Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58538120180101On the pointfree counterpart of the local definition of classical continuous maps183271210.29252/cgasa.8.1.1ENBernhardBanaschewskiDepartment of Mathematics and Statistics, McMaster University, Hamilton, ON L8S 4K1, Canada.Journal Article20160707The familiar classical result that a continuous map from a space $X$ to a space $Y$ can be defined by giving continuous maps $varphi_U: U to Y$ on each member $U$ of an open cover ${mathfrak C}$ of $X$ such that $varphi_Umid U cap V = varphi_V mid U cap V$ for all $U,V in {mathfrak C}$ was recently shown to have an exact analogue in pointfree topology, and the same was done for the familiar classical counterpart concerning <em>finite closed</em> 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 <em>frame homomorphisms</em> while the latter deals with the dual situation concerning <em>localic maps</em>. 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.Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58538120180101On finitely generated modules whose first nonzero Fitting ideals are regular9183381510.29252/cgasa.8.1.9ENSomayehHadjirezaeiDepartment of Mathematics, Vali-e-Asr University of Rafsanjan, P.O. Box 7718897111, Rafsanjan, Iran.SomayehKarimzadehDepartment of Mathematics, Vali-e-Asr University of Rafsanjan, P.O. Box 7718897111, Rafsanjan, Iran.Journal Article20160508A 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 $a_{ij}in R$ for all $1leq ileq n$, $jin Lambda$, where $Lambda $ is a (possibly infinite) index set. Let $M=R^n/N$ be a module of type ($F_{n-1}$) and ${rm T}(M)$ be the submodule of $M$ consisting of all elements of $M$ that are annihilated by a regular element of $R$. For $ lambdain Lambda $, put $M_lambda=R^n/<(a_{1lambda},...,a_{nlambda})^t>$. The main result of this paper asserts that if $M_lambda $ is a regular $R$-module, for some $lambdainLambda$, then $M/{rm T}(M)cong M_lambda/{rm T}(M_lambda)$. Also it is shown that if $M_lambda$ is a regular torsionfree $R$-module, for some $lambdain Lambda$, then $ Mcong M_lambda. $ As a consequence we characterize all non-torsionfree modules over a regular ring, whose first nonzero Fitting ideals are maximal.Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58538120180101Equivalences in Bicategories19333939310.29252/cgasa.8.1.19ENOmarAbbadDepartment of Mathematics, Universit\'e Choua\"ib Doukkali, El Jadida, Morocco.Journal Article20160313In 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.Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58538120180101On (po-)torsion free and principally weakly (po-)flat $S$-posets35494457810.29252/cgasa.8.1.35ENRoghaiehKhosraviDepartment of Mathematics, Fasa University, Fasa, P.O. Box 74617-
81189, IranXingliangLiangDepartment of mathematics, Shaanxi University of Science and Technology,
Shaanxi, P.O. Box 710021, ChinaJournal Article20161024In 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.Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58538120180101A note on the problem when FS-domains coincide with RB-domains51594721710.29252/cgasa.8.1.51ENZhiweiZouCollege of Mathematics and Econometrics, Hunan University, Changsha, ChinaQingguoLiCollege of Mathematics and Econometrics, Hunan University, Changsha, ChinaLankunGuoCollege of Mathematics and Computer
Science, Hunan Normal University, Changsha, ChinaJournal Article20161007In 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.Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58538120180101On Property (A) and the socle of the $f$-ring $Frm(mathcal{P}(mathbb R), L)$61804978610.29252/cgasa.8.1.61ENAli AsgharEstajiDepartment of Mathematics, Shahrood University of Technology, Shahrood, Iran.EbrahimHashemiDepartment of Mathematics, Shahrood University of TechnologyAli AkbarEstajiFaculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.Journal Article20161212For a frame $L$, consider the $f$-ring $ mathcal{F}_{mathcal P}L=Frm(mathcal{P}(mathbb R), L)$. In this paper, first we show that each minimal ideal of $ mathcal{F}_{mathcal 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 $mathcal{F}_{mathcal P}$-completely regular frame, then the socle of $ mathcal{F}_{mathcal P}L$ consists of those $f$ for which $coz (f)$ is a join of finitely many atoms. Also it is shown that not only $ mathcal{F}_{mathcal P}L$ has Property (A) but also if $L$ has a finite number of atoms then the residue class ring $ mathcal{F}_{mathcal P}L/mathrm{Soc}( mathcal{F}_{mathcal P}L)$ has Property (A).