Abstract
Making use of the intimate relations between real places and orderings, we continue studying the spaces of orderings of planar ternary rings (PTRs) with rational prime field. As in the classical case, given a preorderingS of such a PTRT, then the productAS of all natural place ringsAp associated to the orderingsP ∈ X/S is itself a place ring ofT. In particular, ifS is a nontrivial fan ofT thenAs≠ T. Thus L. Brocker's celebrated theorem on the trivialization of fans also applies to our setting:
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