eng
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
2017-07-01
7
Special Issue on the Occasion of Banaschewski's 90th Birthday (II)
9
31
42354
Tangled Closure Algebras
Robert Goldblatt
rob.goldblatt@msor.vuw.ac.nz
1
Ian Hodkinson
i.hodkinson@imperial.ac.uk
2
School of Mathematics and Statistics, Victoria University of Wellington, New Zealand
Department of Computing, Imperial College London, UK.
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
Closure algebra
tangled closure
tangle modality
fixed point
quasi-order
Alexandroff topology
dense-in-itself
dissectable
MacNeille completion
eng
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
2017-07-01
7
Special Issue on the Occasion of Banaschewski's 90th Birthday (II)
33
55
42342
Some Types of Filters in Equality Algebras
Rajabali Borzooei
borzooei@sbu.ac.ir
1
Fateme Zebardast
zebardastfathme@gmail.com
2
Mona Aaly Kologani
mona4011@gmail.com
3
Department of Mathematics, Shahid Beheshti University, Tehran, Iran.
Department of Mathematics, Payam e Noor University, Tehran, Iran.
Payam e Noor University
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
Equality algebra
(positive) implicative filter
fantastic filter
Boolean filter
eng
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
2017-07-01
7
Special Issue on the Occasion of Banaschewski's 90th Birthday (II)
57
88
43180
One-point compactifications and continuity for partial frames
John Frith
john.frith@uct.ac.za
1
Anneliese Schauerte
anneliese.schauerte@uct.ac.za
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.
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.<br />Specifically, we examine here the concept of continuity for partial frames, and compactifications of regular continuous such.<br /><br />Partial frames are meet-semilattices in which not all subsets need have joins.<br />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.<br /><br />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. <br /><br />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
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
eng
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
2017-07-01
7
Special Issue on the Occasion of Banaschewski's 90th Birthday (II)
89
105
43374
Adjoint relations for the category of local dcpos
Bin Zhao
zhaobin@snnu.edu.cn
1
Jing Lu
lujing0926@126.com
2
Kaiyun Wang
wangkaiyun@snnu.edu.cn
3
Shaanxi Normal University
Shaanxi Normal University
Shaanxi Normal University
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:<br /> (1) The forgetful functor $U$ : {bf LDcpo} $longrightarrow$ {bf Pos} has a left adjoint, but does not have a right adjoint;<br />(2) The inclusion functor $I$ : {bf Dcpo} $longrightarrow$ {bf LDcpo} has a left adjoint, but does not have a right adjoint;<br />(3) The forgetful functor $U$ : {bf LDcpo}-$S$ $longrightarrow$ {bf LDcpo} has<br />both left and right adjoints;<br />(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
Dcpo
local dcpo
$S$-ldcpo
forgetful functor
eng
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
2017-07-01
7
Special Issue on the Occasion of Banaschewski's 90th Birthday (II)
107
123
44925
Filters of Coz(X)
Papiya Bhattacharjee
pxb39@psu.edu
1
Kevin M. Drees
kevin.drees@gmail.com
2
School of Science, Penn State Behrend, Erie, PA 16563, USA.
Department of Mathematics and Information Technology, Mercyhurst University, Erie PA.
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
Cozero sets
ultrafilters
minimal prime ideals
$P$-space
$F$-space
inverse topology
$ell$-groups
eng
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
2017-07-01
7
Special Issue on the Occasion of Banaschewski's 90th Birthday (II)
125
140
44926
Perfect secure domination in graphs
S.V. Divya Rashmi
rashmi.divya@gmail.com
1
Subramanian Arumugam
s.arumugam.klu@gmail.com
2
Kiran R. Bhutani
bhutani@cua.edu
3
Peter Gartland
56gartland@cardinalmail.cua.edu
4
Department of Mathematics, Vidyavardhaka College of Engineering, Mysuru 570002, Karnataka, India.
National Centre for Advanced Research in Discrete Mathematics, Kalasalingam University, Anand Nagar, Krishnankoil-626 126, Tamil Nadu, India.
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.
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.
http://cgasa.sbu.ac.ir/article_44926_4a0432bd29e2bbab421183f554f06243.pdf
Secure domination
perfect secure domination
secure domination number
perfect secure domination number
eng
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
2017-07-01
7
Special Issue on the Occasion of Banaschewski's 90th Birthday (II)
141
163
38548
$mathcal{R}L$- valued $f$-ring homomorphisms and lattice-valued maps
Abolghasem Karimi Feizabadi
akarimi@gorganiau.ac.ir
1
Ali Akbar Estaji
aaestaji@hsu.ac.ir
2
Batool Emamverdi
emamverdi55@yahoo.com
3
Department of Mathematics, Gorgan Branch, Islamic Azad University, Gorgan, Iran.
Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.
Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.
In this paper, for each {it lattice-valued 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 lattice-valued map.<br /> We use the notation $delta_{pq}^{a}=(a -p)^{+}wedge (q-a)^{+}$,<br /> where $p, qin mathbb Q$ and $ain A$, that is nominated as {it interval projection}.<br /> 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 $c$,<br /> which are defined and some related results are investigated. On the contrary, we present a cozero lattice-valued 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 lattice-valued maps $Arightarrow L$,<br /> Where the mapping $c^r:Arightarrow L$ is called a {it realization} of $c$. It is shown that $tau_{c^r}=tau_c$ and $c^{rr}=c^r$.<br /> <br /> Finally, we describe how $tau_c$ can be a fundamental tool to extend pointfree version of Gelfand duality constructed by B. Banaschewski.<br />
http://cgasa.sbu.ac.ir/article_38548_d61135e6f18b53e9ac1eb29192263dbc.pdf
frame
cozero lattice-valued map
strong $f$-ring
interval projection
bounded
continuous
$mathbb{Q}$-compatible
coz-compatible
eng
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
2017-07-01
7
Special Issue on the Occasion of Banaschewski's 90th Birthday (II)
165
179
46629
The projectable hull of an archimedean $ell$-group with weak unit
Anthony W. Hager
ahager@wesleyan.edu
1
Warren Wm. McGovern
warren.mcgovern@fau.edu
2
Department of Mathematics and CS, Wesleyan University, Middletown, CT 06459.
H. L. Wilkes Honors College, Florida Atlantic University, Jupiter, FL 33458.
The much-studied 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 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
Archimedean $l$-group
vector lattice
Yosida representation
minimal prime spectrum
principal polar
projectable
principal projection property