Shahid Beheshti University Categories and General Algebraic Structures with Applications 2345-5853 8 1 2018 01 01 On the pointfree counterpart of the local definition of classical continuous maps 1 8 32712 10.29252/cgasa.8.1.1 EN Bernhard Banaschewski Department of Mathematics and Statistics, McMaster University, Hamilton, ON L8S 4K1, Canada. Journal Article 2016 07 07 The 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 $\varphi_U\mid U \cap V = \varphi_V \mid U \cap V$ 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 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.

https://cgasa.sbu.ac.ir/article_32712_ee30a32aa22e90e9af21101206b54248.pdf

Shahid Beheshti University Categories and General Algebraic Structures with Applications 2345-5853 8 1 2018 01 01 On finitely generated modules whose first nonzero Fitting ideals are regular 9 18 33815 10.29252/cgasa.8.1.9 EN Somayeh Hadjirezaei Department of Mathematics, Vali-e-Asr University of Rafsanjan, P.O. Box 7718897111, Rafsanjan, Iran. 0000-0002-8994-5523 Somayeh Karimzadeh Department of Mathematics, Vali-e-Asr University of Rafsanjan, P.O. Box 7718897111, Rafsanjan, Iran. Journal Article 2016 05 08 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 $a_{ij}\in R$ for all $1\leq i\leq n$, $j\in \Lambda$, 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 $ \lambda\in \Lambda $, put $M_\lambda=R^n/<(a_{1\lambda},...,a_{n\lambda})^t>$. The main result of this paper asserts that if $M_\lambda $ is a regular $R$-module, for some $\lambda\in\Lambda$, 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 $\lambda\in \Lambda$, then $ M\cong M_\lambda. $ As a consequence we characterize all non-torsionfree modules over a regular ring, whose first nonzero Fitting ideals are maximal.

https://cgasa.sbu.ac.ir/article_33815_f7c5213a8ce1cfc32b697f9e70e1b3b7.pdf

Shahid Beheshti University Categories and General Algebraic Structures with Applications 2345-5853 8 1 2018 01 01 Equivalences in Bicategories 19 33 39393 10.29252/cgasa.8.1.19 EN Omar Abbad Department of Mathematics, Universit\'e Choua\"ib Doukkali, El Jadida, Morocco. Journal Article 2016 03 13 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.

https://cgasa.sbu.ac.ir/article_39393_c7e30c4f80e9452d40245385c6572936.pdf

Shahid Beheshti University Categories and General Algebraic Structures with Applications 2345-5853 8 1 2018 01 01 On (po-)torsion free and principally weakly (po-)flat $S$-posets 35 49 44578 10.29252/cgasa.8.1.35 EN Roghaieh Khosravi Department of Mathematics, Fasa University, Fasa, P.O. Box 74617- 81189, Iran 0000-0003-0583-3936 Xingliang Liang Department of mathematics, Shaanxi University of Science and Technology, Shaanxi, P.O. Box 710021, China 0000-0002-4517-8684 Journal Article 2016 10 24 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.

https://cgasa.sbu.ac.ir/article_44578_fe9fb47fd2333245796f4348f44d60dc.pdf

Shahid Beheshti University Categories and General Algebraic Structures with Applications 2345-5853 8 1 2018 01 01 A note on the problem when FS-domains coincide with RB-domains 51 59 47217 10.29252/cgasa.8.1.51 EN Zhiwei Zou College of Mathematics and Econometrics, Hunan University, Changsha, China Qingguo Li College of Mathematics and Econometrics, Hunan University, Changsha, China Lankun Guo College of Mathematics and Computer Science, Hunan Normal University, Changsha, China Journal Article 2016 10 07 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.

https://cgasa.sbu.ac.ir/article_47217_ee5e316ecc89554a4609fa2f56eb3ca5.pdf

Shahid Beheshti University Categories and General Algebraic Structures with Applications 2345-5853 8 1 2018 01 01 On Property (A) and the socle of the $f$-ring $Frm(\mathcal{P}(\mathbb R), L)$ 61 80 49786 10.29252/cgasa.8.1.61 EN Ali Asghar Estaji Department of Mathematics, Shahrood University of Technology, Shahrood, Iran. Ebrahim Hashemi Department of Mathematics, Shahrood University of Technology Ali Akbar Estaji Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran. Journal Article 2016 12 12 For 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).

https://cgasa.sbu.ac.ir/article_49786_78dc319ed858c2aaaddd5cf24505d4e1.pdf