A simple system of axioms for spaces of signatures
The author develops a new system of axioms for spaces of signatures, which is easier to check than the original one. Let (G,\(\cdot)\) be an abelian group of finite (even) exponent and X be a non-empty subset of the character group \(\chi\) (G). The pair (X,G) is said to be a space of signatures if it satisfies certain axioms \((S_ 0)-(S_ 4)\). The author introduces the following new \(axioms\) (S\({}^*_ 4) \) \(\cup_{s\in D<1,x>}D<y,s>=\cup_{t\in D<1,y>}D<x,t>\) for any \(x,y\in G.\) (S\({}^*_ 5) \) For any \(\sigma\in \chi (G)\) with \(\sigma (-1)=-1\) and \(1+\sigma (x)=\sigma (y)+\sigma (z)\) for all x,y,z\(\in G\) such that \(<1,x>\cong <y,z>\), holds \(\sigma\in X.\) Theorem 1 states that (X,G) is a space of signatures iff the axioms \(S_ 2,S_ 3,S^*_ 4\) and \(S^*_ 5\) hold. This has the following application to preorderings of division rings: Let K be a formally real division ring and \(T\subset K\) a preordering. Then the set \(X_ T\) of orderings of K containing T can be viewed as a subset of the character group \(\chi (K^*/T^*)\). Theorem 2 states that \((X_ T,K^*/T^*)\) is a space of signatures (which means a space of orderings in the sense of the author [Trans. Am. Math. Soc. 258, 505-521 (1980; Zbl 0427.10015)]).
scientific article; zbMATH DE number 4093565
scientific article