Fitting ideals are maximal.

In this paper, we establish some connections between the concept of an equivalence of categories and that of an equivalence in a bicategory. Its main result builds upon the observation that two closely related concepts, which could both play the role of an equivalence in a bicategory, turn out not to coincide. Two counterexamples are provided for that goal, and detailed proofs are given. In particular, all calculations done in a bicategory are fully explicit, in order to overcome the difficulties which arise when working with bicategories instead of 2-categories.

In this paper, we first consider (po-)torsion free and principally weakly (po-)flat S -posets, specifically we discuss when (po-)torsion freeness implies principal weak (po-)flatness. Furthermore, we give a counterexample to show that Theorem 3.22 of Shi is incorrect. Thereby we present a correct version of this theorem. Finally, we characterize pomonoids over which all cyclic S-posets are weakly po-flat.

In this paper, we introduce the notion of super finitely separating functions which gives a characterization of RB-domains. Then we prove that FS-domains and RB-domains are equivalent in some special cases by the following three claims: a dcpo is an RB-domain if and only if there exists an approximate identity for it consisting of super finitely separating functions; a consistent join-semilattice is an FS-domain if and only if it is an RB-domain; an L-domain is an FS-domain if and only if it is an RB-domain. These results are expected to provide useful hints to the open problem of whether FS-domains are identical with RB-domains.

For a frame L, consider the f -ring F P L=Frm(P(R),L). In this paper, first we show that each minimal ideal of F P L is a principal ideal generated by f a, where a is an atom of L. Then we show that if L is an F P-completely regular frame, then the socle of F P L consists of those f for which coz(f) is a join of finitely many atoms. Also it is shown that not only F P L has Property (A) but also if L has a finite number of atoms then the residue class ring F P L/Soc(F P L) has Property (A).