On the existence of right adjoints for surjective mappings between fuzzy structures0

En este trabajo los autores continúan su estudio de la caracterización de la existencia de adjunciones (conexiones de Galois isótonas) cuyo codominio no está dotado de estructura en principio. En este artículo se considera el caso difuso en el que se tiene un orden difuso R definido en un conjunto A y una aplicación sobreyectiva f:A-> B compatible respecto de dos relaciones de similaridad definidas en el dominio A y en el condominio B, respectivamente. Concretamente, el problema es encontrar un orden difuso S en B y una aplicación g:B-> A compatible también con las correspondientes similaridades definidas en A y en B, de tal forma que el par (f,g) constituya un adjunción.

[n]cycloparaphenylenes with charges: cyclic conjugation at last

Oligophenylenes (polyphenylenes) are constituted by an array of conjugated benzenes where inter-ring electron delocalization tends to extend over the whole chain (linear conjugation) being intrinsically limited, among other factors, by terminal effects. Alternatively, cyclic conjugation is envisaged as the unlimited free-boundary versionofconjugation which will impact the structure of molecules in rather unknown ways. The cyclic version of oligophenylenes, cycloparaphenylenes ([n]CPPs with n the number of phenyl rings) were first synthesized in 2008 by Beztozzi and Jasti.1 Today the whole [n]CPP series from [5]CPP to [18]CPP has been prepared. [n]CPPs represent ideal models to investigate new insights of the electronic structure of molecules and cyclic conjugation when electrons or charges circulate in a closed circuit without boundaries. Radical cations and dications of [n]CPP from n=5 to n=12 have been prepared and studied by Raman spectroscopy.2 Small [n]CPP dications own their stability to the closed-shell electronic configuration imposed by cyclic conjugation. However, in large [n]CPP dications cyclic conjugation is minimal and these divalent species form open-shell biradicals. The Raman spectra reflect the effect of cyclic conjugation in competition with cyclic strain and biradicaloid aromatic stabilization. Cyclic conjugation provokes the existence of a turning point or V-shape behavior of the frequencies of the G bands as a function of n. In this communication we will show the vibrational spectroscopic fingerprint of this rare form of conjugation. [1] R. Jasti, J. Bhattacharjee, J. B. Neaton, C. R. Bertozzi, “Synthesis, Characterization, and Theory of [9]-, [12]-, and [18]Cycloparaphenylene: Carbon Nanohoop Structures”, J. Am. Chem. Soc. 130 (2008), 17646–17647. [2] M. P. Alvarez, P. M. Burrezo, M. Kertesz, T. Iwamoto, S. Yamago, J. Xia, R. Jasti, J. T. L. Navarrete, M. Taravillo, V. G. Baonza, J. Casado, “Properties of Sizeable [n]CycloParaPhenylenes As Molecular Models of Single-Wall Carbon Nanotubes By Raman Spectroscopy: Structural and Electron-Transfer Responses Under Mechanical Stress”, Angew. Chem. Int. Ed. 53, (2014), 7033−7037.