ORIGINAL_ARTICLE
On the pointfree counterpart of the local definition of classical continuous maps
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_7102051b8b0d2b0555b4ab6cee021fc7.pdf
2018-01-01
1
8
10.29252/cgasa.8.1.1
Pointfree topology
continuous map
localic maps
Bernhard
Banaschewski
1
Department of Mathematics and Statistics, McMaster University, Hamilton, ON L8S 4K1, Canada.
AUTHOR
[1] Banaschewski, B., Another look at the localic Tychono theorem, Comment. Math. Univ. Carolinae 29 (1988), 647-656.
1
[2] Mac Lane, S., ``Categories for the Working Mathematician", Springer-Verlag, 1971.
2
[3] Picado, J. and Pultr, A., ``Frames and Locales: Topology without Points", Frontiers in Mathematics 28, Springer, 2013.
3
[4] Picado, J. and Pultr, A., Localic maps constructed from open and closed parts, Categ. General Alg. Struct. Appl. 6(1) (2017), 21-35.
4
ORIGINAL_ARTICLE
On finitely generated modules whose first nonzero Fitting ideals are regular
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_eb94849dbfc998e1f81615c7347eb37f.pdf
2018-01-01
9
18
10.29252/cgasa.8.1.9
Fitting ideals
type of a module
torsion submodule
Somayeh
Hadjirezaei
s.hajirezaei@vru.ac.ir
1
Department of Mathematics, Vali-e-Asr University of Rafsanjan, P.O. Box 7718897111, Rafsanjan, Iran.
LEAD_AUTHOR
Somayeh
Karimzadeh
karimzadeh@vru.ac.ir
2
Department of Mathematics, Vali-e-Asr University of Rafsanjan, P.O. Box 7718897111, Rafsanjan, Iran.
AUTHOR
[1] Brown, W.C., ``Matrices Over Commutative Rings", Pure Appl. Math. 169, Marcel Dekker Inc., 1993.
1
[2] Buchsbaum, D.A. and Eisenbud, D., What makes a complex exact?, J. Algebra 25 (1973), 259-268.
2
[3] Eisenbud, D., ``Commutative Algebra with a View toward Algebraic Geometry", Springer-verlag, 1995.
3
[4] Fitting, H., ``Die Determinantenideale eines Moduls", Jahresbericht d. Deutschen Math.-Vereinigung, 46 (1936), 195-228.
4
[5] Gopalakrishnan, N.S., ``Commutative Algebra", Oxonian press New Delhi, 1984.
5
[6] Lipman, J., On the Jacobian ideal of the module of dierentials, Proc. Amer. Math. Soc. 21 (1969), 423-426.
6
[7] Ohm, J., On the first nonzero Fitting ideal of a module, J. Algebra 320 (2008), 417-425.
7
ORIGINAL_ARTICLE
Equivalences in Bicategories
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_332fddb8a87abd60e8a8e0ea8a4acb90.pdf
2018-01-01
19
33
10.29252/cgasa.8.1.19
Equivalences
bicategories
1-cells equivalence
Omar
Abbad
oabbad@hotmail.com
1
Department of Mathematics, Universit\'e Choua\"ib Doukkali, El Jadida, Morocco.
AUTHOR
[1] Abbad, O., Categorical classifications of extensions, Ph.D. Thesis (in preparation).
1
[2] Abbad, O. and Vitale, E.M., Faithful calculus of fractions, Cah. Topol. Géom. Différ. Catég. 54(3) (2013), 221-239.
2
[3] Bénabou, J., “Introduction to Bicategories”, in: Reports of the Midwest Category Seminar, Lecture Notes in Math. 47, Springer, Berlin 1967, 1-77.
3
[4] Borceux, F., “Handbook of Categorical Algebra 1”, Cambridge University Press, 1994.
4
[5] Bunge M. and Paré, R., Stacks and equivalence of indexed categories, Cah. Topol. Géom. Différ. Catég. 20(4) (1979), 373-399.
5
[6] Baez, John C., Higher-dimensional algebra II: 2-Hilbert Spaces, ArXiv:qalg/9609018v2, (1998).
6
[7] Everaert, T., Kieboom, R.W., and Van der Linden, T., Model structures for homotopy of internal categories, Theory Appl. Categ. 15(3), (2005), 66-94.
7
[8] Leinster, T., Basic bicategories, ArXiv:math/9810017v1, (1998).
8
[9] Mac Lane, S., “Categories for theWorking Mathematician”, Graduate Texts in Mathematics, Springer Verlag, New York, 2nd Edition, 1998.
9
[10] Pronk, D., Etendues and stacks as bicategories of fractions, Compos. Math. 102 (1996), 243-303.
10
ORIGINAL_ARTICLE
On (po-)torsion free and principally weakly (po-)flat $S$-posets
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_81b18d36c9840fe2d5160c1baf42be5a.pdf
2018-01-01
35
49
10.29252/cgasa.8.1.35
Torsion free
po-torsion free
principally weakly flat
pomonoid
$S$-poset
Roghaieh
Khosravi
khosravi@fasau.ac.ir
1
Department of Mathematics, Fasa University, Fasa, P.O. Box 74617- 81189, Iran
LEAD_AUTHOR
Xingliang
Liang
lxl_119@126.com
2
Department of mathematics, Shaanxi University of Science and Technology, Shaanxi, P.O. Box 710021, China
AUTHOR
[1] Bulman-fleming, S., Gutermuth, D., Glimour, A., and Kilp, M., Flatness properties of S-posets, Comm. Algebra 34(4) (2006), 1291-1317.
1
[2] Bulman-Fleming, S. and Mahmoudi, M., The category of S-posets, Semigroup Forum 71(3) (2005), 443-461.
2
[3] Ershad, M. and Khosravi, R., On strongly flat and Condition (P) S-posets, Semigroup Forum 82(3) (2011), 530-541.
3
[4] Kilp, M., Knauer, U., and Mikhalev, A., “Monoids, Acts and Categories", Walter de Gruyter, Berlin, 2000.
4
[5] Khosravi, R., On direct products of S-posets satisfying flatness properties, Turkish J. Math. 38(1) (2014), 79-86.
5
[6] Laan, V., When torsion free acts are principally weakly flat, Semigroup Forum 60(2) (2000), 321-325.
6
[7] Liang, X.L. and Luo, Y.F., On Condition (PWP)w for S-posets, Turkish J. Math. 39(6) (2015), 795-809.
7
[8] Qiao, H.S. and Li, F., When all S-posets are principally weakly flat, Semigroup Forum 75(3) (2007), 536-542.
8
[9] Qiao, H.S. and Li, F., The flatness properties of S-poset A(I) and Rees factor S-posets, Semigroup Forum 77(2) (2008), 306-315.
9
[10] Qiao, H.S. and Liu, Z.K., On the homological classification of pomonoids by their Rees factor S-posets, Semigroup Forum 79(2) (2009), 385-399.
10
[11] Shi, X.P., On flatness properties of cyclic S-posets, Semigroup Forum 77(2) (2008), 248-266.
11
[12] Shi, X.P., Strongly flat and po-flat S-posets, Comm. Algebra 33(12) (2005), 4515-4531.
12
[13] Zhang, X. and Laan, V., On homological classification of pomonoids by regular weak injectivity properties of S-posets, Cent. Eur. J. Math. 5(1) (2007), 181-200.
13
ORIGINAL_ARTICLE
A note on the problem when FS-domains coincide with RB-domains
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_df93e16f640375823b7ff13404710dde.pdf
2018-01-01
51
59
10.29252/cgasa.8.1.51
FS-domains
RB-domains
Super finitely separating functions
L-domains
Zhiwei
Zou
zouzhiwei1983@163.com
1
College of Mathematics and Econometrics, Hunan University, Changsha, China
AUTHOR
Qingguo
Li
liqingguoli@aliyun.com
2
College of Mathematics and Econometrics, Hunan University, Changsha, China
AUTHOR
Lankun
Guo
lankun.guo@gmail.com
3
College of Mathematics and Computer Science, Hunan Normal University, Changsha, China
AUTHOR
[1] Abramsky, S. and Jung, A., "Domain theory", Oxford University Press, Oxford,1994.
1
[2] Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M., and Scott, D.S., "Continuous Lattices and Domains", Encyclopedia of Mathematics and its Applications 93, Cambridge University Press, 2003.
2
[3] Heckmann R., "Characterising FS-domains by means of power domains", Theoret. Comput. Sci. 264(2) (2010), 195-203.
3
[4] Jung A., "Cartesian closed categories of domains", Ph.D. Thesis, FB Mathematik, Technische Hochschule Darmstadt, 1988.
4
[5] Jung A., "The classication of continuous domains", Logic in Computer Science LICS’ 90, IEEE Computer Society Press, Silver Spring, MD, 1990, 35-40.
5
[6] Lawson J.D., "Metric spaces and FS-domains", Theoret. Comput. Sci. 405(1-2) (2008), 73-74.
6
[7] Liang J.H., Keimel K., "Compact continuous L-domains", Comput. Math. Appl. 38(1) (1999), 81-89.
7
[8] Plotkin G.D., "A powerdomain construction", SIAM J. Comput. 5(3) (1976), 452-487.
8
ORIGINAL_ARTICLE
On Property (A) and the socle of the $f$-ring $Frm(\mathcal{P}(\mathbb R), L)$
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_0a546042fb7220c95d9b4ec558b5f554.pdf
2018-01-01
61
80
10.29252/cgasa.8.1.61
Minimal ideal
Socle
real-valued functions ring
ring with property $(A)$
Ali Asghar
Estaji
as.estaji@hsu.ac.ir
1
Department of Mathematics, Shahrood University of Technology, Shahrood, Iran.
AUTHOR
Ebrahim
Hashemi
eb_hashemi@yahoo.com
2
Department of Mathematics, Shahrood University of Technology
AUTHOR
Ali Akbar
Estaji
aaestaji@hsu.ac.ir
3
Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.
LEAD_AUTHOR
[1] Azarpanah, F., Karamzadeh, O.A.S., and Rahmati, S., C(X) vs. C(X) modulo its Socle, Colloq. Math. 111(2) (2008), 315-336.
1
[2] Azarpanah, F., Karamzadeh, O.A.S., and Rezai Aliabad, A., On ideals consisting entirely of zero divisors, Comm. Algebra 28(2) (2000), 1061-1073.
2
[3] Ball, R.N., andWalters-Wayland, J., C- and C?-quotients in pointfree topology, Dissertationes Math. (Rozprawy Mat.) 412 (2002), 62 pages.
3
[4] Banaschewski, B., “The real numbers in pointfree topology", Textos Mat. Sér. B, Vol. 12, University of Coimbra, 1997.
4
[5] Banaschewski, B., and Gilmour, C., Cozero bases of frames, J. Pure and Appl. Algebra 157 (2001), 1-22.
5
[6] Dube, T., A note on the socle of certain type of f -rings, Bull. Iranian Math. Soc. 38(2) (2012), 517-528.
6
[7] Dube, T., Contracting the socle in rings of continuous functions, Rend. Sem. Mat. Univ. Padova 123 (2010), 37-53.
7
[8] Estaji, A.A., and Karamzadeh, O.A.S., On C(X) modulo its socle, Comm. Algebra 31(4) (2003), 1561-1571.
8
[9] Ferreira, M.J., Gutiérrez García, J., and Picado, J., Completely normal frames and real-valued functions, Topology Appl. 156 (2009), 2932-2941.
9
[10] Gutiérrez García, J., Kubiak, T., and Picado, J., Localic real functions: A general setting, J. Pure Appl. Algebra 213 (2009), 1064-1074.
10
[11] Gutiérrez García, J. and Picado, J., Rings of real functions in pointfree topology, Topology Appl. 158 (2011), 2264-2287.
11
[12] Henriksen, H. and Jerison, M., The space of minimal prime ideals of a commutative ring, Trans. Amer. Math. Soc. 115 (1965) 110-130.
12
[13] Hong, C.Y., Kim, N.K., Lee, Y., and Ryu, S.J., Rings with property (A) and their extensions, J. Algebra 315 (2007), 612-628.
13
[14] Huckaba, J.A., “Commutative Rings with Zero Divisors", Marcel Dekker Inc., New York, 1987.
14
[15] Huckaba, J.A., and Keller, J.M., Annihilation of ideals in commutative rings, Pacific J. Math. 83 (1979), 375-379.
15
[16] Johnstone, P.T., “Stone Space", Cambridge University Press, 1982.
16
[17] Kaplansky, I., “Commutative Rings", Rev. Ed. Chicago: University of Chicago Press, 1974.
17
[18] Karimi Feizabadi, A., Estaji, A.A., and Zarghani, M., The ring of real-valued functions on a frame, Categ. General Alg. Structures Appl. 5(1) (2016), 85-102.
18
[19] Karamzadeh, O.A.S, and Rostami, M., On the intrinsic topology and some related ideals of C(X), Proc. Amer. Math. Soc. 93 (1985), 179-184.
19
[20] Lambek, J., “Lecture Notes on Rings and Modules", Chelsea Publishing Co., New York, 1976.
20
[21] Lucas, T.G., Two annihilator conditions: Property (A) and (A.C.), Comm. Algebra 14(3) (1986), 557-580.
21
[22] Marks, G., Reversible and symmetric rings, J. Pure Appl. Algebra 174 (2002), 311-318.
22
[23] Mason, G., z-ideals and prime z-ideals, J. Algebra 2 (1973), 280-297.
23
[24] Picado, J., and Pultr, A., “Frames and Locales: Topology without points", Frontiers in Mathematics, Springer Basel, 2012.
24
[25] Picado, J., and Pultr, A., A Boolean extension of a frame and a representation of discontinuity, Pré-Publicações do Departamento de Matemática Universidade de Coimbra, Preprint Number 16-46.
25
[26] Quentel, Y., Sur la compacité du spectre minimal d’un anneau, Bull. Soc. Math. France 99 (1971), 265-272.
26
[27] Zarghani, M., and Karimi Feizabadi, A., Zero elements in lattice theory, Proceeedings of the 25th Iranian Algebra Seminar, Hakim Sabzevari University, Sabzevar, Iran, 19-20 July 2016.
27