Fans in the theory of real semigroups

Abstract

In [DP1] we introduced the notion of a real semigroup (RS) as an axiomatic framework to study diagonal quadratic forms with arbitrary entries over (commutative, unitary) semireal rings. Two important classes of RSs were studied at length in [DP2], [DP3]. In this paper we introduce and develop the theory of RS-fans, a third class of RSs providing a vast generalization of homonymous notions previously existing in field theory and in the theories of abstract order spaces and of reduced special groups. The Introduction briefly reviews the existing theories of fans and describes the contents of the paper. Acknowledgements. The authors wish to thank F. Miraglia for his careful reading of and fruitful comments on an advanced draft of this paper, as well as for his invaluable help in preparing the final version of the manuscript. Thanks are also due to the referee for a meticulous examination of the submitted text and many useful suggestions, leading to a substantial improvement of the paper and, even, to some new results. The first author was partially supported by a "Projet de mobilité international Sorbonne Paris Cité-Argentine-Brésil 2014".