This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich.© We have been somewhat vague above. In particular we did not make precise the various direct systems which yield the projective limits X and X Q . We only wanted to indicate that in semialgebraic geometry over IR real closed fields may come up in a natural and geometric way.The present lecture notes give a contribution to a basic but rather modest aspect of semialgebraic geometry: the topological phenomena of semialgebraic sets in V(R) for V a variety over a real closed field R.There is a difficulty with the word "topological" here. Thus it is not just for fun or for systematic reasons that we study in Chapter I more general spaces. In the later chapters we are forced to restrict to paracompact spaces, since otherwise our deeper techniques break down.There is one phenomenon in our theory which may seem somewhat unusual Appendix A is of different kind. Here we draw the connections between our theory and "abstract" semialgebraic geometry which, starting from the notion of the real spectrum, now is in a process of rapid development. Appendix A is not needed for our theory in a technical sense, but there we will find the occasion to explain some more points of our philosophy about the "raison d'etre" of locally semialgebraic spaces.We thank the members of the former Regensburger semialgebraic group, in particular Roland Huber and Robby Robson, for stimulating discussions and criticism about the contents of these lecture notes. Special thanks are due to Jose Manuel Gamboa and R. Huber for a penetrating (and very successful) search for mistakes in the final version of the manuscript.We thank Marina Richter for her patience and excellence in typing the book and R. Robson for eliminating some of the most annoying grammatical mistakes. We are well aware that we could have written a better book in our native language, but since the book is designed as a "topologie generale" for semialgebraic geometry which should be useful as a widely accepted reference, we have written in that language which will be understood by the most. Regensburg, July 1985Hans Delfs, Manfred Knebusch **) In order to guarantee that Cov^. is really a set one should only allow subsets of some fixed large set as index sets I. We will ignore all set-theoretic difficulties here. It is evident that T(M) has the following three properties: 1) f (M) e T(M) . 2) X G r(M) Ms X G T(M)3 The union of these sets is M. We apply the procedure of Example 2 -9 fc this family and obtain on the set M a new structure of a locally semi- We come back to the natural map p^ : M io C " >M f°r M a n arbitrary locally complete space over R. K(g) =...
a b s t r a c tWe interpret a valuation v on a ring R as a map v : R → M into a so-called bipotent semiring M (the usual max-plus setting), and then define a supervaluation ϕ as a suitable map into a supertropical semiring U with ghost ideal M (cf. Izhakian and Rowen (2010, in press) [8,9]) covering v via the ghost map U → M. The set Cov(v) of all supervaluations covering v has a natural ordering which makes it a complete lattice. In the case where R is a field, and hence for v a Krull valuation, we give a completely explicit description of Cov(v).The theory of supertropical semirings and supervaluations aims for an algebra fitting the needs of tropical geometry better than the usual max-plus setting. We illustrate this by giving a supertropical version of Kapranov's Lemma.
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