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Quotients of index two and general quotients in a space of orderings

100%

Gładki P. , Marshall M.

Fundamenta Mathematicae

2015

tom 229

nr 3

255-275

We investigate quotient structures and quotient spaces of a space of orderings arising from subgroups of index two. We provide necessary and sufficient conditions for a quotient structure to be a quotient space that, among other things, depend on the stability index of the given space. The case of the space of orderings of the field ℚ(x) is particularly interesting, since then the theory developed simplifies significantly. A part of the theory firstly developed for quotients of index 2 generalizes to quotients of index 2ⁿ for arbitrary finite n. Numerous examples are provided.

On quotients of the space of orderings of the field ℚ(x)

100%

Gładki P. , Jacob B.

Banach Center Publications

2016

tom 108

nr 1

63-84

In this paper we present a method of obtaining new examples of spaces of orderings by considering quotient structures of the space of orderings $(X_{ℚ(x)}, G_{ℚ(x)})$ - it is, in general, nontrivial to determine whether, for a subgroup $G₀ ⊂ G_{ℚ(x)}$ the derived quotient structure $(X_{ℚ(x)}|_{G₀}, G₀)$ is a space of orderings, and we provide some insights into this problem. In particular, we show that if a quotient structure arising from a subgroup of index 2 is a space of orderings, then it necessarily is a profinite one.