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In a previous paper we introduced the notion of a {\it real semigroup} (RS)as an axiomatic framework to study diagonal quadratic forms with arbitraryentries over (commutative, unitary) semi-real rings. Two important classes ofRSs were studied at length in previous papers. In this paper we introduce anddevelop the algebraic theory of {\it RS-fans}, a third class of RSs providing avast generalization of homonymous notions previously existing in field theoryand in the theories of abstract order spaces and of reduced special groups; fora background on fans, see paragraph A of the Introduction, below. The contentsof this paper are briefly reviewed in paragraph B of the Introduction. Thecombinatorial theory of the structures dual to RS-fans, called {\it ARS-fans},is the subject of the paper: M. Dickmann, A. Petrovich, {\em Fans in the Theoryof Real Semigroups. II. Combinatorial Theory}, 20 pp., submitted., acontinuation of the present paper.

Fans in the Theory of Real Semigroups I. Algebraic Theory