Fans in the Theory of Real Semigroups I. Algebraic Theory

TL;DR

This paper develops the algebraic theory of RS-fans, a new class of real semigroups that generalize existing notions in field theory and order space theories, with a focus on their dual combinatorial structures.

Contribution

It introduces and formalizes the concept of RS-fans, expanding the framework of real semigroups and connecting to combinatorial dual structures called ARS-fans.

Findings

Defined and developed the algebraic properties of RS-fans.

Established the duality between RS-fans and ARS-fans.

Extended the theory of real semigroups to include fans.

Abstract

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.