A survey of clones on infinite sets

A clone on a set X is a set of finitary operations on X which contains all projections and which is moreover closed under functional composition. Ordering all clones on X by inclusion, one obtains a complete algebraic lattice, called the clone lattice. We summarize what we know about the clone lattice on an infinite base set X and formulate what we consider the most important open problems.Comment: 37 page

Identifying the Genes of Unconventional High Temperature Superconductors

We elucidate a recently emergent framework in unifying the two families of high temperature (high $T_c$) superconductors, cuprates and iron-based superconductors. The unification suggests that the latter is simply the counterpart of the former to realize robust extended s-wave pairing symmetries in a square lattice. The unification identifies that the key ingredients (gene) of high $T_c$ superconductors is a quasi two dimensional electronic environment in which the d-orbitals of cations that participate in strong in-plane couplings to the p-orbitals of anions are isolated near Fermi energy. With this gene, the superexchange magnetic interactions mediated by anions could maximize their contributions to superconductivity. Creating the gene requires special arrangements between local electronic structures and crystal lattice structures. The speciality explains why high $T_c$ superconductors are so rare. An explicit prediction is made to realize high $T_c$ superconductivity in $Co/Ni$-based materials with a quasi two dimensional hexagonal lattice structure formed by trigonal bipyramidal complexes

Magnetic structures of NaLMnWO_6 perovskites (L=La,Nd,Tb)

The magnetic structures of the perovskites NaLaMnWO_6, NaNdMnWO_6, and NaTbMnWO_6, with rocksalt ordering of the Mn/W ions and layered ordering of Na and the rare-earth ions, have been determined by neutron powder diffraction. The manganese moments in NaLaMnWO_6 order below 10 K with a propagation vector of k_14=(1/2,0,1/2) and a moment of 3.99μB per Mn^2+ ion. The Mn^2+ and Nd^3+ ions order simultaneously in NaNdMnWO_6 at 11 K. The resulting magnetic structure is incommensurate with the underlying crystal structure and has the propagation vector of k_5=(0,0.48,1/2). NaTbMnWO_6 undergoes two magnetic phase transitions at 15 and 9 K. The structure determined at 11 K is based on two propagation vectors of k_14=(1/2,0,1/2) and k_5=(0,0.427,1/2). Upon cooling at 6 K the incommensurate vector is no longer present and the moments order only according to k_14. The moments of the Nd and Tb ions are found to remain within the planes of the A-site cations, and in NaTbMnWO_6 the Mn moments also lie within the xy plane. This study not only reveals magnetic structures with previously unexplored topologies but it also sheds light on the intricate coupling between the two magnetic sublattices

Dicomplemented Lattices: A Contextual Generalization of Boolean Algebras

Das Ziel dieser Arbeit ist es die mathematische Theorie der Begriffsalgebren zu entwickeln. Wir betrachten dabei hauptsaechlich das Repraesentationsproblem dieser vor Kurzem eingefuehrten Strukturen. Motiviert durch die Suche nach einer geeigneten Negation sind die Begriffsalgebren entstanden. Sie sind nicht nur fuer die Philosophie oder die Wissensrepraesentation von Interesse, sondern auch fuer andere Felder, wie zum Beispiel Logik oder Linguistik. Das Problem Negationen geeignet einzufuehren, ist sicher eines der aeltesten der wissenschaftlichen oder philosophischen Gemeinschaft und erregt auch zur Zeit die Aufmerksamkeit vieler Wissenschaftler. Verschiedene Typen von Logik (die sich sehr stark durch die eigefuehrte Negation unterscheiden) unterstreichen die Wichtigkeit dieser Untersuchungen. In dieser Arbeit beschaeftigen wir uns hauptsaechlich mit der kontextuellen Logik, eine Herangehensweise der Formalen Begriffsanalyse, basierend auf der Idee, den Begriff als Einheit des Denkens aufzufassen.The aim of this investigation is to develop a mathematical theory of concept algebras. We mainly consider the representation problem for this recently introduced class of structures. Motivated by the search of a "negation" on formal concepts, "concept algebras" are of considerable interest not only in Philosophy or Knowledge Representation, but also in other fields as Logic or Linguistics. The problem of negation is surely one of the oldest problems of the scientific and philosophic community, and still attracts the attention of many researchers. Various types of Logic (defined according to the behaviour of the corresponding negation) can attest this affirmation. In this thesis we focus on "Contextual Logic", a Formal Concept Analysis approach, based on concepts as units of thought