Fans in the Theory of Real Semigroups II. Combinatorial Theory
In the paper: Fans in the Theory of Real Semigroups. I. Algebraic Theory (submitted) we introduced the notion of fan in the categories of real semigoups and their dual abstract real spectra and developed the algebraic theory of these structures. In this paper we develop the combinatorial theory of ARS-fans, i.e., fans in the dual category of abstract real spectra. Every ARS is a spectral space and hence carries a natural partial order called the {\it specialization partial order}. Our main result shows that the isomorphism type of a finite fan in the category ARS is entirely determined by its order of specialization. The main tools used to prove this result are: (1) Crucial use of the theory of {\it ternary semigroups}, a class of semigroups underlying that of RSs; (2) Every ARS-fan is a disjoint union of abstract order spaces (called {\it levels}); (3) Every level carries a natural involution of abstract order spaces, and (4) The notion of a {\it standard generating system}, a combinatorial tool replacing, in the context of ARSs, the (absent) tools of combinatorial geometry (matroid theory) employed in the cases of fields and of abstract order spaces.