It is known that every preordering of a field is an intersection of all orderings containing it. This fact is regarded by the author as the Intersection Theorem. Many authors proved it in a more general setting: preorderings of 2-power exponent [E. Becker, Hereditary-Pythagorean fields and orderings of higher level, Inst. Mat. Pura Apl., Rio de Janeiro (1978; Zbl 0509.12020)], preorderings of arbitrary even exponent [E. Becker, J. Reine Angew. Math. 307/308, 8-30 (1979; Zbl 0398.12012)], skew fields and preorderings of arbitrary even exponent [V. Powers, J. Algebra 136, 51-59 (1991; Zbl 0715.12002)], and Noetherian rings [the author, Commun. Algebra 27, 5083-5096 (1999; Zbl 0939.16025)].
In the present paper the author steers towards proving a completely general version of the Intersection Theorem, that is, for divisible precones of any exponent on any integral domain. He follows the strategy applied in the special setting mentioned above generalizing some results of Becker. However, to achieve the final aim he needs one more fact, which is formulated as a conjecture: every complete precone on an integral domain is a fan.
In the paper this conjecture is verified in the following two cases: complete precones of 2-power exponent and complete precones on Ore domains. Thus in these cases the author obtains the Intersection Theorem.
In the final section the reader finds the description of all cones on the ring of quantum polynomials.