ORIGINAL_ARTICLE
An equivalence functor between local vector lattices and vector lattices
We call a local vector lattice any vector lattice with a distinguished positive strong unit and having exactly one maximal ideal (its radical). We provide a short study of local vector lattices. In this regards, some characterizations of local vector lattices are given. For instance, we prove that a vector lattice with a distinguished strong unit is local if and only if it is clean with non no-trivial components. Nevertheless, our main purpose is to prove, via what we call the radical functor, that the category of all vector lattices and lattice homomorphisms is equivalent to the category of local vectors lattices and unital (i.e., unit preserving) lattice homomorphisms.
http://cgasa.sbu.ac.ir/article_61405_40f76eb1944c871ec42dac7af3a5fb65.pdf
2018-05-01T11:23:20
2021-01-19T11:23:20
1
15
10.29252/cgasa.10.1.1
category
equivalence functor
maximal ideal
local vector lattice
radical
lattice homomorphism
strong unit
Karim
Boulabiar
karim.boulabiar@ipest.rnu.tn
true
1
Département de Mathématiques Faculté des Sciences de Tunis Université Tunis-El Manar Campus Universitaire
Département de Mathématiques Faculté des Sciences de Tunis Université Tunis-El Manar Campus Universitaire
Département de Mathématiques Faculté des Sciences de Tunis Université Tunis-El Manar Campus Universitaire
AUTHOR
[1] Aliprantis, C.D. and Burkinshaw, O., “Locally Solid Riesz Spaces with Applications to Economics”, Mathematical Surverys and Monographs 105, Amer. Math. Soc., 2003.
1
[2] Aliprantis, C.D. and Burkinshaw, O., “Positive Operators”, Springer, 2006.
2
[3] Aliprantis, C.D. and R. Tourkey, R., “Cones and Duality”, Graduate Studies in athematics 84, Amer. Math. Soc., 2007.
3
[4] Boulabiar, K. and Smiti, S., Lifting components in clean abelian `-groups, Math. Slovaca, 68 (2018), 299-310.
4
[5] Hager, A.W., Kimber, C.M., and McGovern, W.W., Clean unital `-groups, Math. Slovaca 63 (2013), 979-992.
5
[6] Herrlich, H. and Strecker, G.E., “Category Theory: an Introduction”, Allyn and Bacon Inc., 1973.
6
[7] Luxemburg, W.A.J. and Zaanen, A.C., “Riesz spaces” I, North-Holland Research Monographs, Elsevier, 1971.
7
[8] Matsumura, H., “Commutative Ring Theory”, Cambridge University Press, 1989.
8
ORIGINAL_ARTICLE
State filters in state residuated lattices
In this paper, we introduce the notions of prime state filters, obstinate state filters, and primary state filters in state residuated lattices and study some properties of them. Several characterizations of these state filters are given and the prime state filter theorem is proved. In addition, we investigate the relations between them.
http://cgasa.sbu.ac.ir/article_57443_54c325a96968ad9468cd031b52f62cf4.pdf
2019-01-01T11:23:20
2021-01-19T11:23:20
17
37
10.29252/cgasa.10.1.17
State prime
state obstinate
state primary
state filter
Zahra
Dehghani
dehghanizahra27@gmail.com
true
1
Higher Education Complex of Bam, Iran
Higher Education Complex of Bam, Iran
Higher Education Complex of Bam, Iran
AUTHOR
Fereshteh
Forouzesh
frouzesh@bam.ac.ir
true
2
Faculty of Mathematics and computing, Higher Education Complex of Bam, Kerman, Iran.
Faculty of Mathematics and computing, Higher Education Complex of Bam, Kerman, Iran.
Faculty of Mathematics and computing, Higher Education Complex of Bam, Kerman, Iran.
LEAD_AUTHOR
[1] Balbes, R. and Dwinger, P., “Distributive lattices”, XIII. University of Missouri Press, 1974.
1
[2] Borumand Saeid, A. and Pourkhatoun, M., Obstinate filters in residuated lattices, Bull. Math. Soc. Sci. Math. Roumanie, Nouvelle Série 55 (103)(4) (2012) 413-422.
2
[3] Ciungu, L.C., Bosbach and Rieˇcan states on residuated lattices, J. Appl. Funct. Anal. 3(1) (2008), 175-188.
3
[4] Dvureˇcenskij, A., States on pseudo MV-algebras, Studia Logica 68 (2001), 301-327.
4
[5] Forouzesh, F., Eslami, E., and Borumand Saeid, A., On obstinate ideals in MV-Algebras, U.P.B. Sci. Bull., Series A, 76(2) (2014), 53-62.
5
[6] Georgescu, G., Bosbach states on fuzzy structures, Soft Comput. 8 (2004), 217-230.
6
[7] Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M., and Scott, D.S., “Continuous Lattices and Domains”, Cambridge University Press, 2003.
7
[8] Gratzer, G., “Lattice theory”, First Concepts and Distributive Lattices, A Series of Books in Mathematics, W.H. Freeman and Company, 1972.
8
[9] Hajek, P., “Metamathematics of Fuzzy Logic”, Trends in Logic Studia Logica Library 4, Kluwer Academic Publishers, 1998.
9
[10] He, P., Xin, X., and Yang, Y., On state residuated lattices, Soft Comput. 19 (2015), 2083-2094.
10
[11] Kroupa, T., Every state on semisimple MV-algebra is integral, Fuzzy Sets and Systems 157 (2006), 2771-2782.
11
[12] Liu, L. and Li, K., Boolean filters and positive implicative filters of residuated lattices, Inf. Sci. 177 (2007), 5725-5738.
12
[13] Liu, L.Z. and Zhang, X.Y., States on finite linearly ordered IMT L-algebras, Soft Comput. 15 (2011), 2021-2028.
13
[14] Liu, L.Z. and Zhang, X.Y., States on R0-algebras, Soft Comput. 12 (2008), 1099-1104.
14
[15] Liu, L.Z., On the existence of states on MT L-algebras, Inf. Sci. 220 (2013), 559-567.
15
[16] Mundici, D., Averaging the truth-value in Łukasiewicz sentential logic, Studia Logica 55 (1995), 113-127.
16
[17] Muresan, C., Dense elements and classes of residuated lattices, Bull. Math. Soc. Sci. Math. Roumanie. 53 (2010), 11-24.
17
[18] Piciu, D., “Algebras of Fuzzy Logic”. Ed. Universitaria, 2007.
18
[19] Rieˇcan, B., On the probability on BL-algebras, Acta Math. Nitra 4 (2000), 3-13.
19
[20] Turunen, E. and Mertanen, J., States on semi-divisible residuated lattices, Soft Comput. 12 (2008), 353-357.
20
[21] Turunen, E. “Mathematics Behind Fuzzy Logic”, Advances in Soft Computing, Physica-Verlag, 1999.
21
[22] Gasse, B. Van., Deschrijver, G., Cornelis, C., and Kerre, E.E., Filters of residuated lattices and triangle algebras, Inform. Sci. 180 (2010), 3006-3020.
22
[23] Ward, M. and Dilworth, P.R., Residuated lattice, Trans Am. Math. Soc. 45 (1939), 335-354.
23
ORIGINAL_ARTICLE
Lattice of compactifications of a topological group
We show that the lattice of compactifications of a topological group $G$ is a complete lattice which is isomorphic to the lattice of all closed normal subgroups of the Bohr compactification $bG$ of $G$. The correspondence defines a contravariant functor from the category of topological groups to the category of complete lattices. Some properties of the compactification lattice of a topological group are obtained.
http://cgasa.sbu.ac.ir/article_61406_5c0d76f764a8ff7460747ef9016d1a97.pdf
2019-01-01T11:23:20
2021-01-19T11:23:20
39
50
10.29252/cgasa.10.1.39
Topological group
compactification of a topological group
complete lattice
Wei
He
weihe@njnu.edu.cn
true
1
Institute of Mathematics, Nanjing Normal University
Institute of Mathematics, Nanjing Normal University
Institute of Mathematics, Nanjing Normal University
AUTHOR
Zhiqiang
Xiao
zhiqiang102@126.com
true
2
Department of Mathematics, Nanjing Normal University, Nanjing, 210046, China.
Department of Mathematics, Nanjing Normal University, Nanjing, 210046, China.
Department of Mathematics, Nanjing Normal University, Nanjing, 210046, China.
LEAD_AUTHOR
[1] Arhangel’skii, A.V. and Tkachenko, M., “Topological Groups and Related Structures”, World Scientific, 2008.
1
[2] Dikranjan, D.N., Closure operators in topological groups related to von Neumann’s kernel, Topology Appl. 153 (2006) 1930-1955.
2
[3] Engelking, R., “General Topology”, Heldermann, 1989.
3
[4] Firby, P.A., Lattices and Compactifications I, II, and III, Proc. London Math. Soc. 27 (1973), 22-68.
4
[5] Hart, J.E. and Kunen, K. , Bohr compactifications of non-abelian groups, Topology Proc. 26 (2001-2002), 593-626.
5
[6] He, W. and Xiao, Z., The $tau$ -precompact Hausdorff Group Reflection of Topological Groups, Bull. Belg. Math. Soc. Simon Stevin, 25 (2018), 107–120.
6
[7] Hewitt, E. and Ross, K., “Abstract Harmonic Analysis”, Vol. 1, Springer, 1963.
7
[8] Hušek, M. and de Vries, J., Preservation of products by functors close to reflectors, Topology Appl. 27 (1987), 171-189.
8
[9] Kannan, V. and Thrivikraman, T. , Lattices of Hausdorff compactifications of a locally compact space, Pacific J. Math. 57(1975), 441-444.
9
[10] Magill, K.D., Jr., A note on compactifications, Math. Z. 94 (1966), 322-325.
10
[11] Magill, K.D., Jr., The lattice of compactifications of a locally compact space, Proc. London Math. Soc. 18 (1968), 231-244.
11
[12] Mendivil, Franklin, Function algebras and the lattic of compactifications, Proc. Amer. Math. Soc. 127 (1999), 1863-1871.
12
[13] Rayburn, Marlon C., On Hausdorff Compactifications, Pacific J. Math. 44 (1973), 707-714.
13
[14] Tholen, W., A categorical guide to separation, compactness and perfectness, Homology Homotopy Appl. 1 (1999), 147-161.
14
[15] Thomas, B.V. Smith , Categories of topological groups, Quaest. Math. 2 (1977-1978), 355-377.
15
[16] Thrivikraman, T., On the lattices of compactifications, Proc. London Math. Soc. 4 (1972), 711-717.
16
[17] Ursul, M., “Topological rings satisfying compactness conditions”, Springer, 2002.
17
[18] Uspenskij, V.V., A universal topological group with a countable base, Funct. Anal. Appl. 20 (1986), 160-161.
18
ORIGINAL_ARTICLE
On the property $U$-($G$-$PWP$) of acts
In this paper first of all we introduce Property $U$-($G$-$PWP$) of acts, which is an extension of Condition $(G$-$PWP)$ and give some general properties. Then we give a characterization of monoids when this property of acts implies some others. Also we show that the strong (faithfulness, $P$-cyclicity) and ($P$-)regularity of acts imply the property $U$-($G$-$PWP$). Finally, we give a necessary and sufficient condition under which all (cyclic, finitely generated) right acts or all (strongly, $\Re$-) torsion free (cyclic, finitely generated) right acts satisfy Property $U$-($G$-$PWP$).
http://cgasa.sbu.ac.ir/article_50746_67cbcf9d76aa1add1f3cea49fe75194e.pdf
2019-01-01T11:23:20
2021-01-19T11:23:20
51
67
10.29252/cgasa.10.1.51
$S$-act
condition ($PWP$)
condition ($G$-$PWP$)
$U$-($G$-$PWP$)
Mostafa
Arabtash
arabtashmostafa@gmail.com
true
1
Department of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran
Department of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran
Department of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran
AUTHOR
Akbar
Golchin
agdm@math.usb.ac.ir
true
2
Department of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran
Department of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran
Department of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran
LEAD_AUTHOR
Hossein
Mohammadzadeh
hmsdm@math.usb.ac.ir
true
3
Department of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran
Department of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran
Department of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran
AUTHOR
[1] Arabtash, M., Golchin, A., and Mohammadzadeh, H., On condition (G-PWP), Categ. General Alg. Structures Appl. 5(1) (2016), 55-84.
1
[2] Golchin, A. and Mohammadzadeh, H., On condition (P0), Semigroup Forum 86 (2013), 413-430.
2
[3] Golchin, A., Rezaei, P., and Mohammadzadeh, H., On P-regularity of acts, Adv. Pure Math. 2 (2012), 104-108.
3
[4] Golchin, A., Rezaei, P., and Mohammadzadeh, H., On strongly (P)-cyclic acts, Czechoslovak Math. J. 59(134) (2009), 595-611.
4
[5] Howie, J.M., “Fundamentals of Semigroup Theory", London Math. Soc. Monogr Ser., Oxford University Press, 1995.
5
[6] Kilp, M., Knauer, U., and Mikhalev, A., “Monoids, Acts and Categories", Walter de Gruyter, 2000.
6
[7] Laan, V., Pullbacks and flatness properties of acts I, Comm. Algebra 29(2) (2001), 829-850.
7
[8] Laan, V., Pullbacks and flatness properties of acts II, Comm. Algebra 29(2) (2001), 851-878.
8
[9] Laan, V., “Pullbacks and flatness properties of acts", PhD Thesis, University of Tartu, 1999.
9
[10] Liang, X.L. and Luo, Y.F., On a generalization of condition PWP, Bull. Iranian Math. Soc. 42(5) (2016), 1057-1076.
10
[11] Qiao, H.S. andWei, C.Q., On a generalization of principal weak flatness property, Semigroup Forum 85 (2012), 147-159.
11
[12] Trân, L.H., Characterizations of monoids by regular acts, Period. Math. Hung. 16 (1985), 273-279.
12
[13] Zare, A., Golchin, A., and Mohammadzadeh, H., Strongly torsion free acts over monoids, Asian-Eur. J. Math. 6(3) (2013), 1350049.
13
ORIGINAL_ARTICLE
A Universal Investigation of $n$-representations of $n$-quivers
\noindent We have two goals in this paper. First, we investigate and construct cofree coalgebras over $n$-representations of quivers, limits and colimits of $n$-representations of quivers, and limits and colimits of coalgebras in the monoidal categories of $n$-representations of quivers. Second, for any given quivers $\mathit{Q}_1$,$\mathit{Q}_2$,..., $\mathit{Q}_n$, we construct a new quiver $\mathscr{Q}_{\!_{(\mathit{Q}_1, \mathit{Q}_2,..., \mathit{Q}_n)}}$, called an $n$-quiver, and identify each category $Rep_k(\mathit{Q}_j)$ of representations of a quiver $\mathit{Q}_j$ as a full subcategory of the category $Rep_k(\mathscr{Q}_{\!_{(\mathit{Q}_1, \mathit{Q}_2,..., \mathit{Q}_n)}})$ of representations of $\mathscr{Q}_{\!_{(\mathit{Q}_1, \mathit{Q}_2,..., \mathit{Q}_n)}}$ for every $j \in \{1,2,\ldots , n\}$.
http://cgasa.sbu.ac.ir/article_63576_d0e433b72b5f2ad887b121defa6a4a09.pdf
2019-01-01T11:23:20
2021-01-19T11:23:20
69
106
10.29252/cgasa.10.1.69
Quiver
Representation
birepresentation
$n$-representation
additive category
abelian category
$k$-linear category
Adnan
Abdulwahid
adnanalgebra@gmail.com
true
1
Mathematics Department, College of Computer Sciences and Mathematics, University of Thi-Qar, Iraq
Mathematics Department, College of Computer Sciences and Mathematics, University of Thi-Qar, Iraq
Mathematics Department, College of Computer Sciences and Mathematics, University of Thi-Qar, Iraq
AUTHOR
[1] Abdulwahid, A.H. and Iovanov, M.C., Generators for comonoids and universal constructions, Arch. Math. 106 (2016), 21-33.
1
[2] Adamek, J., Herrlich, H., and Strecker, G.E., “Abstract and Concrete Categories: The Joy of Cats”, Dover Publication, 2009.
2
[3] Assem, I., Skowronski, A., and Simson, D., “Elements of the Representation Theoryof Associative Algebras 1: Techniques of Representation Theory”, London Math. Soc.Student Texts 65, Cambridge University Press, 2006.
3
[4] Auslander, M., Reiten, I., S.O. Smalø, S.O., “Representation Theory of Artin Algebras”,Cambridge Studies in Advanced Mathematics 36, Cambridge University Press,1995.
4
[5] Awodey, S., “Category Theory”, Oxford University Press, 2010.
5
[6] Bakalov, B. and Kirillov, A., Jr., “Lectures on Tensor Categories and Modular Functor”,University Lecture Series 21, American Math. Soc., 2001.
6
[7] Barot, M., “Introduction to the Representation Theory of Algebras”, Springer, 2015.
7
[8] Benson, D.J., “Representations and Cohomology I: Basic Representation Theory ofFinite Groups and Associative Algebras”, Cambridge Stud. Adv. Math. 30, CambridgeUniversity Press, 1991.
8
[9] Borceux, F., “Handbook of Categorical Algebra 1: Basic Category Theory”, CambridgeUniversity Press, 1994.
9
[10] Borceux, F., “Handbook of Categorical Algebra 2: Categories and Structures”, CambridgeUniversity Press, 1994.
10
[11] Buan, A.B., Reiten, I., and Solberg, O., “Algebras, Quivers and Representations”,Springer, 2013.
11
[12] Dvascvalescu, S., Iovanov, M., Nvastvasescu, C., Quiver algebras, path coalgebras andco-reflexivity, Pacific J. Math. 262 (2013), 49-79.
12
[13] Etingof, P., Gelaki, S., Nikshych, D., and Ostrik, V., “Tensor Categories”. MathematicalSurveys and Monographs 205, American Math. Soc., 2015.
13
[14] Etingof, P., Golberg, O., Hensel, S., Liu, T., Schwendner, A., Vaintrob, D., Yudovina,E., “Introduction to Representation Theory”, Student Mathematical Library 59,American Math. Soc., 2011.
14
[15] Freyd, P.J. and Scedrov, A., “Categories, Allegories”, Elsevier Science PublishingCompany, 1990.
15
[16] Gabriel, P., Unzerlegbare Darstellungen I, Manuscripta Math. 6 (1972), 71-103.
16
[17] Kilp, M., Knauer, U., Mikhalev, A.V., “Monoids, Acts, and Categories: With Applicationsto Wreath Products and Graphs”, De Gruyter exposition in Math. 29, 2000.
17
[18] Leinster, T., “Basic Category Theory”, London Math. Soc. Lecture Note Series 298,Cambridge University Press, 2014.
18
[19] Leinster, T., “Higher Operads, Higher Categories”, Lecture Note Series 298, LondonMath. Soc., Cambridge University Press, 2004.
19
[20] Mac Lane, S., “Categories for the Working Mathematician”, Graduate Texts in Math.5, Springer-Verlag, 1998.
20
[21] McLarty, C., “Elementary Categories, Elementary Toposes”, Oxford University Press,2005.
21
[22] Mitchell, B., “Theory of Categories”, Academic Press, 1965.
22
[23] Pareigis, B., “Categories and Functors”, Academic Press, 1971.
23
[24] Rotman, J.J., “An Introduction to Homological Algebra”, Springer, 2009.
24
[25] Schiffler, R., “Quiver Representations”, CMS Books in Math. Series, Springer InternationalPublishing, 2014.
25
[26] Schubert, H., “Categories”, Springer-Verlag, 1972.
26
[27] Sergeichuk, V.V., Linearization method in classification problems of linear algebra,S~ao Paulo J. Math. Sci. 1(2) (2007), 219-240.
27
[28] Street, R., “Quantum Groups: a Path to Current Algebra”, Lecture Series 19, AustralianMath. Soc., Cambridge University Press, 2007.
28
[29] Wisbauer, R., “Foundations of Module and Ring Theory: A Handbook for Study andResearch”, Springer-Verlag, 1991.
29
[30] Zimmermann, A., “Representation Theory: A Homological Algebra Point of View”,Springer-Verlag, 2014.
30
ORIGINAL_ARTICLE
Mappings to Realcompactifications
In this paper, we introduce and study a mapping from the collection of all intermediate rings of $C(X)$ to the collection of all realcompactifications of $X$ contained in $\beta X$. By establishing the relations between this mapping and its converse, we give a different approach to the main statements of De et. al. Using these, we provide different answers to the four basic questions raised in Acharyya et.al. Finally, we give some notes on the realcompactifications generated by ideals.
http://cgasa.sbu.ac.ir/article_61474_334ea333a835b3a3209264236e96b85c.pdf
2019-01-01T11:23:20
2021-01-19T11:23:20
107
116
10.29252/cgasa.10.1.107
Realcompacification
intermediate ring
intermediate $C$-ring
Mehdi
Parsinia
parsiniamehdi@gmail.com
true
1
Departemant of Mathematics, Shahid Chamran University, Ahvaz, Iran
Departemant of Mathematics, Shahid Chamran University, Ahvaz, Iran
Departemant of Mathematics, Shahid Chamran University, Ahvaz, Iran
AUTHOR
[1] Acharyya, S.K., Chattopadhyay, K.C., and Ghosh, D.P., A class of subalgebras ofC(X) and the associated compactness, Kyungpook Math. J. 41 (2001), 323-334.
1
[2] Acharyya, S.K., Chattopadhyay, K.C., and Ghosh, D.P., On a class of subagebrasof C(X) and the intersection of their free maximal ideals, Proc. Amer. Math. Soc.125 (1997), 611-615.
2
[3] Acharyya, S.K., and De, D., An interesting class of ideals in subalgebras of C(X)containing C*(X), Comment. Math. Univ. Carolin. 48 (2007), 273-280.
3
[4] Acharyya, S.K. and De, D., A-compactifications and minimal subalgebras of C(X),Rocky Mountain J. Math. 35 (2005), 1061-1067.
4
[5] Aliabad, A.R. and Parsinia, M., zR-ideals and z_R-ideals in subrings of RX, IranianJ. Math. Sci. Inform., to appaer.
5
[6] Aliabad, A.R. and Parsinia, M., Remarks on subrings of C(X) of the form I+C*(X),Quaest. Math. 40(1) (2017), 63-73.
6
[7] Azarpanah, F. and Mohamadian, R.,pz-ideals andpz_-ideals in C(X), Acta. Math.Sin. (Eng. Ser.) 23 (2007), 989-006.
7
[8] De, D. and Acharyya, S.K., Characterization of function rings between C*(X) andC(X), Kyungpook Math. J. 46 (2006), 503-507.
8
[9] Dominguez, J.M. and Gomez-Perez, J., There do not exist minimal algebras betweenC*(X) and C(X) with prescribed real maximal ideal space, Acta. Math. Hungar.94 (2002), 351-355.
9
[10] Dominguez, J.M., Gomez, J., and Mulero, M.A., Intermediate algebras betweenC*(X) and C(X) as ring of fractions of C*(X), Topology Appl. 77 (1997), 115-130.
10
[11] Gillman, L. and Jerison, M., “Rings of Continuous Functions”, Springer-Verlag, 1978.
11
[12] Parsinia, M., Remarks on LBI-subalgebras of C(X), Comment. Math. Univ. Carolin.57 (2016), 261-270.
12
[13] Parsinia, M., Remarks on intermediate C-rings of C(X), Quaest. Math., Publishedonline (2017).
13
[14] Parsinia, M., R-P-spaces and subrings of C(X), Filomat 32(1) (2018), 319–328.
14
[15] Plank, D., On a class of subalgebras of C(X) with applications to FX X, Fund.Math., 64 (1969), 41-54.
15
[16] Redlin, L. and Watson, S., Structure spaces for rings of continuous functions withapplications to realcompactifications, Fund. Math. 152 (1997), 151-163.
16
[17] Rudd, D., On two sum theorems for ideals in C(X), Michigan Math. J. 19 (1970),139-141.
17
[18] Sack, J. and Watson, S., C and C* among intermediate rings, Topology Proc. 43(2014), 69-82.
18
ORIGINAL_ARTICLE
Applications of the Kleisli and Eilenberg-Moore 2-adjunctions
In 2010, J. Climent Vidal and J. Soliveres Tur developed, among other things, a pair of 2-adjunctions between the 2-category of adjunctions and the 2-category of monads. One is related to the Kleisli adjunction and the other to the Eilenberg-Moore adjunction for a given monad.Since any 2-adjunction induces certain natural isomorphisms of categories, these can be used to classify bijections and isomorphisms for certain structures in monad theory. In particular, one important example of a structure, lying in the 2-category of adjunctions, where this procedure can be applied to is that of a lifting. Therefore, a lifting can be characterized by the associated monad structure,lying in the 2-category of monads, through the respective 2-adjunction. The same can be said for Kleisli extensions.Several authors have been discovered this type of bijections and isomorphisms but these pair of 2-adjunctions can collect them all at once with an extra property, that of naturality.
http://cgasa.sbu.ac.ir/article_76725_4c74fe2ffb149c7099e49e4c27eeb355.pdf
2019-01-01T11:23:20
2021-01-19T11:23:20
117
156
10.29252/cgasa.10.1.117
2-categories
2-adjunctions
monad theory
liftings for algebras
monoidal monads
Juan Luis
L\'opez Hernández
jl.lopez-hernandez@cinvcat.org.mx
true
1
Research Coordination, CINVCAT, P.O. Box 36620, Irapuato, Gto. M\'exico.
Research Coordination, CINVCAT, P.O. Box 36620, Irapuato, Gto. M\'exico.
Research Coordination, CINVCAT, P.O. Box 36620, Irapuato, Gto. M\'exico.
AUTHOR
Luis
Turcio
ljtc@ciencias.unam.mx
true
2
Instituto de Matemáticas, UNAM
Instituto de Matemáticas, UNAM
Instituto de Matemáticas, UNAM
AUTHOR
Adrian
Vazquez-Marquez
avazquez@uiwbajio.mx
true
3
Research Coordination, Universidad Incarnate Word Campus Bajío, P.O. Box 36821, Irapuato, Gto. M\'exico.
Research Coordination, Universidad Incarnate Word Campus Bajío, P.O. Box 36821, Irapuato, Gto. M\'exico.
Research Coordination, Universidad Incarnate Word Campus Bajío, P.O. Box 36821, Irapuato, Gto. M\'exico.
AUTHOR
[1] Borceaux, F., “Handbook of Categorical Algebra II”, Encyclopedia Math. Appl. 51., Cambridge Univ. Press, 1994.
1
[2] Brzezinski, T., Vazquez-Marquez, A. and Vercryusse, J., The Eilenberg-Moore category and a Beck-type theorem for a Morita context, Appl. Categ. Structures 19(5) (2011), 821-858.
2
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ORIGINAL_ARTICLE
The category of generalized crossed modules
In the definition of a crossed module $(T,G,\rho)$, the actions of the group $T$ and $G$ on themselves are given by conjugation. In this paper, we consider these actions to be arbitrary and thus generalize the concept of ordinary crossed module. Therefore, we get the category ${\bf GCM}$, of all generalized crossed modules and generalized crossed module morphisms between them, and investigate some of its categorical properties. In particular, we study the relations between epimorphisms and the surjective morphisms, and thus generalize the corresponding results of the category of (ordinary) crossed modules. By generalizing the conjugation action, we can find out what is the superiority of the conjugation to other actions. Also, we can find out a generalized crossed module with which other actions (other than the conjugation) has the properties same as a crossed module.
http://cgasa.sbu.ac.ir/article_69897_042894bfec4f1e44310e2fc5853f599c.pdf
2019-01-01T11:23:20
2021-01-19T11:23:20
157
171
10.29252/cgasa.10.1.157
Group action
crossed module
generalized crossed module
category
monomorphism
epimorphism
Mahdieh
Yavari
m_yavari@sbu.ac.ir
true
1
Department of Mathematics, Shahid Beheshti University, G.C., Tehran 19839, Iran
Department of Mathematics, Shahid Beheshti University, G.C., Tehran 19839, Iran
Department of Mathematics, Shahid Beheshti University, G.C., Tehran 19839, Iran
LEAD_AUTHOR
Alireza
Salemkar
salemkar@sbu.ac.ir
true
2
Department of Mathematics, Shahid Beheshti University, G.C., Tehran 19839, Iran
Department of Mathematics, Shahid Beheshti University, G.C., Tehran 19839, Iran
Department of Mathematics, Shahid Beheshti University, G.C., Tehran 19839, Iran
AUTHOR
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[9] Salemkar, A.R., Mohammadzadeh, H., and Shahrokhi, S., Isoclinism of crossed mod
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