  In a previous paper we introduced the notion of a {\it 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 {\it 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 {\it ARS-fans},
is the subject of the paper: M. Dickmann, A. Petrovich, {\em Fans in the Theory
of Real Semigroups. II. Combinatorial Theory}, 20 pp., submitted., a
continuation of the present paper.
