On Matthews’ Relationship Between Quasi-Metrics and Partial Metrics: An Aggregation Perspective
Borsík and Doboš studied the problem of how to merge a family of metric spaces into a single one through a function. They called such functions metric preserving and provided a characterization of them in terms of the so-called triangle triplets. Since then, different papers have extended their stud...
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| Abstract | Borsík and Doboš studied the problem of how to merge a family of metric spaces into a single one through a function. They called such functions metric preserving and provided a characterization of them in terms of the so-called triangle triplets. Since then, different papers have extended their study to the case of generalized metric spaces. Concretely, Mayor and Valero (Inf Sci 180:803–812, 2010) provided two characterizations of those functions, called quasi-metric aggregation functions, that allows us to merge a collection of quasi-metric spaces into a new one. In Massanet and Valero (in: Sainz-Palmero et al (eds) Proceedings of the 16th Spanish conference on fuzzy technology and fuzzy logic, European Society for Fuzzy Logic and Techonology, Valladolid, 2012) gave a characterization of the functions, called partial metric aggregation function, that are useful for merging a collection of partial metric spaces into single one as final output. Inspired by the preceding work, Martín et al. (in: Bustince et al (eds) Aggregation functions in theory and in practice. Advances in intelligent systems and computing, vol 228, Springer, Berlin, 2013) addressed the problem of constructing metrics from quasi-metrics, in a general way, using a class of functions that they called metric generating functions. In particular, they solved the posed problem providing a characterization of such functions and, thus, all ways under which a metric can be induced from a quasi-metric from an aggregation viewpoint. Following this idea, we propose the same problem in the framework of partial metric spaces. So, we characterize those functions that are able to generate a quasi-metric from a partial metric, and conversely, in such a way that Matthews’ relationship between both type of generalized metrics is retrieved as a particular case. Moreover, we study if both, the partial order and the topology induced by a partial metric or a quasi-metric, respectively, are preserved by the new method in the spirit of Matthews. Furthermore, we discuss the relationship between the new functions and those families introduced in the literature, i.e., metric preserving functions, quasi-metric aggregation functions, partial metric aggregation functions and metric generating functions. |
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| AbstractList | Borsík and Doboš studied the problem of how to merge a family of metric spaces into a single one through a function. They called such functions metric preserving and provided a characterization of them in terms of the so-called triangle triplets. Since then, different papers have extended their study to the case of generalized metric spaces. Concretely, Mayor and Valero (Inf Sci 180:803–812, 2010) provided two characterizations of those functions, called quasi-metric aggregation functions, that allows us to merge a collection of quasi-metric spaces into a new one. In Massanet and Valero (in: Sainz-Palmero et al (eds) Proceedings of the 16th Spanish conference on fuzzy technology and fuzzy logic, European Society for Fuzzy Logic and Techonology, Valladolid, 2012) gave a characterization of the functions, called partial metric aggregation function, that are useful for merging a collection of partial metric spaces into single one as final output. Inspired by the preceding work, Martín et al. (in: Bustince et al (eds) Aggregation functions in theory and in practice. Advances in intelligent systems and computing, vol 228, Springer, Berlin, 2013) addressed the problem of constructing metrics from quasi-metrics, in a general way, using a class of functions that they called metric generating functions. In particular, they solved the posed problem providing a characterization of such functions and, thus, all ways under which a metric can be induced from a quasi-metric from an aggregation viewpoint. Following this idea, we propose the same problem in the framework of partial metric spaces. So, we characterize those functions that are able to generate a quasi-metric from a partial metric, and conversely, in such a way that Matthews’ relationship between both type of generalized metrics is retrieved as a particular case. Moreover, we study if both, the partial order and the topology induced by a partial metric or a quasi-metric, respectively, are preserved by the new method in the spirit of Matthews. Furthermore, we discuss the relationship between the new functions and those families introduced in the literature, i.e., metric preserving functions, quasi-metric aggregation functions, partial metric aggregation functions and metric generating functions. |
| ArticleNumber | 47 |
| Author | Valero, Oscar Miñana, Juan-José |
| Author_xml | – sequence: 1 givenname: Juan-José orcidid: 0000-0001-9835-0700 surname: Miñana fullname: Miñana, Juan-José email: jj.minana@uib.es organization: Departament de Ciències Matemàtiques i Informàtica, Universitat de les Illes Balears, Institut d’Investigació Sanitària Illes Balears (IdISBa) – sequence: 2 givenname: Oscar surname: Valero fullname: Valero, Oscar organization: Departament de Ciències Matemàtiques i Informàtica, Universitat de les Illes Balears, Institut d’Investigació Sanitària Illes Balears (IdISBa) |
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| References | MatthewsSGAn extensional treatment of lazy data flow deadlockTheor. Comput. Sci.1995151195205136215310.1016/0304-3975(95)00051-W RomagueraSSchellekensMPQuasi-metric properties of complexity spacesTopol. Its Appl.199998311322172000910.1016/S0166-8641(98)00102-3 García-RaffiLMRomagueraSSánchez-PérezEASequence spaces and asymmetric norms in the theory of computational complexityMath. Comput. Modell.200236111192505510.1016/S0895-7177(02)00100-0 ShahzadNValeroOFixed point theorems in quasi-metric spaces and the specialization partial orderFixed Point Theory2018192733750382179410.24193/fpt-ro.2018.2.55 Goubault-LarrecqJNon-Hausdorff Topology and Domain Theory2013New YorkCambridge University Press10.1017/CBO9781139524438 RomagueraSTiradoPValeroONew results on mathematical foundations of asymptotic complexity analysis of algorithms via complexity spaceInt. J. Comput. Math.20128917281741296751310.1080/00207160.2012.659246 MayorGValeroOAggregation of asymmetric distances in computer scienceInf. Sci.2010180803812257834610.1016/j.ins.2009.06.020 RomagueraSSchellekensMPValeroOThe complexity space of partial functions: a connection between complexity analysis and denotational semanticsInt. J. Comput. Math.20118818191829281086410.1080/00207161003631885 HitzlerPSedaAKMathematical Aspects of Logic Programming Semantics2010Boca RatonCRC Press1219.68085 MartínJMayorGValeroOOn quasi-metric aggregation functions and fixed point theoremsFuzzy Sets Syst.201322888104309241810.1016/j.fss.2012.08.009 BorsíkJDobošJOn a product of metric spacesMathematica Slovaka1981311932050562.54016 MatthewsSGPartial metric topologyAnn. N. Y. Acad. Sci.1994728183197146777310.1111/j.1749-6632.1994.tb44144.x SchellekensMPThe Smyth completion: a common foundation for denotational semantics and complexity analysisElectron. Notes Theor. Comput. Sci.19951211232148686410.1016/S1571-0661(04)00029-5 AlghamdiMAShahzadNValeroOFixed point theorems in generalized metric spaces with applications to computer scienceFixed Point Theory Appl.20132013118306401210.1186/1687-1812-2013-118 DezaMMDezaEEncyclopedia of Distances2009BerlinSpringer10.1007/978-3-642-00234-2 Massanet, S., Valero, O.: New results on metrics aggregation. In: Sainz-Palmero, G.I. et al. (eds.) Proceedings of the 16th Spanish Conference on Fuzzy Technology and Fuzzy Logic, European Society for Fuzzy Logic and Techonology, Valladolid, pp. 558–563 (2012) RomagueraSValeroOAfzalMAsymptotic complexity analysis and denotational semantics for recursive programs based on complexity spacesSemantics Advances in Theories and Mathematical Models2012RijekaInTech Open Science99120 PestovVStojmirovićAIndexing schemes for similarity search: an illustrated paradigmFundamenta Informaticae20067036738522469801095.68580 García-RaffiLMRomagueraSSánchez-PérezEAThe supremum asymmetric norm on sequence algebras: a general framework to measure complexity spacesElectron. Notes Theor. Comput. Sci.200374395010.1016/S1571-0661(04)80764-3 HaghiRHRezapourShShahzadNBe careful on partial metric fixed point resultsTopol. Its Appl.20131603450454301035010.1016/j.topol.2012.11.004 DobošJMetric Preserving Functions1998KošiceŠtroffek0942.26010 Shahzad, N., Valero, O.: On 0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0$$\end{document}-complete partial metric spaces and quantitative fixed point techniques in Denotational Semantics. Abstr. Appl. Anal. 2013, 11, Article ID 985095 RomagueraSSánchez-PérezEAValeroOComputing complexity distances between algorithmsKybernetika20033956958220423421249.54069 PestovVStojmirovićAIndexing schemes for similarity search in datasets of short protein fragmentsInf. Syst.2007321145116510.1016/j.is.2007.03.001 AliprantisCDBorderKCInfinite Dimensional Analysis1999HeidelbergSpringer10.1007/978-3-662-03961-8 MartínJMayorGValeroOBustinceHOn the symmetrization of quasi-metrics: an aggregation perspectiveAggregation Functions in Theory and in Practice. Advances in Intelligent Systems and Computing2013BerlinSpringer3193311277.28025 J Doboš (1173_CR5) 1998 G Mayor (1173_CR16) 2010; 180 S Romaguera (1173_CR22) 2011; 88 N Shahzad (1173_CR26) 2018; 19 J Borsík (1173_CR3) 1981; 31 J Martín (1173_CR11) 2013 SG Matthews (1173_CR15) 1995; 151 S Romaguera (1173_CR24) 2012 MM Deza (1173_CR4) 2009 P Hitzler (1173_CR10) 2010 S Romaguera (1173_CR23) 2012; 89 S Romaguera (1173_CR20) 2003; 39 MA Alghamdi (1173_CR2) 2013; 2013 V Pestov (1173_CR19) 2007; 32 V Pestov (1173_CR18) 2006; 70 J Goubault-Larrecq (1173_CR8) 2013 RH Haghi (1173_CR9) 2013; 160 SG Matthews (1173_CR14) 1994; 728 S Romaguera (1173_CR21) 1999; 98 J Martín (1173_CR12) 2013; 228 LM García-Raffi (1173_CR7) 2002; 36 LM García-Raffi (1173_CR6) 2003; 74 1173_CR17 MP Schellekens (1173_CR25) 1995; 1 CD Aliprantis (1173_CR1) 1999 1173_CR13 |
| References_xml | – reference: RomagueraSValeroOAfzalMAsymptotic complexity analysis and denotational semantics for recursive programs based on complexity spacesSemantics Advances in Theories and Mathematical Models2012RijekaInTech Open Science99120 – reference: MayorGValeroOAggregation of asymmetric distances in computer scienceInf. Sci.2010180803812257834610.1016/j.ins.2009.06.020 – reference: MatthewsSGAn extensional treatment of lazy data flow deadlockTheor. Comput. Sci.1995151195205136215310.1016/0304-3975(95)00051-W – reference: RomagueraSSchellekensMPValeroOThe complexity space of partial functions: a connection between complexity analysis and denotational semanticsInt. J. Comput. Math.20118818191829281086410.1080/00207161003631885 – reference: AlghamdiMAShahzadNValeroOFixed point theorems in generalized metric spaces with applications to computer scienceFixed Point Theory Appl.20132013118306401210.1186/1687-1812-2013-118 – reference: RomagueraSSchellekensMPQuasi-metric properties of complexity spacesTopol. Its Appl.199998311322172000910.1016/S0166-8641(98)00102-3 – reference: RomagueraSTiradoPValeroONew results on mathematical foundations of asymptotic complexity analysis of algorithms via complexity spaceInt. J. Comput. Math.20128917281741296751310.1080/00207160.2012.659246 – reference: ShahzadNValeroOFixed point theorems in quasi-metric spaces and the specialization partial orderFixed Point Theory2018192733750382179410.24193/fpt-ro.2018.2.55 – reference: MartínJMayorGValeroOBustinceHOn the symmetrization of quasi-metrics: an aggregation perspectiveAggregation Functions in Theory and in Practice. Advances in Intelligent Systems and Computing2013BerlinSpringer3193311277.28025 – reference: SchellekensMPThe Smyth completion: a common foundation for denotational semantics and complexity analysisElectron. Notes Theor. Comput. Sci.19951211232148686410.1016/S1571-0661(04)00029-5 – reference: RomagueraSSánchez-PérezEAValeroOComputing complexity distances between algorithmsKybernetika20033956958220423421249.54069 – reference: PestovVStojmirovićAIndexing schemes for similarity search: an illustrated paradigmFundamenta Informaticae20067036738522469801095.68580 – reference: MartínJMayorGValeroOOn quasi-metric aggregation functions and fixed point theoremsFuzzy Sets Syst.201322888104309241810.1016/j.fss.2012.08.009 – reference: PestovVStojmirovićAIndexing schemes for similarity search in datasets of short protein fragmentsInf. Syst.2007321145116510.1016/j.is.2007.03.001 – reference: DezaMMDezaEEncyclopedia of Distances2009BerlinSpringer10.1007/978-3-642-00234-2 – reference: DobošJMetric Preserving Functions1998KošiceŠtroffek0942.26010 – reference: Shahzad, N., Valero, O.: On 0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0$$\end{document}-complete partial metric spaces and quantitative fixed point techniques in Denotational Semantics. Abstr. Appl. Anal. 2013, 11, Article ID 985095 – reference: García-RaffiLMRomagueraSSánchez-PérezEASequence spaces and asymmetric norms in the theory of computational complexityMath. Comput. Modell.200236111192505510.1016/S0895-7177(02)00100-0 – reference: Massanet, S., Valero, O.: New results on metrics aggregation. In: Sainz-Palmero, G.I. et al. (eds.) Proceedings of the 16th Spanish Conference on Fuzzy Technology and Fuzzy Logic, European Society for Fuzzy Logic and Techonology, Valladolid, pp. 558–563 (2012) – reference: HitzlerPSedaAKMathematical Aspects of Logic Programming Semantics2010Boca RatonCRC Press1219.68085 – reference: AliprantisCDBorderKCInfinite Dimensional Analysis1999HeidelbergSpringer10.1007/978-3-662-03961-8 – reference: HaghiRHRezapourShShahzadNBe careful on partial metric fixed point resultsTopol. Its Appl.20131603450454301035010.1016/j.topol.2012.11.004 – reference: MatthewsSGPartial metric topologyAnn. N. Y. Acad. Sci.1994728183197146777310.1111/j.1749-6632.1994.tb44144.x – reference: Goubault-LarrecqJNon-Hausdorff Topology and Domain Theory2013New YorkCambridge University Press10.1017/CBO9781139524438 – reference: García-RaffiLMRomagueraSSánchez-PérezEAThe supremum asymmetric norm on sequence algebras: a general framework to measure complexity spacesElectron. Notes Theor. Comput. Sci.200374395010.1016/S1571-0661(04)80764-3 – reference: BorsíkJDobošJOn a product of metric spacesMathematica Slovaka1981311932050562.54016 – volume: 39 start-page: 569 year: 2003 ident: 1173_CR20 publication-title: Kybernetika – volume-title: Infinite Dimensional Analysis year: 1999 ident: 1173_CR1 doi: 10.1007/978-3-662-03961-8 – volume: 70 start-page: 367 year: 2006 ident: 1173_CR18 publication-title: Fundamenta Informaticae – volume: 31 start-page: 193 year: 1981 ident: 1173_CR3 publication-title: Mathematica Slovaka – volume: 19 start-page: 733 issue: 2 year: 2018 ident: 1173_CR26 publication-title: Fixed Point Theory doi: 10.24193/fpt-ro.2018.2.55 – start-page: 99 volume-title: Semantics Advances in Theories and Mathematical Models year: 2012 ident: 1173_CR24 – volume-title: Encyclopedia of Distances year: 2009 ident: 1173_CR4 doi: 10.1007/978-3-642-00234-2 – volume: 2013 start-page: 118 year: 2013 ident: 1173_CR2 publication-title: Fixed Point Theory Appl. doi: 10.1186/1687-1812-2013-118 – volume: 36 start-page: 1 year: 2002 ident: 1173_CR7 publication-title: Math. Comput. Modell. doi: 10.1016/S0895-7177(02)00100-0 – volume: 728 start-page: 183 year: 1994 ident: 1173_CR14 publication-title: Ann. N. Y. Acad. Sci. doi: 10.1111/j.1749-6632.1994.tb44144.x – volume-title: Metric Preserving Functions year: 1998 ident: 1173_CR5 – volume: 74 start-page: 39 year: 2003 ident: 1173_CR6 publication-title: Electron. Notes Theor. Comput. Sci. doi: 10.1016/S1571-0661(04)80764-3 – start-page: 319 volume-title: Aggregation Functions in Theory and in Practice. Advances in Intelligent Systems and Computing year: 2013 ident: 1173_CR11 – volume: 228 start-page: 88 year: 2013 ident: 1173_CR12 publication-title: Fuzzy Sets Syst. doi: 10.1016/j.fss.2012.08.009 – volume-title: Non-Hausdorff Topology and Domain Theory year: 2013 ident: 1173_CR8 doi: 10.1017/CBO9781139524438 – volume: 89 start-page: 1728 year: 2012 ident: 1173_CR23 publication-title: Int. J. Comput. Math. doi: 10.1080/00207160.2012.659246 – volume: 1 start-page: 211 year: 1995 ident: 1173_CR25 publication-title: Electron. Notes Theor. Comput. Sci. doi: 10.1016/S1571-0661(04)00029-5 – ident: 1173_CR17 doi: 10.1155/2013/985095 – volume: 88 start-page: 1819 year: 2011 ident: 1173_CR22 publication-title: Int. J. Comput. Math. doi: 10.1080/00207161003631885 – volume-title: Mathematical Aspects of Logic Programming Semantics year: 2010 ident: 1173_CR10 – volume: 180 start-page: 803 year: 2010 ident: 1173_CR16 publication-title: Inf. Sci. doi: 10.1016/j.ins.2009.06.020 – volume: 32 start-page: 1145 year: 2007 ident: 1173_CR19 publication-title: Inf. Syst. doi: 10.1016/j.is.2007.03.001 – volume: 160 start-page: 450 issue: 3 year: 2013 ident: 1173_CR9 publication-title: Topol. Its Appl. doi: 10.1016/j.topol.2012.11.004 – volume: 98 start-page: 311 year: 1999 ident: 1173_CR21 publication-title: Topol. Its Appl. doi: 10.1016/S0166-8641(98)00102-3 – ident: 1173_CR13 – volume: 151 start-page: 195 year: 1995 ident: 1173_CR15 publication-title: Theor. Comput. Sci. doi: 10.1016/0304-3975(95)00051-W |
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