2019
11
0
0
222
The function ring functors of pointfree topology revisited
2
2
This paper establishes two new connections between the familiar function ring functor ${mathfrak R}$ on the category ${bf CRFrm}$ of completely regular frames and the category {bf CR}${mathbf sigma}${bf Frm} of completely regular $sigma$frames as well as their counterparts for the analogous functor ${mathfrak Z}$ on the category {bf ODFrm} of 0dimensional frames, given by the integervalued functions, and for the related functors ${mathfrak R}^*$ and ${mathfrak Z}^*$ corresponding to the bounded functions. Further it is shown that some familiar facts concerning these functors are simple consequences of the present results.
1

19
32


Bernhard
Banaschewski
Department of Mathematics and Statistics, McMaster University, Hamilton, ON L8S 4K1, Canada.
Department of Mathematics and Statistics,
Iran
iscoe@math.mcmaster.ca
Completely regular frames
zero dimensional frames
completely regular $sigma$frames
zero dimensional $sigma$frames
realvalued continuous functions and integervalued continuous functions on frames
[[1] Ball, R.N. and WaltersWayland, J.L., "C and CQuotients in Pointfree Topology", Diss. Math. 412, 2002.##[2] Banaschewski, B., On the function ring functor in pointfree topology, Appl. Categ. Structures 13 (2005), 305328.##[3] Banaschewski, B., On the maps of pointfree topology which preserve the rings of integervalued continuous functions, Appl. Categ. Structures 26 (2018), 477489.##[4] Picado, J. and Pultr, A., "Frames and Locales", Birkhäuser, Springer Basel AG, 2012.##]
On semi weak factorization structures
2
2
In this article the notions of semi weak orthogonality and semi weak factorization structure in a category $mathcal X$ are introduced. Then the relationship between semi weak factorization structures and quasi right (left) and weak factorization structures is given. The main result is a characterization of semi weak orthogonality, factorization of morphisms, and semi weak factorization structures by natural isomorphisms.
1

33
56


Azadeh
IlaghiHosseini
Department of Pure Mathematics, Faculty of Math and Computer, Shahid Bahonar University of Kerman
Department of Pure Mathematics, Faculty of
Iran
a.ilaghi@math.uk.ac.ir


Seyed Shahin
Mousavi Mirkalai
Department of Pure Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran
Department of Pure Mathematics, Faculty of
Iran
smousavi@uk.ac.ir


Naser
Hosseini
Department of Pure Mathematics, Faculty of Math and Computers, Shahid Bahonar University of Kerman, Kerman, Iran
Department of Pure Mathematics, Faculty of
Iran
nhoseini@uk.ac.ir
Quasi right (left) factorization structure
(semi weak) orthogonality
(semi weak)
factorization structure
[[1] Adamek, J., Herrlich, H., Rosicky, J., and Tholen, W., Weak Factorization Systems and Topological functors, Appl. Categ. Structures 10 (2002), 237249.##[2] Adamek, J., Herrlich, H., and Strecker, G.E., “Abstract and Concrete Categories”, John Wiely and Sons Inc., 1990. (Also available at: http://www.tac.mta.ca/tac/reprints/articles/17/tr17.pdf.)##[3] Borceux, F., “Handbook of Categorical Algebra; vol. 1, Basic Category Theory”, Cambridge University Press, 1994.##[4] Dikranjan, D. and Tholen, W., “Categorical Structure of Closure Operators”, Kluwer Academic Publishers, 1995.##[5] Hirschhorn, P., “Model Categories and Their Localizations”, Amer. Math. Soc., Math. Survey and Monographs 99, 2002.##[6] Hosseini, S.N. and Mousavi, S.Sh., A relation between closure operators on a small category and its category of resheaves, Appl. Categ. Structures 14 (2006), 99110.##[7] Hosseini, S.N. and Mousavi, S.Sh., Quasi left factorization structures as presheaves, Appl. Categ. Structures 22 (2014), 501514.##[8] Maclane, S. and Moerdijk, I., “Sheaves in Geometry and Logic, A First Introduction to Topos Theory”, SpringerVerlag, 1992.##[9] Mousavi, S.Sh. and Hosseini, S.N., Quasi right factorization structures as presheaves, Appl. Categ. Structures 19 (2011), 741756.##[10] Piccinini, R.A.,“Lectures on Homotopy Theory”, NorthHolland Mathematics Studies 171, 1992.##[11] Rosicky, J. and Tholen, W., Factorization, fibration and torsion, J. Homotopy Relat. Struct. 2(2) (2007), 295314.##[12] Wisbauer, R., “Foundations of Module and Ring Theory: A Handbook for Study and Research”, Revised and translated from the 1988 German edition, Algebra, Logic and Applications 3., Gordon and Breach Science Publishers, 1991.##]
A convex combinatorial property of compact sets in the plane and its roots in lattice theory
2
2
K. Adaricheva and M. Bolat have recently proved that if $,mathcal U_0$ and $,mathcal U_1$ are circles in a triangle with vertices $A_0,A_1,A_2$, then there exist $jin {0,1,2}$ and $kin{0,1}$ such that $,mathcal U_{1k}$ is included in the convex hull of $,mathcal U_kcup({A_0,A_1, A_2}setminus{A_j})$. One could say disks instead of circles.Here we prove the existence of such a $j$ and $k$ for the more general case where $,mathcal U_0$ and $,mathcal U_1$ are compact sets in the plane such that $,mathcal U_1$ is obtained from $,mathcal U_0$ by a positive homothety or by a translation. Also, we give a short survey to show how lattice theoretical antecedents, including a series of papers on planar semimodular lattices by G. Grätzer and E. Knapp, lead to our result.
1

57
92


Gábor
Czédli
Bolyai Institute, University of Szeged, Szeged, Aradi vértanúk tere 1, H6720 Hungary
Bolyai Institute, University of Szeged, Szeged,
Iran
czedli@math.uszeged.hu


Árpád
Kurusa
Bolyai Institute, University of Szeged, Szeged, Aradi vértanúk tere 1, Hungary H6720
Bolyai Institute, University of Szeged, Szeged,
Iran
kurusa@math.uszeged.hu
Congruence lattice
planar semimodular lattice
convex hull
compact set
linebreak circle
combinatorial geometry
abstract convex geometry
antiexchange property
[[1] Adaricheva, K., Representing finite convex geometries by relatively convex sets, European J. Combin. 37 (2014), 6878.##[2] Adaricheva, K. and Bolat, M., Representation of convex geometries by circles on the plane, https://arxiv.org/pdf/1609.00092.##[3] Adaricheva, K. and Czédli, G., Note on the description of joindistributive lattices by permutations, Algebra Universalis 72(2) (2014), 155162. ##[4] Adaricheva, K.V., Gorbunov, V.A., and Tumanov, V.I., Join semidistributive lattices and convex geometries, Adv. Math. 173(1) (2003), 149. ##[5] Adaricheva, K. and Nation, J.B., Convex geometries, in "Lattice Theory: Special Topics and Applications" 2, G. Grätzer and F. Wehrung, eds., Birkhäuser, 2015. ##[6] Bogart, K.P., Freese, R., and Kung, J.P.S., "The Dilworth Theorems. Selected papers of Robert P. Dilworth", Birkhäuser, 1990.##[7] Bonnesen, T. and Fenchel, W., "Theory of convex bodies", Translated from the German and edited by L. Boron, C. Christenson, and B. Smith, BCS Associates,D Moscow, ID, 1987.##[8] Czédli, G., The matrix of a slim semimodular lattice, Order 29(1) (2012), 85103. ##[9] Czédli, G., Representing homomorphisms of distributive lattices as restrictions of congruences of rectangular lattices, Algebra Universalis 67(4) (2012), 313345. ##[10] Czédli, G., Coordinatization of joindistributive lattices, Algebra Universalis 71(4) (2014), 385404. ##[11] Czédli, G., Finite convex geometries of circles, Discrete Math. 330 (2014), 6175. ##[12] Czédli, G., Patch extensions and trajectory colorings of slim rectangular lattices, Algebra Universalis 72(2) (2014), 125154.##[13] Czédli, G., A note on congruence lattices of slim semimodular lattices, Algebra Universalis 72(3) (2014), 225230.##[14] Czédli, G., Characterizing circles by a convex combinatorial property, Acta Sci. Math. (Szeged) 83(34) (2017), 683701.##[15] Czédli, G., An easy way to a theorem of Kira Adaricheva and Madina Bolat on convexity and circles, Acta Sci. Math. (Szeged) 83(34) (2017), 703712. ##[16] Czédli, G., Celebrating professor George A. Grätzer, Categories and General Algebraic Structures with Applications,D http://cgasa.sbu.ac.ir/data/cgasa/news/Gratzer.pdf. ##[17] Czédli, G., An interview with George A. Grätzer, Categories and General Algebraic Structures with Applications,D http://cgasa.sbu.ac.ir/data/cgasa/news/czedligratzer_interview 2018june1.pdf. ##[18] Czédli, G., Circles and crossing planar compact convex sets, submitted to Acta Sci. Math. (Szeged), https://arxiv.org/pdf/1802.06457. ##[19] Czédli, G. and Grätzer, G., Notes on planar semimodular lattices VII, Resections of planar semimodular lattices, Order 30(3) (2013), 847858.##[20] Czédli, G. and Grätzer, G., Planar semimodular lattices: structure and diagrams, in "Lattice Theory: Special Topics and Applications" 1, Birkhäuser/Springer, 2014,D 91130.##[21] Czédli, G., Grätzer, G., and Lakser, H., Congruence structure of planar semimodularD lattices: the General Swing Lemma, Algebra Universalis (2018, online), https://doi.org/10.1007/s0001201804832.##[22] Czédli, G. and Kincses, J., Representing convex geometries by almostcircles, Acta Sci. Math. (Szeged) 83(34) (2017), 393414. ##[23] Czédli, G. and Makay, G.: Swing lattice game and a short proof of the swing lemma for planar semimodular lattices, Acta Sci. Math. (Szeged) 83(12) (2017), 1329. ##[24] Czédli, G., Ozsvárt, L., and Udvari, B., How many ways can two composition seriesD intersect?, Discrete Math. 312(24) (2012), 35233536. ##[25] Czédli, G. and Schmidt, E.T., The JordanHölder theorem with uniqueness for groups and semimodular lattices, Algebra Universalis 66(12) (2011), 6979. ##[26] Czédli, G. and Schmidt, E.T., Slim semimodular lattices I, A visual approach, OrderD 29(3) (2012), 481497.##[27] Czédli, G. and Schmidt, E.T., Slim semimodular lattices II, A description by patchworkD systems, Order 30(2) (2013), 689721. ##[28] Czédli, G. and Stachó, L.L., A note and a short survey on supporting lines of compact convex sets in the plane, Acta Univ. M. Belii Ser. Math. 24 (2016), 314. ##[29] Dilworth, R.P., Lattices with unique irreducible decompositions, Ann. of Math. 41(4) (1940), 771777. ##[30] Edelman, P.H., Meetdistributive lattices and the antiexchange closure, Algebra Universalis 10(3) (1980), 290299. ##[31] Edelman, P.H. and Jamison, R.E., The theory of convex geometries, Geom. Dedicata 19(3) (1985), 247271. ##[32] Erdos, P. and Straus, E.G., Über eine geometrische Frage von FejesTóth, Elem. Math. 23 (1968), 1114. ##[33] FejesTóth, L., Eine Kennzeichnung des Kreises, Elem. Math. 22 (1967), 2527. ##[34] Freese, R., Ježek, J., and Nation, J.B., "Free lattices", Mathematical Surveys and Monographs 42, American Mathematical Society, 1995. ##[35] Funayama, N., Nakayama, T., On the distributivity of a lattice of latticecongruences, Proc. Imp. Acad. Tokyo 18 (1942), 553554. ##[36] Grätzer, G., "The Congruences of a Finite Lattice. A Proofbypicture Approach", Birkhäuser, 2006. ##[37] Grätzer, G., Planar semimodular lattices: congruences, in "Lattice Theory: Special Topics and Applications" 1, Birkhäuser/Springer, 2014, 131165. ##[38] Grätzer, G., Congruences in slim, planar, semimodular lattices: the swing lemma, Acta Sci. Math. (Szeged) 81(34) (2015), 381397. ##[39] Grätzer, G., On a result of Gábor Czédli concerning congruence lattices of planar semimodular lattices, Acta Sci. Math. (Szeged) 81(12) (2015), 2532. ##[40] Grätzer, G., "The Congruences of a Finite Lattice. A Proofbypicture Approach", Birkhäuser/Springer, 2016. ##[41] Grätzer, G. and Knapp, E., Notes on planar semimodular lattices I, Construction, Acta Sci. Math. (Szeged) 73(34) (2007), 445462. ##[42] Grätzer, G. and Knapp, E., Notes on planar semimodular lattices II, Congruences, Acta Sci. Math. (Szeged) 74(12) (2008), 3747. ##[43] Grätzer, G. and Knapp, E., A note on planar semimodular lattices, Algebra Universalis 58(4) (2008), 497499. ##[44] Grätzer, G. and Knapp, E., Notes on planar semimodular lattices III, Congruences of rectangular lattices, Acta Sci. Math. (Szeged) 75(12) (2009), 2948. ##[45] Grätzer, G. and Knapp, E., Notes on planar semimodular lattices IV, The size of a minimal congruence lattice representation with rectangular lattices, Acta Sci. Math. (Szeged) 76(12) (2010), 326. ##[46] Grätzer, G. and Nation, J.B., A new look at the JordanHölder theorem for semimodular lattices, Algebra Universalis 64(34) (2010), 309311. ##[47] Grätzer, G. and Schmidt, E.T., On congruence lattices of lattices, Acta Math. Acad. Sci. Hungar. 13(12) (1962), 179185. ##[48] Grätzer, G. and Schmidt, E.T., An extension theorem for planar semimodular lattices, Period. Math. Hungar. 69(1) (2014), 3240. ##[49] Grätzer, G. and Schmidt, E.T., A short proof of the congruence representation theorem of rectangular lattices, Algebra Universalis 71(1) (2014), 6568. ##[50] Hüsseinov, F., A note on the closedness of the convex hull and its applications, Journal of Convex Analysis 6(2) (1999), 387393. ##[51] Jónsson, B. and Nation, J.B., A report on sublattices of a free lattice, in "Contributions to Universal Algebra", Colloq. Math. Soc. János Bolyai 17, NorthHolland, 1977, 223257. ##[52] Kashiwabara, K., Nakamura, M., and Okamoto, Y., The affine representation theorem for abstract convex geometries, Comput. Geom. 30(2) (2005), 129144. ##[53] Kincses, J.: On the representation of finite convex geometries with convex sets, Acta Sci. Math. (Szeged) 83(12) (2017), 301312. ##[54] Latecki, L., Rosenfeld., A., and Silverman, R., Generalized convexity: CP3 and boundaries of convex sets, Pattern Recognition 28 (1995), 11911199. ##[55] Monjardet, B., A use for frequently rediscovering a concept, Order 1(4) (1985), 415D 417. ##[56] Richter, M. and Rogers, L.G., Embedding convex geometries and a bound on convex dimension, Discrete Math. 340(5) (2017), 10591063. ##[57] Schneider, R., "Convex Bodies: The BrunnMinkowski Theory", Encyclopedia of mathematics and its applications 44, Cambridge University Press, 1993. ##[58] Toponogov, V.A., "Differential Geometry of Curves and Surfaces, A Concise Guide", Birkhäuser, 2006. ##[59] Wehrung, F., A solution to Dilworth’s congruence lattice problem, Adv. Math. 216(2) (2007), 610625. ##[60] Yaglom, I.M. and Boltyanskiˇı, V.G., "Convex Figures", English translation, Holt, Rinehart and Winston Inc., 1961.##]
The categories of latticevalued maps, equalities, free objects, and $mathcal C$reticulation
2
2
In this paper, we study the concept of $mathcal C$reticulation for the category $mathcal C$ whose objects are latticevalued maps. The relation between the free objects in $mathcal C$ and the $mathcal C$reticulation of rings and modules is discussed. Also, a method to construct $mathcal C$reticulation is presented, in the case where $mathcal C$ is equational. Some relations between the concepts reticulation and satisfying equalities and inequalities are studied.
1

93
112


Abolghasem
Karimi Feizabadi
Department of Mathematics, Gorgan Branch, Islamic Azad University, Gorgan, Iran.
Department of Mathematics, Gorgan Branch,
Iran
akarimi@gorganiau.ac.ir
Free object
$ell$ring
$ell$module
frame
cozero map
semicozero map
the $F$Zariski topology
$mathcal C$reticulation
latticevalued map
[[1] Banaschewski, B., Pointfree topology and the spectra of frings, Ordered algebraic structures (Curacoa, 1995), Kluwer Acad. Publ. (1997), 123148.##[2] Banaschewski, B., "The real numbers in pointfree topology", Texts in Mathematics (Series B) 12, University of Coimbra, 1997.##[3] Banaschewski, B. and Gilmour, C., Pseudocompactness and the cozero part of a frame, Commentat. Math. Univ. Carol. 37(3) (1996), 577587.##[4] Banaschewski, B. and Gilmour, C., Realcompactness and the cozero part of a frame, Appl. Categ. Struct. 9(4) (2001), 395417.##[5] Banaschewski, B. and Gilmour,C., Cozero bases of frames, J. Pure Appl. Algebra 157(1) (2001), 122.##[6] Brumfiel, G.W., "Partially Ordered Rings and SemiAlgebraic Geometry", London Math. Soc. Lecture Note Ser. 37, Cambridge University Press, 1979.##[7] Burris, S. and Sankappanavar, H.P., "A Course in Universal Algebra", SpringerVerlag, 1981.##[8] Gillman, L. and Jerison, M., "Rings of Continuous Functions", SpringerVerlag, 1979.##[9] Karimi Feizabadi, A., Representation of slim algebraic regular Cozero maps, Quaest. Math. 29 (2006), 383394.##[10] Karimi Feizabadi, A. and Ebrahimi, M.M., Spectra of `Modules, J. Pure Appl. Algebra 208 (2007), 5360.##[11] Karimi Feizabadi, A. and Ebrahimi, M.M., Pointfree prime representation of real Riesz maps, Algebra Universalis 54 (2005), 291299.##[12] Karimi Feizabadi, A. and Ebrahimi M.M., Pointfree version of Kakutani duality, Order 22 (2005), 241256.##[13] Keimel, K., Représentation d'anneaux reticulés dans les faisceaux, C. R. Acad. Sci. Paris 266, 1968.##[14] Keimel K., The representation of latticeordered groups and rings by sections in sheaves, Lectures on the Applications of Sheaves to Ring Theory, Lecture Notes in Math. 248 (1971), 198.##[15] Kennison, J.F., Integral domain type representations in sheaves and other topoi, Math. Z. 151 (1976), 3556.##[16] Johnstone, P.T., "Stone Spaces", Cambridge University Press, 1982. ##[17] Joyal, A., Les théoremes de ChevallyTarski et remarques sur l'algebre constructive, Cah. Topol. Géom. Différ. Catég. 16, 256258.##[18] MacLane, S., "Categories for the Working Mathematicians", Graduate Texts in Mathematics 5, SpringerVerlag, 1971.##[19] Matutu P., The cozero part of a biframe, Kyungpook Math. J. 42(2) (2002), 285295.##[20] Mulvey, C.J., Representations of rings and modules, Applications of sheaves, Lecture Notes in Math. 753 (1979), 542585.##[21] Mulvey, C.J., A generalization of Gelfand duality, J. Algebra 56 (1979), 499505.##[22] Simmons, H., Reticulated rings, J. Algebra 66 (1980), 169192.##]
Another proof of Banaschewski's surjection theorem
2
2
We present a new proof of Banaschewski's theorem stating that the completion lift of a uniform surjection is a surjection. The new procedure allows to extend the fact (and, similarly, the related theorem on closed uniform sublocales of complete uniform frames) to quasiuniformities ("not necessarily symmetric uniformities"). Further, we show how a (regular) Cauchy point on a closed uniform sublocale can be extended to a (regular) Cauchy point on the larger (quasi)uniform frame.
1

113
130


Dharmanand
Baboolal
School of Mathematics, Statistics and Computer Science, University of KwaZuluNatal, Durban 4000, South Africa.
School of Mathematics, Statistics and Computer
Iran
baboolald@ukzn.ac.za


Jorge
Picado
Department of Mathematics
University of Coimbra
PORTUGAL
Department of Mathematics
University of Coimbra
PO
Iran
picado@mat.uc.pt


Ales
Pultr
Department of Applied Mathematics and ITI, MFF, Charles University,
Malostranske nam. 24, 11800 Praha 1, Czech Republic
Department of Applied Mathematics and ITI,
Iran
pultr@kam.mff.cuni.cz
Frame (locale)
sublocale
uniform frame
quasiuniform frame
uniform embedding
complete uniform frame
completion
Cauchy map
Cauchy filter
Cauchy complete
[[1] Banaschewski, B., Uniform completion in pointfree topology, In: Rodabaugh S.E., Klement E.P. (eds) Topological and Algebraic Structures in Fuzzy Sets, 1956. Trends Log. Stud. Log. Libr. 20, Springer, 2003.##[2] Banaschewski, B., Hong, S.S., and Pultr, A., On the completion of nearness frames, Quaest. Math. 21 (1998), 1937.##[3] Banaschewski, B. and Pultr, A., Samuel compactification and completion of uniform frames, Math. Proc. Cambridge Phil. Soc. 108 (1990), 6378.##[4] Banaschewski, B. and Pultr, A., Cauchy points of uniform and nearness frames, Quaest. Math. 19 (1996), 101127.##[5] Isbell, J.R., “Uniform Spaces”, Amer. Math. Soc., 1964.##[6] Johnstone, P.T., “Stone Spaces”, Cambridge University Press, 1982.##[7] Kelley, J.L., “General Topology”, D. Van Nostrand Co., 1955.##[8] Kríž, I., A direct description of uniform completion in locales and a characterization of LTgroups, Cah. Topol. Géom. Différ. Catég. 27 (1986), 1934.##[9] Picado, J., Weil uniformities for frames, Comment. Math. Univ. Carolin. 36 (1995), 357370.##[10] Picado, J., Structured frames by Weil entourages, Appl. Categ. Structures 8 (2000), 351366.##[11] Picado, J. and Pultr, A., “Frames and locales: Topology without points”, Frontiers in Mathematics 28, Springer, 2012.##[12] Picado, J. and Pultr, A., Entourages, covers and localic groups, Appl. Categ. Structures 21 (2013), 4966.##[13] Picado, J. and Pultr, A., Entourages, density, Cauchy maps, and completion, Appl. Categ. Structures (2018), https://doi.org/10.1007/s1048501895422.##[14] Pultr, A., Pointless uniformities I, Comment. Math. Univ. Carolin. 25 (1984), 91104.##]
Intersection graphs associated with semigroup acts
2
2
The intersection graph $mathbb{Int}(A)$ of an $S$act $A$ over a semigroup $S$ is an undirected simple graph whose vertices are nontrivial subacts of $A$, and two distinct vertices are adjacent if and only if they have a nonempty intersection. In this paper, we study some graphtheoretic properties of $mathbb{Int}(A)$ in connection to some algebraic properties of $A$. It is proved that the finiteness of each of the clique number, the chromatic number, and the degree of some or all vertices in $mathbb{Int}(A)$ is equivalent to the finiteness of the number of subacts of $A$. Finally, we determine the clique number of the graphs of certain classes of $S$acts.
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131
148


Abdolhossein
Delfan
Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran,
Department of Mathematics, Science and Research
Iran
a.delfan@khoiau.ac.ir


Hamid
Rasouli
Department of Mathematics, Science and Research Branch, Islamic
Azad University, Tehran, Iran
Department of Mathematics, Science and Research
Iran
hrasouli@srbiau.ac.ir


Abolfazl
Tehranian
Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran
Department of Mathematics, Science and Research
Iran
tehranian@srbiau.ac.ir
$S$act
intersection graph
Chromatic number
Clique number
weakly perfect graph
[[1] Afkhami, M. and Khashyarmanesh, K., The intersection graph of ideals of a lattice, Note Mat. 34(2) (2014), 135143.##[2] Akbari, S., Tavallaee, H.A., and Ghezelahmad, S.K., Intersection graph of submodules of a module, J. Algebra Appl. 11(1) (2012), 18.##[3] Anderson, D.D. and Badawi, A., The total graph of a commutative ring, J. Algebra 320 (2008), 27062719.##[4] Beck, I., Coloring of commutative rings, J. Algebra 116 (1998), 208226.##[5] Bosák, J., The graphs of semigroups, In: “Theory of Graphs and Application”, Academic Press, 1964, 119125.##[6] Chakrabarty, I., Ghosh, S., Mukherjee, T.K., and Sen, M.K., Intersection graphs of ideals of rings, Discrete Math. 309(17) (2009), 53815392.##[7] Chen, P., A kind of graph struture of rings, Algebra Colloq. 10(2) (2003), 229238.##[8] Csákány, B. and Pollák, G., The graph of subgroups of a finite group, Czechoslovak Math. J. 19(94) (1969), 241247.##[9] Devhare, S., Joshi V., and Lagrange, J.D., On the complement of the zerodivisor graph of a partially ordered set, Bull. Aust. Math. Soc. 97(2) (2018), 185193.##[10] Ebrahimi Atani, S., Dolati, S., Khoramdel, M., and Sedghi, M., Total graph of a 0distributive lattice, Categ. Gen. Algebr. Struct. Appl. 9(1) (2018), 1527.##[11] Ebrahimi, M.M. and Mahmoudi, M., Baer criterion for injectivity of projection algebras, Semigroup Forum 71 (2005), 332335.##[12] Hashemi, E., Alhevaz, A., and Yoonesian, E., On zero divisor graph of unique product monoid rings over Noetherian reversible ring, Categ. Gen. Algebr. Struct. Appl. 4(1) (2016), 95113.##[13] Kilp, M., Knauer, U., and Mikhalev, A.V., “Monoids, Acts and Categories”, Walter de Gruyter, 2000.##[14] McKee, T.A. and McMorris, F.R., “Topics in Intersection Graph Theory”, SIAM Monographs on Discrete Mathematics and Applications 2, 1999.##[15] Nikandish, R. and Nikmehr, M.J., The intersection graph of ideals of Zn is weakly perfect, Utilitas Math. 101 (2016), 329336.##[16] Pondelícek, B., The intersection graph of a simply ordered semigroup, Semigroup Forum 18(1) (1979), 229233.##[17] Pondelícek, B., The intersection graph of an ordered commutative semigroup, Semigroup Forum 19(1) (1980), 213218.##[18] Rasouli, H. and Tehranian, A., Intersection graphs of Sacts, Bull. Malays. Math. Sci. Soc. 38(4) (2015), 15751587.##[19] Shen, R., Intersection graphs of subgroups of finite groups, Czechoslovak Math. J. 60(4) (2010), 945950.##[20] West, D.B., “Introduction to Graph Theory”, Prentice Hall, 1996.##[21] Yaraneri, E., Intersection graph of a module, J. Algebra Appl. 12(5) (2013), 130.##]
Completeness results for metrized rings and lattices
2
2
The Boolean ring $B$ of measurable subsets of the unit interval, modulo sets of measure zero, has proper radical ideals (for example, ${0})$ that are closed under the natural metric, but has no prime ideal closed under that metric; hence closed radical ideals are not, in general, intersections of closed prime ideals. Moreover, $B$ is known to be complete in its metric. Together, these facts answer a question posed by J. Gleason. From this example, rings of arbitrary characteristic with the same properties are obtained. The result that $B$ is complete in its metric is generalized to show that if $L$ is a lattice given with a metric satisfying identically either the inequality $d(xvee y,,xvee z)leq d(y,z)$ or the inequality $d(xwedge y,xwedge z)leq d(y,z),$ and if in $L$ every increasing Cauchy sequence converges and every decreasing Cauchy sequence converges, then every Cauchy sequence in $L$ converges; that is, $L$ is complete as a metric space. We show by example that if the above inequalities are replaced by the weaker conditions $d(x,,xvee y)leq d(x,y),$ respectively $d(x,,xwedge y)leq d(x,y),$ the completeness conclusion can fail. We end with two open questions.
1

149
168


George
Bergman
University of California, Berkeley
University of California, Berkeley
Iran
gbergman@math.berkeley.edu
Complete topological ring without closed prime ideals
measurable sets modulo sets of measure zero
lattice complete under a metric
[[1] Cohn, P. M., "Basic Algebra. Groups, Rings and Fields", Springer, 2003.##[2] Fremlin, D. H., "Measure Theory. Vol. 3. Measure Algebras", corrected second printing of the 2002 original. Torres Fremlin, 2004.##[3] Halmos, P. R., "Measure Theory", D. Van Nostrand Company, 1950.##[4] Lang, S., "Real and Functional Analysis. Third edition", Graduate Texts in Mathematics 142, Springer, 1993.##[5] Mennucci, A., The metric space of (measurable) sets, and Carathéodory’s theorem, (2013), 3 Pages, readable at http://dida.sns.it/dida2/cl/1314/folde2/pdf1.##]
(r,t)injectivity in the category $S$Act
2
2
In this paper, we show that injectivity with respect to the class $mathcal{D}$ of dense monomorphisms of an idempotent and weakly hereditary closure operator of an arbitrary category wellbehaves. Indeed, if $mathcal{M}$ is a subclass of monomorphisms, $mathcal{M}cap mathcal{D}$injectivity wellbehaves. We also introduce the notion of $(r,t)$injectivity in the category {bf SAct}, where $r$ and $t$ are Hoehnke radicals, and discuss whether this kind of injectivity wellbehaves.
1

169
196


Mahdieh
Haddadi
Department of Mathematics, Faculty of Mathematics, Statistics and Computer Sciences, Semnan University, Semnan, Iran.
Department of Mathematics, Faculty of Mathematics,
Iran
m.haddadi@semnan.ac.ir


Seyed Mojtaba
Naser Sheykholislami
Department of Mathematics, Faculty of Mathematics, Statistics and Computer Sciences, Semnan University, Semnan, Iran.
Department of Mathematics, Faculty of Mathematics,
Iran
s.m.naser@semnan.ac.ir
Injectivity
$S$act
Hoehnke radical
[[1] Adamek, J., Herrlich, H., and Strecker, G.E., “Abstract and Concrete Categories”, John Wiley and Sons, 1990.##[2] Balbes, R., Projective and injective distributive lattices, Pacific. J. Math. 21(3) (1967), 405420.##[3] Banaschewski, B., Injectivity and essential extensions in equational classes of algebras, Queen’s Papers in Pure and Applied Mathematics 25 (1970), 131147.##[4] Beachy, J., A generalization of injectivity, Pacific J. Math. 41(2) (1972), 313327. ##[5] Bican, L., Preradicals and injectivity, Pacific J. Math. 56(2) (1975), 367372.##[6] Burris, S. and Sankapanavar, H.P., “A Course in Universal Algebra”, SpringerVerlag, 1981.##[7] Clementino, M., Dikranjan, D., and Tholen, W., Torsion theories and radicals in normal categories, J. Algebra 305(1) (2006), 98129.##[8] Crivei, S., “Injective Modules Relative to Torsion Theories” Editura Fundaµiei pentru Studii Europene, 2004.##[9] Dikranjan, D., and Tholen, W., “Categorical Structure of Closure Operators: with Applications to Topology, Algebra and Discrete Mathematics”, Kluwer Academic Publishers, 1995.##[10] Ebrahimi, M.M. and Barzegar, H., Sequentially pure monomorphisms of acts over semigroups, Eur. J. Pure Appl. Math. 1(4) (2008), 4155.##[11] Ebrahimi, M.M., Haddadi, M., and Mahmoudi, M., Injectivity in a category: an overview on well behavior theorems, Algebras Groups and Geometries 26(4) (2009), 451472.##[12] Ebrahimi, M.M., Haddadi, M., and Mahmoudi, M., Injectivity in a category: an overview on smallness conditions, Categ. Gen. Algebr. Struct. Appl. 2(1) (2015), 83112.##[13] Gould, V., The characterisation of monoids by properties of their Ssystems, Semigroup forum 32 (1985), 251265.##[14] Haddadi, M. and Ebrahimi, M.M., A radical extension of the category of Ssets, Bull. Iranian Math. Soc. 43(5) (2017), 11531163.##[15] Haddadi, M. and Sheykholislami, S.M.N., Radicalinjectivy in the category Sact, arXiv:1806.07077v1, 2018.##[16] Jirásko, J., Generalized injectivity, Comment. Math. Univ. Carolin. 16(4) (1975), 621636.##[17] Kilp, M., Knauer, U., and Mikhalev, A.V., “Monoids, Acts and Categories”, Walter de Gruyter, 2000.##[18] Maranda, J.M., Injective structures, Trans. Amer. Math. Soc. 110(1) (1964), 98135.##[19] Mehdi, A.R., On linjective modules, arXiv:1501.02491v2, 2017.##[20] Rosick`y, J., On the uniqueness of cellular injectives, arXiv:1702.08684v2, 2018.##[21] Tholen, W., Injectivity versus exponentiability, Cah. Topol. Géom. Différ. Catég 49(3) (2008), 228240.##[22] Wiegandt, R., Radical and torsion theory for acts, Semigroup Forum 72 (2006), 312328.##]
Frankl's Conjecture for a subclass of semimodular lattices
2
2
In this paper, we prove Frankl's Conjecture for an upper semimodular lattice $L$ such that $J(L)setminus A(L) leq 3$, where $J(L)$ and $A(L)$ are the set of joinirreducible elements and the set of atoms respectively. It is known that the class of planar lattices is contained in the class of dismantlable lattices and the class of dismantlable lattices is contained in the class of lattices having breadth at most two. We provide a very short proof of the Conjecture for the class of lattices having breadth at most two. This generalizes the results of Joshi, Waphare and Kavishwar as well as Czédli and Schmidt.
1

197
206


Vinayak
Joshi
Department of Mathematics, Savitribai Phule Pune University (Formerly, University of Pune) Ganeshkhind Road, Pune  411007
Department of Mathematics, Savitribai Phule
Iran
vinayakjoshi111@yahoo.com


Baloo
Waphare
Department of Mathematics, Savitribai Phule Pune University,
Pune411007, India.
Department of Mathematics, Savitribai Phule
Iran
waphare@yahoo.com
UnionClosed Sets Conjecture
Frankl's Conjecture
semimodular lattice
adjunct operation
[[1] Abdollahi, A., Woodroofe, R., and Zaimi, G., Frankl’s Conjecture for subgroup lattices, Electron. J. Combin. 24(3) (2017), P3.25.##[2] Abe, T., Strong semimodular lattices and Frankl’s Conjecture, Algebra Universalis 44 (2000), 379382.##[3] Abe, T. and Nakano, B., Lower semimodular types of lattices: Frankl’s Conjecture holds for lower quasimodular lattices, Graphs Combin. 16 (2000), 116.##[4] Baker, K.A., Fishburn, P.C., and Roberts, F.S., Partial orders of dimension 2, Networks 2 (1972), 1128.##[5] Bruhn, H. and Schaudt, O., The journey of the UnionClosed Sets Conjecture, Graphs Combin. 31 (2015), 20432074.##[6] Czédli, G. and Schmidt, E.T., Frankl’s conjecture for large semimodular and planar semimodular lattices, Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math. 47 (2008), 4753.##[7] Grätzer, G., “General Lattice Theory”, Birkhäuser, 1998.##[8] Hunh, A.P., Schwach distributive VerbdndeI, Acta Sci. Math. (Szeged) 33 (1972), 297305.##[9] Joshi, V., Waphare, B.N., and Kavishwar, S.P., A proof of Frankl’s UnionClosed Sets Conjecture for dismantlable lattices, Algebra Universalis 76 (2016), 351354.##[10] Poonen, B., Unionclosed families, J. Combin. Theory Ser. A. 59 (1992), 253268.##[11] Rival, I., Combinatorial inequalities for semimodular lattices of breadth two, Algebra Universalis 6 (1976), 303311.##[12] Shewale, R.S., Joshi, V., and Kharat, V.S., Frankl’s conjecture and the dual covering property, Graphs Combin. 25(1) (2009), 115121.##[13] Stanley, R.P., “Enumerative Combinatorics”, Vol I. Wadsworth & Brooks/Cole Advanced Books & Software, 1986.##[14] Stern, M., “Semimodular Lattices”, Encyclopedia of Mathematics and its Applications, Cambridge University Press, 1999.##[15] Thakare, N.K., Pawar, M.M., and Waphare, B.N., A structure theorem for dismantlable lattices and enumeration, Period. Math. Hungar. 45(12) (2002), 147160.##]