Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58537Special Issue on the Occasion of Banaschewski's 90th Birthday (II)20170701Tangled Closure Algebras93142354ENRobert GoldblattSchool of Mathematics and Statistics, Victoria University of Wellington, New ZealandIan HodkinsonDepartment of Computing, Imperial College London, UK.Journal Article20160920The 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.Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58537Special Issue on the Occasion of Banaschewski's 90th Birthday (II)20170701Some Types of Filters in Equality Algebras335542342ENRajabali BorzooeiDepartment of Mathematics, Shahid Beheshti University, Tehran, Iran.Fateme ZebardastDepartment of Mathematics, Payam e Noor University, Tehran, Iran.Mona Aaly KologaniPayam e Noor UniversityJournal Article20161111Equality 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.Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58537Special Issue on the Occasion of Banaschewski's 90th Birthday (II)20170701One-point compactifications and continuity for partial frames578843180ENJohn FrithDepartment of Mathematics and Applied Mathematics, University of Cape Town, Private Bag Rondebosch, 7701,
South Africa.Anneliese SchauerteDepartment of Mathematics and Applied Mathematics, University of Cape Town, Private Bag Rondebosch, 7701, South Africa.Journal Article20161020Locally 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.Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58537Special Issue on the Occasion of Banaschewski's 90th Birthday (II)20170701Adjoint relations for the category of local dcpos8910543374ENBin ZhaoShaanxi Normal UniversityJing LuShaanxi Normal UniversityKaiyun WangShaanxi Normal UniversityJournal Article20161020In 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.Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58537Special Issue on the Occasion of Banaschewski's 90th Birthday (II)20170701Filters of Coz(X)10712344925ENPapiya BhattacharjeeSchool of Science, Penn State Behrend, Erie, PA 16563, USA.Kevin M. DreesDepartment of Mathematics and Information Technology, Mercyhurst University, Erie PA.Journal Article20161211 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.Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58537Special Issue on the Occasion of Banaschewski's 90th Birthday (II)20170701Perfect secure domination in graphs12514044926ENS.V. Divya RashmiDepartment of Mathematics, Vidyavardhaka College of Engineering, Mysuru 570002, Karnataka, India.Subramanian ArumugamNational Centre for Advanced Research in Discrete Mathematics, Kalasalingam University, Anand Nagar, Krishnankoil-626 126, Tamil Nadu, India.Kiran R. BhutaniDepartment of Mathematics, The Catholic University of America, Washington, D.C. 20064, USA.Peter GartlandDepartment of Mathematics, The Catholic University of America, Washington, D.C. 20064, USA.Journal Article20161016Let $G=(V,E)$ 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.Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58537Special Issue on the Occasion of Banaschewski's 90th Birthday (II)20170701$mathcal{R}L$- valued $f$-ring homomorphisms and lattice-valued maps14116338548ENAbolghasem Karimi FeizabadiDepartment of Mathematics, Gorgan Branch, Islamic Azad University, Gorgan, Iran.Ali Akbar EstajiFaculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.Batool EmamverdiFaculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.Journal Article20160728In 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 /> Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58537Special Issue on the Occasion of Banaschewski's 90th Birthday (II)20170701The projectable hull of an archimedean $ell$-group with weak unit16517946629ENAnthony W. HagerDepartment of Mathematics and CS, Wesleyan University, Middletown, CT 06459.Warren Wm. McGovernH. L. Wilkes Honors College, Florida Atlantic University, Jupiter, FL 33458.Journal Article20160810The 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.