Shahid Beheshti University Categories and General Algebraic Structures with Applications 2345-5853 7 Special Issue on the Occasion of Banaschewski's 90th Birthday (II) 2017 07 01 Tangled Closure Algebras 9 31 42354 EN Robert Goldblatt School of Mathematics and Statistics, Victoria University of Wellington, New Zealand Ian Hodkinson Department of Computing, Imperial College London, UK. Journal Article 2016 09 20 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.

https://cgasa.sbu.ac.ir/article_42354_09def3b31ada32383d6d12c9644168af.pdf

Shahid Beheshti University Categories and General Algebraic Structures with Applications 2345-5853 7 Special Issue on the Occasion of Banaschewski's 90th Birthday (II) 2017 07 01 Some Types of Filters in Equality Algebras 33 55 42342 EN Rajabali Borzooei Department of Mathematics, Shahid Beheshti University, Tehran, Iran. Fateme Zebardast Department of Mathematics, Payam e Noor University, Tehran, Iran. Mona Aaly Kologani Payam e Noor University Journal Article 2016 11 11 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.

https://cgasa.sbu.ac.ir/article_42342_cf5624efc3f4dd8d61c28cc7af659734.pdf

Shahid Beheshti University Categories and General Algebraic Structures with Applications 2345-5853 7 Special Issue on the Occasion of Banaschewski's 90th Birthday (II) 2017 07 01 One-point compactifications and continuity for partial frames 57 88 43180 EN John Frith Department of Mathematics and Applied Mathematics, University of Cape Town, Private Bag Rondebosch, 7701, South Africa. Anneliese Schauerte Department of Mathematics and Applied Mathematics, University of Cape Town, Private Bag Rondebosch, 7701, South Africa. Journal Article 2016 10 20 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.

https://cgasa.sbu.ac.ir/article_43180_02e474fcbfa63e236d1fbd237390dba8.pdf

Shahid Beheshti University Categories and General Algebraic Structures with Applications 2345-5853 7 Special Issue on the Occasion of Banaschewski's 90th Birthday (II) 2017 07 01 Adjoint relations for the category of local dcpos 89 105 43374 EN Bin Zhao Shaanxi Normal University Jing Lu Shaanxi Normal University Kaiyun Wang Shaanxi Normal University Journal Article 2016 10 20 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} has both 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.

https://cgasa.sbu.ac.ir/article_43374_e3ba4928af107559409d8a2f182b5716.pdf

Shahid Beheshti University Categories and General Algebraic Structures with Applications 2345-5853 7 Special Issue on the Occasion of Banaschewski's 90th Birthday (II) 2017 07 01 Filters of Coz(X) 107 123 44925 EN Papiya Bhattacharjee School of Science, Penn State Behrend, Erie, PA 16563, USA. Kevin M. Drees Department of Mathematics and Information Technology, Mercyhurst University, Erie PA. Journal Article 2016 12 11 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.

https://cgasa.sbu.ac.ir/article_44925_013d795e3961eac0b6b094ece513d81e.pdf

Shahid Beheshti University Categories and General Algebraic Structures with Applications 2345-5853 7 Special Issue on the Occasion of Banaschewski's 90th Birthday (II) 2017 07 01 Perfect secure domination in graphs 125 140 44926 EN S.V. Divya Rashmi Department of Mathematics, Vidyavardhaka College of Engineering, Mysuru 570002, Karnataka, India. Subramanian Arumugam National Centre for Advanced Research in Discrete Mathematics, Kalasalingam University, Anand Nagar, Krishnankoil-626 126, Tamil Nadu, India. Kiran R. Bhutani Department of Mathematics, The Catholic University of America, Washington, D.C. 20064, USA. Peter Gartland Department of Mathematics, The Catholic University of America, Washington, D.C. 20064, USA. Journal Article 2016 10 16 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.

https://cgasa.sbu.ac.ir/article_44926_4a0432bd29e2bbab421183f554f06243.pdf

Shahid Beheshti University Categories and General Algebraic Structures with Applications 2345-5853 7 Special Issue on the Occasion of Banaschewski's 90th Birthday (II) 2017 07 01 $\mathcal{R}L$- valued $f$-ring homomorphisms and lattice-valued maps 141 163 38548 EN Abolghasem Karimi Feizabadi Department of Mathematics, Gorgan Branch, Islamic Azad University, Gorgan, Iran. Ali Akbar Estaji Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran. Batool Emamverdi Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran. Journal Article 2016 07 28 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.

https://cgasa.sbu.ac.ir/article_38548_d61135e6f18b53e9ac1eb29192263dbc.pdf

Shahid Beheshti University Categories and General Algebraic Structures with Applications 2345-5853 7 Special Issue on the Occasion of Banaschewski's 90th Birthday (II) 2017 07 01 The projectable hull of an archimedean $\ell$-group with weak unit 165 179 46629 EN Anthony W. Hager Department of Mathematics and CS, Wesleyan University, Middletown, CT 06459. Warren Wm. McGovern H. L. Wilkes Honors College, Florida Atlantic University, Jupiter, FL 33458. Journal Article 2016 08 10 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.

https://cgasa.sbu.ac.ir/article_46629_aaedb6eda82247753a33798657cb5075.pdf