ORIGINAL_ARTICLE
Tangled Closure Algebras
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 McKinsey-Tarski 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.
http://cgasa.sbu.ac.ir/article_42354_09def3b31ada32383d6d12c9644168af.pdf
2017-07-01T11:23:20
2020-10-24T11:23:20
9
31
Closure algebra
tangled closure
tangle modality
fixed point
quasi-order
Alexandroff topology
dense-in-itself
dissectable
MacNeille completion
Robert
Goldblatt
rob.goldblatt@msor.vuw.ac.nz
true
1
School of Mathematics and Statistics, Victoria University of Wellington, New Zealand
School of Mathematics and Statistics, Victoria University of Wellington, New Zealand
School of Mathematics and Statistics, Victoria University of Wellington, New Zealand
AUTHOR
Ian
Hodkinson
i.hodkinson@imperial.ac.uk
true
2
Department of Computing, Imperial College London, UK.
Department of Computing, Imperial College London, UK.
Department of Computing, Imperial College London, UK.
AUTHOR
[1] Banaschewski, B., Hullensysteme und Erweiterung von Quasi-Ordnungen, Z. Math. Log. Grundlagen Math. 2 (1956), 117-130.
1
[2] Banaschewski, B. and Bruns, G., Categorical characterization of the MacNeille completion, Arch. Math. 18 (1967), 369-377.
2
[3] Davey, B.A. and Priestley, H.A., "Introduction to Lattices and Order", Cambridge University Press, 1990.
3
[4] Dawar, A. and Otto, M., Modal characterisation theorems over special classes of frames, Ann. Pure Appl. Logic 161 (2009), 1-42.
4
[5] Dummett, M.A.E. and Lemmon, E.J., Modal logics between S4 and S5, Z. Math. Log. Grundlagen Math., 5 (1959), 250-264.
5
[6] Fernandez-Duque, D., Tangled modal logic for spatial reasoning, In Toby Walsh, editor, Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence (IJCAI), AAAI Press/IJCAI (2011), 857-862.
6
[7] Fernandez-Duque, D., Tangled modal logic for topological dynamics, Ann. Pure Appl. Logic 163 (2012), 467-481.
7
[8] Givant, S. and Halmos, P., "Introduction to Boolean Algebras", Springer, 2009.
8
[9] Goldblatt, R. and Hodkinson I., The finite model property for logics with the tangle modality, Submitted.
9
[10] Goldblatt, R. and Hodkinson I., Spatial logic of modal mu-calculus and tangled closure operators, arxiv.org/abs/1603.01766, 2016.
10
[11] Goldblatt, R. and Hodkinson I., Spatial logic of tangled closure operators and modal mu-calculus, Ann. Pure Appl. Logic, Available online, Nov. 2016: http://dx.doi.org/10.1016/j.apal.2016.11.006.
11
[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, 342-361.
12
[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, 835-851.
13
[14] Jonsson, B. and Tarski, A., Boolean algebras with operators, I, Amer. J. Math. 73 (1951), 891-939.
14
[15] Kuratowski, C., Sur l'operation A de l'Analysis Situs, Fund. Math. 3 (1922), 182-199.
15
[16] MacNeille, H.M., Partially ordered sets, Trans. Amer. Math. Soc. 42 (1937), 416-460.
16
[17] McKinsey, J.C.C. and Tarski, A., The algebra of topology, Ann. Math. 45 (1944), 141-191.
17
[18] McKinsey, J.C.C. and Tarski, A., On closed elements in closure algebras, Ann. Math. 47 (1946), 122-162.
18
[19] Monk, J.D., Completions of Boolean algebras with operators, Math. Nachr. 46 (1970), 47-55.
19
[20] Rasiowa, H., Algebraic treatment of the functional calculi of Heyting and Lewis, Fund. Math. 38 (1951), 99-126.
20
[21] Rasiowa, H. and Sikorski, R., "The Mathematics of Metamathematics", PWN{Polish Scientific Publishers, Warsaw, 1963.
21
[22] Sikorski, R., "Boolean Algebras", Springer-Verlag, Berlin, 1964.
22
[23] Tarski, A., Der Aussagenkalkul und die Topologie, Fund. Math. 31 (1938), 103-134. English translation by J.H. Woodger as Sentential calculus and topology in [24], 421-454.
23
[24] Tarski, A., Logic, Semantics, Metamathematics: Papers from 1923 to 1938", Oxford University Press, 1956. Translated into English and edited by J.H. Woodger.
24
[25] Theunissen, M. and Venema, Y., MacNeille completions of lattice expansions, Algebra Universalis 57 (2007), 143-193.
25
[26] Van Benthem, J.F.A.K., "Modal correspondence theory", PhD Thesis, University of Amsterdam, 1976.
26
[27] Van Benthem, J.F.A.K., "Modal Logic and Classical Logic", Bibliopolis, Naples, 1983.
27
ORIGINAL_ARTICLE
Some Types of Filters in Equality Algebras
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.
http://cgasa.sbu.ac.ir/article_42342_cf5624efc3f4dd8d61c28cc7af659734.pdf
2017-07-01T11:23:20
2020-10-24T11:23:20
33
55
Equality algebra
(positive) implicative filter
fantastic filter
Boolean filter
Rajabali
Borzooei
borzooei@sbu.ac.ir
true
1
Department of Mathematics, Shahid Beheshti University, Tehran, Iran.
Department of Mathematics, Shahid Beheshti University, Tehran, Iran.
Department of Mathematics, Shahid Beheshti University, Tehran, Iran.
AUTHOR
Fateme
Zebardast
zebardastfathme@gmail.com
true
2
Department of Mathematics, Payam e Noor University, Tehran, Iran.
Department of Mathematics, Payam e Noor University, Tehran, Iran.
Department of Mathematics, Payam e Noor University, Tehran, Iran.
AUTHOR
Mona
Aaly Kologani
mona4011@gmail.com
true
3
Payam e Noor University
Payam e Noor University
Payam e Noor University
AUTHOR
[1] Borzooei, R.A., Khosravi Shoar, S., and Ameri, R., Some types of filters in MTL-algebras, Fuzzy Sets and Systems 187(1) (2012), 92-102.
1
[2] Cignoli, R., D'ottaviano, I., and Mundici, D., "Algebraic Foundations of Many-Valued Reasoning", Springer, Trends in Logic 7, 2000.
2
[3] Ciungu, L.C., Internal states on equality algebras, Soft Computing 19(4) (2015), 939-953.
3
[4] Esteva, F. and Godo, L., Monoidal t-normbased logic: towards a logic for left-continuous t-norms, Fuzzy Sets and Systems, 124(3) (2001), 271-288.
4
[5] H'ajek, P., "Metamathematics of Fuzzy Logic", Springer, Trends in Logic 4, 1998.
5
[6] Haveshki, M., Borumand Saeid, A., and Eslami, E., Some types of filters in BL algebras, Soft Computing 10(8) (2006), 657-664.
6
[7] Jenei, S., Equality algebras, Studia Logica 100(6) (2012), 1201-1209.
7
[8] Jenei, S. and K'or'odi, L., On the variety of equality algebras, Fuzzy Logic and Technology (2011), 153-155.
8
[9] Liu, L., On the existence of states on MTL-algebras, Inform. Sci. 220 (2013), 559-567.
9
[10] Liu, L. and Zhang, X., Implicative and positive implicative prefilters of EQ-algebras, J. Intell. Fuzzy Systems 26(5) (2014), 2087-2097.
10
[11] Nov'ak, V. and De Baets, B., EQ-algebras, Fuzzy Sets and Systems 160(20) (2009), 2956-2978.
11
[12] Rezaei, A., Borumand Saeid, A., and Borzooei, R.A., Some types of filters in BE-algebras, Math. Comput. Sci. 7(3) (2013), 341-352.
12
[13] Van Gasse, B., Deschrijver, G., Cornelis, C., and Kerre, E.E., Filters of residuated lattices and triangle algebras, Inform. Sci. 180(16) (2010), 3006-3020.
13
[14] Zebardast, F., Borzooei, R.A., and Aaly Kologani, M., Results on Equality algebras, Inform. Sci. 381 (2017), 270-282.
14
ORIGINAL_ARTICLE
One-point compactifications and continuity for partial frames
Locally compact Hausdorff spaces and their one-point 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 lattice-theoretic 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 meet-semilattices 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 ``way-below'' 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 one-point compactification for a regular continuous partial frame using generators and relations. We use strong inclusions to link continuity and one-point compactifications to least compactifications. As an application, we show that a one-point compactification of a zero-dimensional continuous partial frame is again zero-dimensional. 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 one-point compactification.
http://cgasa.sbu.ac.ir/article_43180_02e474fcbfa63e236d1fbd237390dba8.pdf
2017-07-01T11:23:20
2020-10-24T11:23:20
57
88
frame
partial frame
$sels$-frame
$kappa$-frame
$sigma$-frame
$mathcal{Z}$-frame
compactification
one-point compactification
strong inclusion
strongly regular ideal
continuous lattice
locally compact
John
Frith
john.frith@uct.ac.za
true
1
Department of Mathematics and Applied Mathematics, University of Cape Town, Private Bag Rondebosch, 7701, South Africa.
Department of Mathematics and Applied Mathematics, University of Cape Town, Private Bag Rondebosch, 7701, South Africa.
Department of Mathematics and Applied Mathematics, University of Cape Town, Private Bag Rondebosch, 7701, South Africa.
AUTHOR
Anneliese
Schauerte
anneliese.schauerte@uct.ac.za
true
2
Department of Mathematics and Applied Mathematics, University of Cape Town, Private Bag Rondebosch, 7701, South Africa.
Department of Mathematics and Applied Mathematics, University of Cape Town, Private Bag Rondebosch, 7701, South Africa.
Department of Mathematics and Applied Mathematics, University of Cape Town, Private Bag Rondebosch, 7701, South Africa.
AUTHOR
[1] Adamek, J., Herrlich, H., and Strecker, G., "Abstract and Concrete Categories", John Wiley & Sons Inc., New York, 1990.
1
[2] Baboolal, D., Conditions under which the least compactification of a regular continuous frame is perfect, Czechoslovak Math. J. 62(137) (2012), 505-515.
2
[3] Baboolal, D., N-star compactifications of frames, Topology Appl. 168 (2014), 8-15.
3
[4] Banaschewski, B., The duality of distributive $sigma$-continuous lattices, in: "Continuous lattices", Lecture Notes in Math. 871 (1981), 12-19.
4
[5] Banaschewski, B., Compactification of frames, Math. Nachr. 149 (1990), 105-116.
5
[6] Banaschewski, B., $sigma$-frames, unpublished manuscript, 1980. Available at http://mathcs.chapman.edu/CECAT/members/Banaschewski publications
6
[7] Banaschewski B. and Gilmour, C.R.A., Stone-Cech compactification and dimension theory for regular $sigma$-Frames, J. London Math. Soc. 39(2) (1989), 1-8.
7
[8] Banaschewski, B. and Gilmour, C.R.A., Realcompactness and the cozero part of a frame, Appl. Categ. Structures 9 (2001), 395-417.
8
[9] Banaschewski, B. and Gilmour, C.R.A., Cozero bases of frames, J. Pure Appl. Algebra 157 (2001), 1-22.
9
[10] Banaschewski, B. and Matutu, P., Remarks on the frame envelope of a $sigma$-frame, J. Pure Appl. Algebra 177(3) (2003), 231-236.
10
[11] Erne, M. and Zhao, D., Z-join spectra of Z-supercompactly generated lattices, Appl. Categ. Structures 9(1) (2001), 41-63.
11
[12] Frith, J. and Schauerte, A., The Samuel compactification for quasi-uniform biframes, Topology Appl. 156 (2009), 2116-2122.
12
[13] Frith, J. and Schauerte, A., Uniformities and covering properties for partial frames (I), Categ. General Alg. Struct. Appl. 2(1) (2014), 1-21.
13
[14] Frith, J. and Schauerte, A., Uniformities and covering properties for partial frames (II), Categ. General Alg. Struct. Appl. 2(1) (2014), 23-35.
14
[15] Frith, J. and Schauerte, A., The Stone-Cech compactification of a partial frame via ideals and cozero elements, Quaest. Math. 39(1) (2016), 115-134.
15
[16] Frith, J. and Schauerte, A., Completions of uniform partial frames, Acta Math. Hungar. 147(1) (2015), 116-134.
16
[17] Frith, J. and Schauerte, A., Coverages give free constructions for partial frames, Appl. Categ. Structures, Available online (2015), DOI: 10.1007/s10485-015-9417-8
17
[18] Frith, J. and Schauerte, A., Compactifications of partial frames via strongly regular ideals, Math. Slovaca, accepted June 2016.
18
[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), 976-1001.
19
[20] Johnstone, P.T., "Stone Spaces", Cambridge University Press, 1982.
20
[21] Lee, S.O., Countably approximating frames, Commun. Korean Math. Soc. 17(2) (2002), 295-308.
21
[22] Mac Lane, S., "Categories for the Working Mathematician", Springer-Verlag, 1971.
22
[23] Madden, J.J., -frames, J. Pure Appl Algebra 70 (1991), 107-127.
23
[24] Paseka, J., Covers in generalized frames, in: "General Algebra and Ordered Sets" (Horni Lipova 1994), Palacky Univ. Olomouc, Olomouc, 84-99.
24
[25] Paseka, J. and Smarda, B., On some notions related to compactness for locales, Acta Univ. Carolin. Math. Phys. 29(2) (1988), 51-65.
25
[26] Picado, J. and Pultr, A., "Frames and Locales", Springer, 2012.
26
[27] Walters, J.L., Compactifications and uniformities on $sigma$-frames, Comment. Math. Univ. Carolin. 32(1) (1991), 189-198.
27
[28] Zenk, E.R., Categories of partial frames, Algebra Universalis 54 (2005), 213-235.
28
[29] Zhao, D., Nuclei on Z-frames, Soochow J. Math. 22(1) (1996), 59-74.
29
[30] Zhao, D., On Projective Z-frames, Canad. Math. Bull. 40(1) (1997), 39-46.
30
[31] Zhao, D., Closure spaces and completions of posets, Semigroup Forum 90(2) (2015), 545-555.
31
ORIGINAL_ARTICLE
Adjoint relations for the category of local dcpos
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 ldcpo-monoid, then the forgetful functor $U$: {\bf LDcpo}-$S$ $\longrightarrow$ {\bf Pos}-$S$ has a left adjoint.
http://cgasa.sbu.ac.ir/article_43374_e3ba4928af107559409d8a2f182b5716.pdf
2017-07-01T11:23:20
2020-10-24T11:23:20
89
105
Dcpo
local dcpo
$S$-ldcpo
forgetful functor
Bin
Zhao
zhaobin@snnu.edu.cn
true
1
Shaanxi Normal University
Shaanxi Normal University
Shaanxi Normal University
AUTHOR
Jing
Lu
lujing0926@126.com
true
2
Shaanxi Normal University
Shaanxi Normal University
Shaanxi Normal University
AUTHOR
Kaiyun
Wang
wangkaiyun@snnu.edu.cn
true
3
Shaanxi Normal University
Shaanxi Normal University
Shaanxi Normal University
LEAD_AUTHOR
[1] Adamek, J., Herrlich, H., and Strecker, G.E., "Abstract and Concrete Categories: The Joy of Cats", John Wiley & Sons, New York, 1990.
1
[2] Bulman-Fleming, S. and Mahmoudi, M., The category of S-posets, Semigroup Forum 71 (2005), 443-461.
2
[3] Crole, R.L., "Categories for Types", Cambridge University Press, Cambridge, 1994.
3
[4] Erne, M., Minimal bases, ideal extensions, and basic dualities, Topology Proc. 29 (2005), 445-489.
4
[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.
5
[6] Keimel, K. and Lawson, J.D., D-completions and the d-topology, Ann. Pure Appl. Logic 159(3) (2009), 292-306.
6
[7] Mahmoudi, M. and Moghbeli, H., Free and cofree acts of dcpo-monoids on directed complete posets, Bull. Malaysian Math. Sci. Soc. 39 (2016), 589-603.
7
[8] Mislove, M.W., Local dcpos, local cpos, and local completions, Electron. Notes Theor. Comput. Sci. 20 (1999), 399-412.
8
[9] Xu, L. and Mao, X., Strongly continuous posets and the local Scott topology, J. Math. Anal. Appl. 345 (2008), 816-824.
9
[10] Zhao, D. and Fan, T., Dcpo-completion of posets, Theoret. Comput. Sci. 411 (2010), 2167-2173.
10
[11] Zhao, D., Partial dcpo's and some applications, Arch. Math. (Brno) 48 (2012), 243-260.
11
ORIGINAL_ARTICLE
Filters of Coz(X)
In this article we investigate filters of cozero sets for real-valued 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 hull-kernel and the inverse topologies placed on the collection of minimal prime ideals of $C(X)$ and general lattice-ordered groups.
http://cgasa.sbu.ac.ir/article_44925_013d795e3961eac0b6b094ece513d81e.pdf
2017-07-01T11:23:20
2020-10-24T11:23:20
107
123
Cozero sets
ultrafilters
minimal prime ideals
$P$-space
$F$-space
inverse topology
$ell$-groups
Papiya
Bhattacharjee
pxb39@psu.edu
true
1
School of Science, Penn State Behrend, Erie, PA 16563, USA.
School of Science, Penn State Behrend, Erie, PA 16563, USA.
School of Science, Penn State Behrend, Erie, PA 16563, USA.
LEAD_AUTHOR
Kevin
M. Drees
kevin.drees@gmail.com
true
2
Department of Mathematics and Information Technology, Mercyhurst University, Erie PA.
Department of Mathematics and Information Technology, Mercyhurst University, Erie PA.
Department of Mathematics and Information Technology, Mercyhurst University, Erie PA.
AUTHOR
[1] Atiyah, M. and MacDonald, I., "Introduction to Commutative Algebra", Addison-Wesley Publishing Co., 1969.
1
[2] Bhattacharjee, P. and McGovern, W., Lamron `-groups, in preparation, 2017.
2
[3] Bhattacharjee, P. and McGovern, W., When Min(A)1 is Hausdorf, Comm. Algebra 41(1) (2013), 99-108.
3
[4] Brooks, R., On Wallman compactification, Fund. Math. 60 (1967), 157-173.
4
[5] Conrad, P. and Martinez, J., Complemented lattice-ordered groups, Indag. Math. (N.S.) 1(3) (1990), 281-297.
5
[6] Darnel, M., "Theory of Lattice-Ordered Groups", Marcel Dekker, 1995.
6
[7] Dashiell, F., Hager, A., and Henriksen, M., Order-Cauchy completions of rings and vector lattices of continuous functions, Canad. J. Math XXXII(3) (1980), 657-685.
7
[8] Engelking, R., "General Topology", Helderman Verlag, 1989.
8
[9] Fine, N., Gilman, L., and Lambek, J., "Rings of Quotients of Rings of Functions", McGill University Press, 1966.
9
[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), 366-391.
10
[11] Gillman, L. and Jerison, M., "Rings of Continuous Functions", D. Van Nostrand Co., 1960.
11
[12] Henriksen, M. and Jerison, M., The space of minimal prime ideals of a commutative ring, Trans. Amer. Math. Soc. 115 (1965), 110-130.
12
[13] Huckaba, J., Commutative Rings with Zero Divisors, Marcel Dekker, 1988.
13
[14] Kaplansky, I., "Commutative Rings (Revised ed.)", University of Chicago Press, 1974.
14
[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), 1-24.
15
[16] Lang, S., "Algebra", Springer, 2002.
16
[17] McGovern, W., Neat rings, J. Pure Appl. Algebra 205(2) (2006), 243-265.
17
[18] Samuel, P., Ultrafilters and compactification of uniform spaces, Trans. Amer. Math. Soc. 64 (1948), 100-132.
18
[19] Wallman, H., Lattice and topological spaces, Ann. of Math. 39(2) (1938), 112-126.
19
ORIGINAL_ARTICLE
Perfect secure domination in graphs
Let $G=(V,E)$ be a graph. A subset $S$ of $V$ is a dominating set of $G$ if every vertex in $V\setminus S$ is adjacent to a vertex in $S.$ A dominating set $S$ is called a secure dominating set if for each $v\in V\setminus S$ there exists $u\in S$ such that $v$ is adjacent to $u$ and $S_1=(S\setminus\{u\})\cup \{v\}$ is a dominating set. If further the vertex $u\in S$ 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.
http://cgasa.sbu.ac.ir/article_44926_4a0432bd29e2bbab421183f554f06243.pdf
2017-07-01T11:23:20
2020-10-24T11:23:20
125
140
Secure domination
perfect secure domination
secure domination number
perfect secure domination number
S.V. Divya
Rashmi
rashmi.divya@gmail.com
true
1
Department of Mathematics, Vidyavardhaka College of Engineering, Mysuru 570002, Karnataka, India.
Department of Mathematics, Vidyavardhaka College of Engineering, Mysuru 570002, Karnataka, India.
Department of Mathematics, Vidyavardhaka College of Engineering, Mysuru 570002, Karnataka, India.
AUTHOR
Subramanian
Arumugam
s.arumugam.klu@gmail.com
true
2
National Centre for Advanced Research in Discrete Mathematics, Kalasalingam University, Anand Nagar, Krishnankoil-626 126, Tamil Nadu, India.
National Centre for Advanced Research in Discrete Mathematics, Kalasalingam University, Anand Nagar, Krishnankoil-626 126, Tamil Nadu, India.
National Centre for Advanced Research in Discrete Mathematics, Kalasalingam University, Anand Nagar, Krishnankoil-626 126, Tamil Nadu, India.
AUTHOR
Kiran R.
Bhutani
bhutani@cua.edu
true
3
Department of Mathematics, The Catholic University of America, Washington, D.C. 20064, USA.
Department of Mathematics, The Catholic University of America, Washington, D.C. 20064, USA.
Department of Mathematics, The Catholic University of America, Washington, D.C. 20064, USA.
AUTHOR
Peter
Gartland
56gartland@cardinalmail.cua.edu
true
4
Department of Mathematics, The Catholic University of America, Washington, D.C. 20064, USA.
Department of Mathematics, The Catholic University of America, Washington, D.C. 20064, USA.
Department of Mathematics, The Catholic University of America, Washington, D.C. 20064, USA.
AUTHOR
[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), 159-175.
1
[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), 179-194.
2
[3] Burger, A.P., Henning, M.A., and van Vuuren, J.H., Vertex covers and secure domination in graphs, Quaest. Math. 31 (2008), 163-171.
3
[4] Chartrand, G., Lesniak, L., and Zhang, P., "Graphs & Digraphs", Fourth Edition, Chapman and Hall/CRC, 2005.
4
[5] Cockayne, E.J., Irredundance, secure domination and maximum degree in trees, Discrete Math. 307 (2007), 12-17.
5
[6] Cockayne, E.J., Favaron, O., and Mynhardt, C.M., Secure domination, weak Roman domination and forbidden subgraph, Bull. Inst. Combin. Appl. 39 (2003), 87-100.
6
[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), 19-32.
7
[8] Haynes, T.W., Hedetniemi, S.T., and Slater, P.J., "Fundamentals of Domination in Graphs", Marcel Dekker, Inc. New York, 1998.
8
[9] Henning, M.A., and Hedetniemi, S.M., Defending the Roman empire-A new stategy, Discrete Math. 266 (2003), 239-251.
9
[10] Mynhardt, C.M., Swart, H.C., and Ungerer, E., Excellent trees and secure domination, Util. Math. 67 (2005), 255-267.
10
[11] Weichsel, P.M., Dominating sets in n-cubes, J. Graph Theory 18(5) (1994), 479-488.
11
ORIGINAL_ARTICLE
$\mathcal{R}L$- valued $f$-ring homomorphisms and lattice-valued maps
In this paper, for each {\it lattice-valued map} $A\rightarrow L$ with some properties, a ring representation $A\rightarrow \mathcal{R}L$ 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 lattice-valued map. We use the notation $\delta_{pq}^{a}=(a -p)^{+}\wedge (q-a)^{+}$, where $p, q\in \mathbb Q$ and $a\in A$, that is nominated as {\it interval projection}. To get a well-defined $f$-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 lattice-valued map $c_{\phi}:A\rightarrow L $ for each $f$-ring homomorphism $\phi: A\rightarrow \mathcal{R}L$. 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 $A\rightarrow \mathcal{R}L$ and the lattice-valued maps $A\rightarrow L$, Where the mapping $c^r:A\rightarrow L$ 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.
http://cgasa.sbu.ac.ir/article_38548_d61135e6f18b53e9ac1eb29192263dbc.pdf
2017-07-01T11:23:20
2020-10-24T11:23:20
141
163
frame
cozero lattice-valued map
strong $f$-ring
interval projection
bounded
continuous
$mathbb{Q}$-compatible
coz-compatible
Abolghasem
Karimi Feizabadi
akarimi@gorganiau.ac.ir
true
1
Department of Mathematics, Gorgan Branch, Islamic Azad University, Gorgan, Iran.
Department of Mathematics, Gorgan Branch, Islamic Azad University, Gorgan, Iran.
Department of Mathematics, Gorgan Branch, Islamic Azad University, Gorgan, Iran.
AUTHOR
Ali Akbar
Estaji
aaestaji@hsu.ac.ir
true
2
Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.
Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.
Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.
AUTHOR
Batool
Emamverdi
emamverdi55@yahoo.com
true
3
Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.
Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.
Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.
AUTHOR
[1] Banaschewski, B., Pointfree topology and the spectra of f-rings, Ordered algebraic structures (Curacao, 1995), Kluwer Acad. Publ., Dordrecht, (1997), 123-148.
1
[2] Banaschewski, B., The real numbers in pointfree topology, Texts in Mathematics (Series B), 12, University of Coimbra, 1997.
2
[3] Bigard, A., K. Keimel, and S. Wolfenstein, Groups et anneaux reticules, Lecture Notes in Math. 608, Springer-Verlag, 1977.
3
[4] Ebrahimi, M.M. and A. Karimi Feizabadi, Pointfree prime representation of real Riesz maps, Algebra Universalis 54 (2005), 291-299.
4
[5] Gillman, L. and M. Jerison, "Rings of Continuous Function", Graduate Texts in Mathematics 43, Springer-Verlag, 1979.
5
[6] Karimi Feizabadi, A., Representation of slim algebraic regular cozero maps, Quaest. Math. 29 (2006), 383-394.
6
[7] Karimi Feizabadi, A., Free lattice-valued functions, reticulation of rings and modules, submitted.
7
[8] Picado, J. and A. Pultr, "Frames and Locales: Topology without Points", Frontiers in Mathematics 28, Springer, Basel, 2012.
8
ORIGINAL_ARTICLE
The projectable hull of an archimedean $\ell$-group with weak unit
The much-studied projectable hull of an $\ell$-group $G\leq pG$ is an essential extension, so that, in the case that $G$ is archimedean with weak unit, ``$G\in {\bf W}$", 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 $U(g)=\{P\in Min(G):g\notin P\}$ ($g\in G$), 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 pre-existing 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.
http://cgasa.sbu.ac.ir/article_46629_aaedb6eda82247753a33798657cb5075.pdf
2017-07-01T11:23:20
2020-10-24T11:23:20
165
179
Archimedean $l$-group
vector lattice
Yosida representation
minimal prime spectrum
principal polar
projectable
principal projection property
Anthony W.
Hager
ahager@wesleyan.edu
true
1
Department of Mathematics and CS, Wesleyan University, Middletown, CT 06459.
Department of Mathematics and CS, Wesleyan University, Middletown, CT 06459.
Department of Mathematics and CS, Wesleyan University, Middletown, CT 06459.
AUTHOR
Warren Wm.
McGovern
warren.mcgovern@fau.edu
true
2
H. L. Wilkes Honors College, Florida Atlantic University, Jupiter, FL 33458.
H. L. Wilkes Honors College, Florida Atlantic University, Jupiter, FL 33458.
H. L. Wilkes Honors College, Florida Atlantic University, Jupiter, FL 33458.
AUTHOR
[1] Ball, R.N., Marra, V., McNeill, D., and Pedrini, A., From Freudenthal's Spectral Theorem to projectable hulls of unital archimedean lattice-groups, through compactification of minimal spectra, arXiv: 1406-1352 V2.
1
[2] Bigard, A., Keimel, K., and Wolfenstein, S., "Groupes et Anneaux R'eticul'es", Lecture Notes in Math. 608, Springer-Verlag, Berlin-New York, 1977.
2
[3] Carral, M. and Coste, M., Normal spectral spaces and their dimensions, J. Pure Appl. Algebra 30(3) (1983), 227-235.
3
[4] Darnel, M., "Theory of Lattice-Ordered Groups", Monographs and Textbooks in Pure and Applied Mathematics 187, Marcel Dekker, Inc., New York, 1995.
4
[5] Engelking, R., "General Topology". Second edition. Sigma Series in Pure Mathematics 6, Heldermann Verlag, Berlin, 1989.
5
[6] Fine, N.J., Gillman, L., and Lambek, J., "Rings of Quotients of Rings of Functions", McGill Univ. Press, Montreal, 1966.
6
[7] Hager, A.W., Minimal covers of topological spaces, Papers on general topology and related category theory and topological algebra (New York, 1985/1987), 44-59, Ann. New York Acad. Sci. 552, New York Acad. Sci., New York, 1989.
7
[8] Hager, A.W., Kimber, C.M., and McGovern, W.Wm., Weakly least integer closed groups, Rend. Circ. Mat. Palermo (2), 52(3) (2003), 453-480.
8
[9] Hager, A.W. and McGovern, W.Wm., The Yosida space and representation of the projectable hull of an archimedean `-group with weak unit, Quaest. Math., 40(1) (2017), 57-63.
9
[10] Hager, A.W. and Robertson, L., Representing and ringifying a Riesz space, Symp. Math 21 (1977), 411-431.
10
[11] Hager, A.W. and L. Robertson, On the embedding into a ring of an archimedean-group, Canad. J. Math. 31 (1979), 1-8.
11
[12] Johnson, D.G. and Kist, J.E., Prime ideals in vector lattices, Canad. J. Math. 14 (1962), 517-528.
12
[13] Luxemburg, W.A.J. and Zaanen, A.C., "Riesz Spaces". Vol. I., North-Holland Publishing Co., Amsterdam-London, 1971.
13
[14] Mart'inez, J., Hull classes of Archimedean lattice-ordered groups with unit: a survey, Ordered algebraic structures, 89-121, Dev. Math. 7, Kluwer Acad. Publ., Dordrecht, 2002.
14
[15] Porter, J. and Woods, R.G., "Extensions and Absolutes of Hausdorf Spaces", Springer-Verlag, New York, 1988.
15
[16] Sikorski, R., "Boolean Algebras", Third edition. 25 Springer-Verlag New York Inc., New York, 1969.
16
[17] Veksler, A.I. and Gev{i}ler, V.A., Order and disjoint completeness of linear partially ordered spaces, Sib. Math. J. 13 (1972), 30-35. (Plenum translation).
17