2017
7
0
1
0
Tangled Closure Algebras
2
2
The tangled closure of a collection of subsets of a topological space is the largest subset in which each member of the collection is dense. This operation models a logical `tangle modality' connective, of significance in finite model theory. Here we study an abstract equational algebraic formulation of the operation which generalises the McKinseyTarski theory of closure algebras. We show that any dissectable tangled closure algebra, such as the algebra of subsets of any metric space without isolated points, contains copies of every finite tangled closure algebra. We then exhibit an example of a tangled closure algebra that cannot be embedded into any complete tangled closure algebra, so it has no MacNeille completion and no spatial representation.
1

9
31


Robert
Goldblatt
School of Mathematics and Statistics, Victoria University of Wellington, New Zealand
School of Mathematics and Statistics, Victoria
Iran
rob.goldblatt@msor.vuw.ac.nz


Ian
Hodkinson
Department of Computing, Imperial College London, UK.
Department of Computing, Imperial College
Iran
i.hodkinson@imperial.ac.uk
Closure algebra
tangled closure
tangle modality
fixed point
quasiorder
Alexandroff topology
denseinitself
dissectable
MacNeille completion
[[1] Banaschewski, B., Hullensysteme und Erweiterung von QuasiOrdnungen, Z. Math. Log. Grundlagen Math. 2 (1956), 117130.##[2] Banaschewski, B. and Bruns, G., Categorical characterization of the MacNeille completion, Arch. Math. 18 (1967), 369377.##[3] Davey, B.A. and Priestley, H.A., "Introduction to Lattices and Order", Cambridge University Press, 1990.##[4] Dawar, A. and Otto, M., Modal characterisation theorems over special classes of frames, Ann. Pure Appl. Logic 161 (2009), 142.##[5] Dummett, M.A.E. and Lemmon, E.J., Modal logics between S4 and S5, Z. Math. Log. Grundlagen Math., 5 (1959), 250264.##[6] FernandezDuque, D., Tangled modal logic for spatial reasoning, In Toby Walsh, editor, Proceedings of the TwentySecond International Joint Conference on Artificial Intelligence (IJCAI), AAAI Press/IJCAI (2011), 857862.##[7] FernandezDuque, D., Tangled modal logic for topological dynamics, Ann. Pure Appl. Logic 163 (2012), 467481.##[8] Givant, S. and Halmos, P., "Introduction to Boolean Algebras", Springer, 2009.##[9] Goldblatt, R. and Hodkinson I., The finite model property for logics with the tangle modality, Submitted.##[10] Goldblatt, R. and Hodkinson I., Spatial logic of modal mucalculus and tangled closure operators, arxiv.org/abs/1603.01766, 2016.##[11] Goldblatt, R. and Hodkinson I., Spatial logic of tangled closure operators and modal mucalculus, Ann. Pure Appl. Logic, Available online, Nov. 2016: http://dx.doi.org/10.1016/j.apal.2016.11.006.##[12] Goldblatt, R. and Hodkinson, I., The tangled derivative logic of the real line and zerodimensional spaces, In "Advances in Modal Logic" Volume 11, Lev Beklemishev, Stephane Demri, and Andras Mate, editors, College Publications, 2016, 342361.##[13] Johnstone P., Elements of the history of locale theory, In ``Handbook of the History of General Topology" Volume 3, C.E. Aull and R. Lowen, editors, Kluwer Academic Publishers, 2001, 835851.##[14] Jonsson, B. and Tarski, A., Boolean algebras with operators, I, Amer. J. Math. 73 (1951), 891939.##[15] Kuratowski, C., Sur l'operation A de l'Analysis Situs, Fund. Math. 3 (1922), 182199.##[16] MacNeille, H.M., Partially ordered sets, Trans. Amer. Math. Soc. 42 (1937), 416460.##[17] McKinsey, J.C.C. and Tarski, A., The algebra of topology, Ann. Math. 45 (1944), 141191.##[18] McKinsey, J.C.C. and Tarski, A., On closed elements in closure algebras, Ann. Math. 47 (1946), 122162.##[19] Monk, J.D., Completions of Boolean algebras with operators, Math. Nachr. 46 (1970), 4755.##[20] Rasiowa, H., Algebraic treatment of the functional calculi of Heyting and Lewis, Fund. Math. 38 (1951), 99126.##[21] Rasiowa, H. and Sikorski, R., "The Mathematics of Metamathematics", PWN{Polish Scientific Publishers, Warsaw, 1963.##[22] Sikorski, R., "Boolean Algebras", SpringerVerlag, Berlin, 1964.##[23] Tarski, A., Der Aussagenkalkul und die Topologie, Fund. Math. 31 (1938), 103134. English translation by J.H. Woodger as Sentential calculus and topology in [24], 421454.##[24] Tarski, A., Logic, Semantics, Metamathematics: Papers from 1923 to 1938", Oxford University Press, 1956. Translated into English and edited by J.H. Woodger.##[25] Theunissen, M. and Venema, Y., MacNeille completions of lattice expansions, Algebra Universalis 57 (2007), 143193.##[26] Van Benthem, J.F.A.K., "Modal correspondence theory", PhD Thesis, University of Amsterdam, 1976.##[27] Van Benthem, J.F.A.K., "Modal Logic and Classical Logic", Bibliopolis, Naples, 1983.##]
Some Types of Filters in Equality Algebras
2
2
Equality algebras were introduced by S. Jenei as a possible algebraic semantic for fuzzy type theory. In this paper, we introduce some types of filters such as (positive) implicative, fantastic, Boolean, and prime filters in equality algebras and we prove some results which determine the relation between these filters. We prove that the quotient equality algebra induced by an implicative filter is a Boolean algebra, by a fantastic filter is a commutative equality algebra, and by a prime filter is a chain, under suitable conditions. Finally, we show that positive implicative, implicative, and Boolean filters are equivalent on bounded commutative equality algebras.
1

33
55


Rajabali
Borzooei
Department of Mathematics, Shahid Beheshti University, Tehran, Iran.
Department of Mathematics, Shahid Beheshti
Iran
borzooei@sbu.ac.ir


Fateme
Zebardast
Department of Mathematics, Payam e Noor University, Tehran, Iran.
Department of Mathematics, Payam e Noor University
Iran
zebardastfathme@gmail.com


Mona
Aaly Kologani
Payam e Noor University
Payam e Noor University
Iran
mona4011@gmail.com
Equality algebra
(positive) implicative filter
fantastic filter
Boolean filter
[[1] Borzooei, R.A., Khosravi Shoar, S., and Ameri, R., Some types of filters in MTLalgebras, Fuzzy Sets and Systems 187(1) (2012), 92102.##[2] Cignoli, R., D'ottaviano, I., and Mundici, D., "Algebraic Foundations of ManyValued Reasoning", Springer, Trends in Logic 7, 2000.##[3] Ciungu, L.C., Internal states on equality algebras, Soft Computing 19(4) (2015), 939953.##[4] Esteva, F. and Godo, L., Monoidal tnormbased logic: towards a logic for leftcontinuous tnorms, Fuzzy Sets and Systems, 124(3) (2001), 271288.##[5] H'ajek, P., "Metamathematics of Fuzzy Logic", Springer, Trends in Logic 4, 1998.##[6] Haveshki, M., Borumand Saeid, A., and Eslami, E., Some types of filters in BL algebras, Soft Computing 10(8) (2006), 657664.##[7] Jenei, S., Equality algebras, Studia Logica 100(6) (2012), 12011209.##[8] Jenei, S. and K'or'odi, L., On the variety of equality algebras, Fuzzy Logic and Technology (2011), 153155.##[9] Liu, L., On the existence of states on MTLalgebras, Inform. Sci. 220 (2013), 559567.##[10] Liu, L. and Zhang, X., Implicative and positive implicative prefilters of EQalgebras, J. Intell. Fuzzy Systems 26(5) (2014), 20872097.##[11] Nov'ak, V. and De Baets, B., EQalgebras, Fuzzy Sets and Systems 160(20) (2009), 29562978.##[12] Rezaei, A., Borumand Saeid, A., and Borzooei, R.A., Some types of filters in BEalgebras, Math. Comput. Sci. 7(3) (2013), 341352.##[13] Van Gasse, B., Deschrijver, G., Cornelis, C., and Kerre, E.E., Filters of residuated lattices and triangle algebras, Inform. Sci. 180(16) (2010), 30063020.##[14] Zebardast, F., Borzooei, R.A., and Aaly Kologani, M., Results on Equality algebras, Inform. Sci. 381 (2017), 270282.##]
Onepoint compactifications and continuity for partial frames
2
2
Locally compact Hausdorff spaces and their onepoint compactifications are much used in topology and analysis; in lattice and domain theory, the notion of continuity captures the idea of local compactness. Our work is located in the setting of pointfree topology, where latticetheoretic methods can be used to obtain topological results.Specifically, we examine here the concept of continuity for partial frames, and compactifications of regular continuous such.Partial frames are meetsemilattices in which not all subsets need have joins.A distinguishing feature of their study is that a small collection of axioms of an elementary nature allows one to do much that is traditional for frames or locales. The axioms are sufficiently general to include as examples $sigma$frames, $kappa$frames and frames.In this paper, we present the notion of a continuous partial frame by means of a suitable ``waybelow'' relation; in the regular case this relation can be characterized using separating elements, thus avoiding any use of pseudocomplements (which need not exist in a partial frame). Our first main result is an explicit construction of a onepoint compactification for a regular continuous partial frame using generators and relations. We use strong inclusions to link continuity and onepoint compactifications to least compactifications. As an application, we show that a onepoint compactification of a zerodimensional continuous partial frame is again zerodimensional. We next consider arbitrary compactifications of regular continuous partial frames. In full frames, the natural tools to use are right and left adjoints of frame maps; in partial frames these are, in general, not available. This necessitates significantly different techniques to obtain largest and smallest elements of fibres (which we call balloons); these elements are then used to investigate the structure of the compactifications. We note that strongly regular ideals play an important r^{o}le here. The paper concludes with a proof of the uniqueness of the onepoint compactification.
1

57
88


John
Frith
Department of Mathematics and Applied Mathematics, University of Cape Town, Private Bag Rondebosch, 7701,
South Africa.
Department of Mathematics and Applied Mathematics,
Iran
john.frith@uct.ac.za


Anneliese
Schauerte
Department of Mathematics and Applied Mathematics, University of Cape Town, Private Bag Rondebosch, 7701, South Africa.
Department of Mathematics and Applied Mathematics,
Iran
anneliese.schauerte@uct.ac.za
Frame
partial frame
$sels$frame
$kappa$frame
$sigma$frame
$mathcal{Z}$frame
compactification
onepoint compactification
strong inclusion
strongly regular ideal
continuous lattice
locally compact
[[1] Adamek, J., Herrlich, H., and Strecker, G., "Abstract and Concrete Categories", John Wiley & Sons Inc., New York, 1990.##[2] Baboolal, D., Conditions under which the least compactification of a regular continuous frame is perfect, Czechoslovak Math. J. 62(137) (2012), 505515.##[3] Baboolal, D., Nstar compactifications of frames, Topology Appl. 168 (2014), 815.##[4] Banaschewski, B., The duality of distributive $sigma$continuous lattices, in: "Continuous lattices", Lecture Notes in Math. 871 (1981), 1219.##[5] Banaschewski, B., Compactification of frames, Math. Nachr. 149 (1990), 105116.##[6] Banaschewski, B., $sigma$frames, unpublished manuscript, 1980. Available at http://mathcs.chapman.edu/CECAT/members/Banaschewski publications##[7] Banaschewski B. and Gilmour, C.R.A., StoneCech compactification and dimension theory for regular $sigma$Frames, J. London Math. Soc. 39(2) (1989), 18.##[8] Banaschewski, B. and Gilmour, C.R.A., Realcompactness and the cozero part of a frame, Appl. Categ. Structures 9 (2001), 395417.##[9] Banaschewski, B. and Gilmour, C.R.A., Cozero bases of frames, J. Pure Appl. Algebra 157 (2001), 122.##[10] Banaschewski, B. and Matutu, P., Remarks on the frame envelope of a $sigma$frame, J. Pure Appl. Algebra 177(3) (2003), 231236.##[11] Erne, M. and Zhao, D., Zjoin spectra of Zsupercompactly generated lattices, Appl. Categ. Structures 9(1) (2001), 4163.##[12] Frith, J. and Schauerte, A., The Samuel compactification for quasiuniform biframes, Topology Appl. 156 (2009), 21162122.##[13] Frith, J. and Schauerte, A., Uniformities and covering properties for partial frames (I), Categ. General Alg. Struct. Appl. 2(1) (2014), 121.##[14] Frith, J. and Schauerte, A., Uniformities and covering properties for partial frames (II), Categ. General Alg. Struct. Appl. 2(1) (2014), 2335.##[15] Frith, J. and Schauerte, A., The StoneCech compactification of a partial frame via ideals and cozero elements, Quaest. Math. 39(1) (2016), 115134.##[16] Frith, J. and Schauerte, A., Completions of uniform partial frames, Acta Math. Hungar. 147(1) (2015), 116134.##[17] Frith, J. and Schauerte, A., Coverages give free constructions for partial frames, Appl. Categ. Structures, Available online (2015), DOI: 10.1007/s1048501594178##[18] Frith, J. and Schauerte, A., Compactifications of partial frames via strongly regular ideals, Math. Slovaca, accepted June 2016.##[19] Gutierrez Garcia, J., Mozo Carollo, I., and Picado, J., Presenting the frame of the unit circle, J. Pure and Appl. Algebra 220(3) (2016), 9761001.##[20] Johnstone, P.T., "Stone Spaces", Cambridge University Press, 1982.##[21] Lee, S.O., Countably approximating frames, Commun. Korean Math. Soc. 17(2) (2002), 295308.##[22] Mac Lane, S., "Categories for the Working Mathematician", SpringerVerlag, 1971.##[23] Madden, J.J., frames, J. Pure Appl Algebra 70 (1991), 107127.##[24] Paseka, J., Covers in generalized frames, in: "General Algebra and Ordered Sets" (Horni Lipova 1994), Palacky Univ. Olomouc, Olomouc, 8499.##[25] Paseka, J. and Smarda, B., On some notions related to compactness for locales, Acta Univ. Carolin. Math. Phys. 29(2) (1988), 5165.##[26] Picado, J. and Pultr, A., "Frames and Locales", Springer, 2012.##[27] Walters, J.L., Compactifications and uniformities on $sigma$frames, Comment. Math. Univ. Carolin. 32(1) (1991), 189198.##[28] Zenk, E.R., Categories of partial frames, Algebra Universalis 54 (2005), 213235.##[29] Zhao, D., Nuclei on Zframes, Soochow J. Math. 22(1) (1996), 5974.##[30] Zhao, D., On Projective Zframes, Canad. Math. Bull. 40(1) (1997), 3946.##[31] Zhao, D., Closure spaces and completions of posets, Semigroup Forum 90(2) (2015), 545555.##]
Adjoint relations for the category of local dcpos
2
2
In this paper, we consider the forgetful functor from the category {bf LDcpo} of local dcpos (respectively, {bf Dcpo} of dcpos) to the category {bf Pos} of posets (respectively, {bf LDcpo} of local dcpos), and study the existence of its left and right adjoints. Moreover, we give the concrete forms of free and cofree $S$ldcpos over a local dcpo, where $S$ is a local dcpo monoid. The main results are: (1) The forgetful functor $U$ : {bf LDcpo} $longrightarrow$ {bf Pos} has a left adjoint, but does not have a right adjoint;(2) The inclusion functor $I$ : {bf Dcpo} $longrightarrow$ {bf LDcpo} has a left adjoint, but does not have a right adjoint;(3) The forgetful functor $U$ : {bf LDcpo}$S$ $longrightarrow$ {bf LDcpo} hasboth left and right adjoints;(4) If $(S,cdot,1)$ is a good ldcpomonoid, then the forgetful functor $U$: {bf LDcpo}$S$ $longrightarrow$ {bf Pos}$S$ has a left adjoint.
1

89
105


Bin
Zhao
Shaanxi Normal University
Shaanxi Normal University
Iran
zhaobin@snnu.edu.cn


Jing
Lu
Shaanxi Normal University
Shaanxi Normal University
Iran
lujing0926@126.com


Kaiyun
Wang
Shaanxi Normal University
Shaanxi Normal University
Iran
wangkaiyun@snnu.edu.cn
Dcpo
local dcpo
$S$ldcpo
forgetful functor
[[1] Adamek, J., Herrlich, H., and Strecker, G.E., "Abstract and Concrete Categories: The Joy of Cats", John Wiley & Sons, New York, 1990.##[2] BulmanFleming, S. and Mahmoudi, M., The category of Sposets, Semigroup Forum 71 (2005), 443461.##[3] Crole, R.L., "Categories for Types", Cambridge University Press, Cambridge, 1994.##[4] Erne, M., Minimal bases, ideal extensions, and basic dualities, Topology Proc. 29 (2005), 445489.##[5] 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.##[6] Keimel, K. and Lawson, J.D., Dcompletions and the dtopology, Ann. Pure Appl. Logic 159(3) (2009), 292306.##[7] Mahmoudi, M. and Moghbeli, H., Free and cofree acts of dcpomonoids on directed complete posets, Bull. Malaysian Math. Sci. Soc. 39 (2016), 589603.##[8] Mislove, M.W., Local dcpos, local cpos, and local completions, Electron. Notes Theor. Comput. Sci. 20 (1999), 399412. ##[9] Xu, L. and Mao, X., Strongly continuous posets and the local Scott topology, J. Math. Anal. Appl. 345 (2008), 816824.##[10] Zhao, D. and Fan, T., Dcpocompletion of posets, Theoret. Comput. Sci. 411 (2010), 21672173.##[11] Zhao, D., Partial dcpo's and some applications, Arch. Math. (Brno) 48 (2012), 243260.##]
Filters of Coz(X)
2
2
In this article we investigate filters of cozero sets for realvalued continuous functions, called $coz$filters. Much is known for $z$ultrafilters and their correspondence with maximal ideals of $C(X)$. Similarly, a correspondence will be established between $coz$ultrafilters and minimal prime ideals of $C(X)$. We will further notice various properties of $coz$ultrafilters in relation to $P$spaces and $F$spaces. In the last two sections, the collection of $coz$ultrafilters will be topologized, and then compared to the hullkernel and the inverse topologies placed on the collection of minimal prime ideals of $C(X)$ and general latticeordered groups.
1

107
123


Papiya
Bhattacharjee
School of Science, Penn State Behrend, Erie, PA 16563, USA.
School of Science, Penn State Behrend, Erie,
Iran
pxb39@psu.edu


Kevin
M. Drees
Department of Mathematics and Information Technology, Mercyhurst University, Erie PA.
Department of Mathematics and Information
Iran
kevin.drees@gmail.com
Cozero sets
ultrafilters
minimal prime ideals
$P$space
$F$space
inverse topology
$ell$groups
[[1] Atiyah, M. and MacDonald, I., "Introduction to Commutative Algebra", AddisonWesley Publishing Co., 1969.##[2] Bhattacharjee, P. and McGovern, W., Lamron `groups, in preparation, 2017.##[3] Bhattacharjee, P. and McGovern, W., When Min(A)1 is Hausdorf, Comm. Algebra 41(1) (2013), 99108.##[4] Brooks, R., On Wallman compactification, Fund. Math. 60 (1967), 157173.##[5] Conrad, P. and Martinez, J., Complemented latticeordered groups, Indag. Math. (N.S.) 1(3) (1990), 281297.##[6] Darnel, M., "Theory of LatticeOrdered Groups", Marcel Dekker, 1995.##[7] Dashiell, F., Hager, A., and Henriksen, M., OrderCauchy completions of rings and vector lattices of continuous functions, Canad. J. Math XXXII(3) (1980), 657685.##[8] Engelking, R., "General Topology", Helderman Verlag, 1989.##[9] Fine, N., Gilman, L., and Lambek, J., "Rings of Quotients of Rings of Functions", McGill University Press, 1966.##[10] Gillman, L. and Henriksen, M., Rings of continuous functions in which every finitely generated ideal is principal, Trans. Amer. Math. Soc. 82(2) (1956), 366391.##[11] Gillman, L. and Jerison, M., "Rings of Continuous Functions", D. Van Nostrand Co., 1960.##[12] Henriksen, M. and Jerison, M., The space of minimal prime ideals of a commutative ring, Trans. Amer. Math. Soc. 115 (1965), 110130.##[13] Huckaba, J., Commutative Rings with Zero Divisors, Marcel Dekker, 1988.##[14] Kaplansky, I., "Commutative Rings (Revised ed.)", University of Chicago Press, 1974.##[15] Knox, M., Levy, R., McGovern, W., and Shapiro, J., Generalizations of complemented rings with applications to rings of functions, J. Algebra Appl. 7(6) (2008), 124.##[16] Lang, S., "Algebra", Springer, 2002.##[17] McGovern, W., Neat rings, J. Pure Appl. Algebra 205(2) (2006), 243265.##[18] Samuel, P., Ultrafilters and compactification of uniform spaces, Trans. Amer. Math. Soc. 64 (1948), 100132.##[19] Wallman, H., Lattice and topological spaces, Ann. of Math. 39(2) (1938), 112126.##]
Perfect secure domination in graphs
2
2
Let $G=(V,E)$ be a graph. A subset $S$ of $V$ is a dominating set of $G$ if every vertex in $Vsetminus S$ is adjacent to a vertex in $S.$ A dominating set $S$ is called a secure dominating set if for each $vin Vsetminus S$ there exists $uin S$ such that $v$ is adjacent to $u$ and $S_1=(Ssetminus{u})cup {v}$ is a dominating set. If further the vertex $uin S$ is unique, then $S$ is called a perfect secure dominating set. The minimum cardinality of a perfect secure dominating set of $G$ is called the perfect secure domination number of $G$ and is denoted by $gamma_{ps}(G).$ In this paper we initiate a study of this parameter and present several basic results.
1

125
140


S.V. Divya
Rashmi
Department of Mathematics, Vidyavardhaka College of Engineering, Mysuru 570002, Karnataka, India.
Department of Mathematics, Vidyavardhaka
Iran
rashmi.divya@gmail.com


Subramanian
Arumugam
National Centre for Advanced Research in Discrete Mathematics, Kalasalingam University, Anand Nagar, Krishnankoil626 126, Tamil Nadu, India.
National Centre for Advanced Research in
Iran
s.arumugam.klu@gmail.com


Kiran R.
Bhutani
Department of Mathematics, The Catholic University of America, Washington, D.C. 20064, USA.
Department of Mathematics, The Catholic University
Iran
bhutani@cua.edu


Peter
Gartland
Department of Mathematics, The Catholic University of America, Washington, D.C. 20064, USA.
Department of Mathematics, The Catholic University
Iran
56gartland@cardinalmail.cua.edu
Secure domination
perfect secure domination
secure domination number
perfect secure domination number
[[1] Burger, A.P., Cockayne, E.J., Grundlingh, W.R., Mynhardt, C.M., van Vuuren, J.H., and Winterbach, W., Finite order domination in graph, J. Combin. Math. Combin. Comput. 49 (2004), 159175.##[2] Burger, A.P., Cockayne, E.J., Grundlingh, W.R., Mynhardt, C.M., van Vuuren, J.H., and Winterbach, W., Infinite order domination in graphs, J. Combin. Math. Combin. Comput. 50 (2004), 179194.##[3] Burger, A.P., Henning, M.A., and van Vuuren, J.H., Vertex covers and secure domination in graphs, Quaest. Math. 31 (2008), 163171.##[4] Chartrand, G., Lesniak, L., and Zhang, P., "Graphs & Digraphs", Fourth Edition, Chapman and Hall/CRC, 2005.##[5] Cockayne, E.J., Irredundance, secure domination and maximum degree in trees, Discrete Math. 307 (2007), 1217.##[6] Cockayne, E.J., Favaron, O., and Mynhardt, C.M., Secure domination, weak Roman domination and forbidden subgraph, Bull. Inst. Combin. Appl. 39 (2003), 87100.##[7] Cockayne, E.J., Grobler, P.J.P., Grundlingh, W.R., Munganga, J., and van Vuuren, J.H., Protection of a graph, Util. Math. 67 (2005), 1932.##[8] Haynes, T.W., Hedetniemi, S.T., and Slater, P.J., "Fundamentals of Domination in Graphs", Marcel Dekker, Inc. New York, 1998.##[9] Henning, M.A., and Hedetniemi, S.M., Defending the Roman empireA new stategy, Discrete Math. 266 (2003), 239251.##[10] Mynhardt, C.M., Swart, H.C., and Ungerer, E., Excellent trees and secure domination, Util. Math. 67 (2005), 255267.##[11] Weichsel, P.M., Dominating sets in ncubes, J. Graph Theory 18(5) (1994), 479488.##]
$mathcal{R}L$ valued $f$ring homomorphisms and latticevalued maps
2
2
In this paper, for each {it latticevalued map} $Arightarrow L$ with some properties, a ring representation $Arightarrow mathcal{R}L$ is constructed. This representation is denoted by $tau_c$ which is an $f$ring homomorphism and a $mathbb Q$linear map, where its index $c$, mentions to a latticevalued map. We use the notation $delta_{pq}^{a}=(a p)^{+}wedge (qa)^{+}$, where $p, qin mathbb Q$ and $ain A$, that is nominated as {it interval projection}. To get a welldefined $f$ring homomorphism $tau_c$, we need such concepts as {it bounded}, {it continuous}, and $mathbb Q${it compatible} for $c$, which are defined and some related results are investigated. On the contrary, we present a cozero latticevalued map $c_{phi}:Arightarrow L $ for each $f$ring homomorphism $phi: Arightarrow mathcal{R}L$. It is proved that $c_{tau_c}=c^r$ and $tau_{c_{phi}}=phi$, which they make a kind of correspondence relation between ring representations $Arightarrow mathcal{R}L$ and the latticevalued maps $Arightarrow L$, Where the mapping $c^r:Arightarrow L$ is called a {it realization} of $c$. It is shown that $tau_{c^r}=tau_c$ and $c^{rr}=c^r$. Finally, we describe how $tau_c$ can be a fundamental tool to extend pointfree version of Gelfand duality constructed by B. Banaschewski.
1

141
163


Abolghasem
Karimi Feizabadi
Department of Mathematics, Gorgan Branch, Islamic Azad University, Gorgan, Iran.
Department of Mathematics, Gorgan Branch,
Iran
akarimi@gorganiau.ac.ir


Ali Akbar
Estaji
Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.
Faculty of Mathematics and Computer Sciences,
Iran
aaestaji@hsu.ac.ir


Batool
Emamverdi
Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.
Faculty of Mathematics and Computer Sciences,
Iran
emamverdi55@yahoo.com
Frame
cozero latticevalued map
strong $f$ring
interval projection
bounded
continuous
$mathbb{Q}$compatible
cozcompatible
[[1] Banaschewski, B., Pointfree topology and the spectra of frings, Ordered algebraic structures (Curacao, 1995), Kluwer Acad. Publ., Dordrecht, (1997), 123148.##[2] Banaschewski, B., The real numbers in pointfree topology, Texts in Mathematics (Series B), 12, University of Coimbra, 1997.##[3] Bigard, A., K. Keimel, and S. Wolfenstein, Groups et anneaux reticules, Lecture Notes in Math. 608, SpringerVerlag, 1977.##[4] Ebrahimi, M.M. and A. Karimi Feizabadi, Pointfree prime representation of real Riesz maps, Algebra Universalis 54 (2005), 291299.##[5] Gillman, L. and M. Jerison, "Rings of Continuous Function", Graduate Texts in Mathematics 43, SpringerVerlag, 1979.##[6] Karimi Feizabadi, A., Representation of slim algebraic regular cozero maps, Quaest. Math. 29 (2006), 383394.##[7] Karimi Feizabadi, A., Free latticevalued functions, reticulation of rings and modules, submitted.##[8] Picado, J. and A. Pultr, "Frames and Locales: Topology without Points", Frontiers in Mathematics 28, Springer, Basel, 2012.##]
The projectable hull of an archimedean $ell$group with weak unit
2
2
The muchstudied projectable hull of an $ell$group $Gleq pG$ is an essential extension, so that, in the case that $G$ is archimedean with weak unit, ``$Gin {bf W}$", we have for the Yosida representation spaces a ``covering map" $YG leftarrow YpG$. We have earlier cite{hkm2} shown that (1) this cover has a characteristic minimality property, and that (2) knowing $YpG$, one can write down $pG$. We now show directly that for $mathscr{A}$, the boolean algebra in the power set of the minimal prime spectrum $Min(G)$, generated by the sets $U(g)={Pin Min(G):gnotin P}$ ($gin G$), the Stone space $mathcal{A}mathscr{A}$ is a cover of $YG$ with the minimal property of (1); this extends the result from cite{bmmp} for the strong unit case. Then, applying (2) gives the preexisting description of $pG$, which includes the strong unit description of cite{bmmp}. The present methods are largely topological, involving details of covering maps and Stone duality.
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Anthony W.
Hager
Department of Mathematics and CS, Wesleyan University, Middletown, CT 06459.
Department of Mathematics and CS, Wesleyan
Iran
ahager@wesleyan.edu


Warren Wm.
McGovern
H. L. Wilkes Honors College, Florida Atlantic University, Jupiter, FL 33458.
H. L. Wilkes Honors College, Florida Atlantic
Iran
warren.mcgovern@fau.edu
Archimedean $l$group
vector lattice
Yosida representation
minimal prime spectrum
principal polar
projectable
principal projection property
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