Submission date: 21 March 2017

Abstract:

In a previous paper we introduced the notion of a real semigroup (RS) as an axiomatic framework to study diagonal quadratic forms with arbitrary entries over (commutative, unitary) semi-real rings. Two important classes of RSs were studied at length in previous papers. In this paper we introduce and develop the algebraic 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; for a background on fans, see paragraph A of the Introduction, below. The contents of this paper are briefly reviewed in paragraph B of the Introduction. The combinatorial theory of the structures dual to RS-fans, called ARS-fans, is the subject of the paper: M. Dickmann, A. Petrovich, Fans in the Theory of Real Semigroups. II. Combinatorial Theory, 20 pp., submitted., a continuation of the present paper.

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Full text arXiv 1703.07434: pdf, ps.