Orders and signatures of higher level on a commutative ring

One obtains orders of higher level in a commutative ring A by pulling back the higher level orders in the residue fields of its prime ideals. Since inclusion relationships can hold amongst the higher level orders in a field (unlike the level 1 situation), there may exist orders in the ring A which are not contained in a unique order maximal with respect to inclusion. However, if the specializations of an order P are defined to be those orders Q ⊇ P such that Q \ P ⊆ Q ⋂ – Q, every higher level order in A is contained in a unique maximal specialization. The real spectrum of A relative to a higher level preorder T is defined to be the set SperTA of all orders in A containing T. As with the ordinary real spectrum of Coste and Roy, SperTA is given a compact topology in which the closed points are precisely the orders in A maximal with respect to specialization. For 2-primary level, we show that an abstract higher level version of the Hormander-Lojasiewicz Inequality holds and use it to characterize the basic sets in SperTA. A higher level signature on a commutative ring A is a pull-back σ of a higher level signature on the residue field of some prime ideal with σ () = 0. If T is a higher level preorder in A and σ(T) = {0, 1} then σ is called a T-signature. Specializations of T-signatures are defined just as for orders and every T-signature is shown to have a unique maximal specialization. Each T-signature a determines a unique order in A containing T which is maximal with respect to specialization iff σ is. Generalizing a result of M. Marshall, we show for a higher level preorder T in a commutative ring satisfying a certain simple axiom, the space XT of all maximal T-signatures can be embedded in the character group of a suitable abelian group GT of finite even exponent and under this embedding, the pair (XT, GT) is a space of signatures in the sense of Mulcahy and Marshall