naa a22 4500 510782515 CHVBK 20180411083252.0 cr unu---uuuuu 180411e20130101xx s 000 0 eng 10.1007/s11856-012-0079-z doi (NATIONALLICENCE)springer-10.1007/s11856-012-0079-z Topological and algebraic structures on the ring of Fermat reals [Elektronische Daten] [Paolo Giordano, Michael Kunzinger] The ring of Fermat reals is an extension of the real field containing nilpotent infinitesimals, and represents an alternative to Synthetic Differential Geometry in classical logic. In the present paper, our first aim is to study this ring by using standard topological and algebraic structures. We present the Fermat topology, generated by a complete pseudo-metric, and the omega topology, generated by a complete metric. The first one is closely related to the differentiation of (non-standard) smooth functions defined on open sets of Fermat reals. The second one is connected to the differentiation of smooth functions defined on infinitesimal sets. Subsequently, we prove that every (proper) ideal is a set of infinitesimals whose order is less than or equal to some real number. Finally, we define and study roots of infinitesimals. A computer implementation as well as an application to infinitesimal Taylor formulas with fractional derivatives are presented. Hebrew University Magnes Press, 2012 Giordano Paolo Faculty of Mathematics, University of Vienna, Nordbergstr. 15, A-1090, Vienna, Austria aut Kunzinger Michael Faculty of Mathematics, University of Vienna, Nordbergstr. 15, A-1090, Vienna, Austria aut Israel Journal of Mathematics The Hebrew University Magnes Press 193/1(2013-01-01), 459-505 0021-2172 193:1<459 2013 193 11856 https://doi.org/10.1007/s11856-012-0079-z text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s11856-012-0079-z text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Giordano Paolo Faculty of Mathematics, University of Vienna, Nordbergstr. 15, A-1090, Vienna, Austria aut NATIONALLICENCE 700 1- Kunzinger Michael Faculty of Mathematics, University of Vienna, Nordbergstr. 15, A-1090, Vienna, Austria aut NATIONALLICENCE 773 0- Israel Journal of Mathematics The Hebrew University Magnes Press 193/1(2013-01-01), 459-505 0021-2172 193:1<459 2013 193 11856 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer