Orderings and Signatures of Higher Level on Multirings and Hyperfields

Multirings are objects like rings but with multi-valued addition. They are a variant of other objects called hyperrings, defined by Krasner [12], [13]. In [16] the second author defines multirings, introduces a certain special class of multirings called real reduced multirings, defines a natural reflection A Ã Qred(A) from the category of multirings satisfying −1 / ∈ ∑ A2 to the full subcategory of real reduced multirings, provides an elementary first-order description of these objects, and proves that these objects are precisely the spaces of signs, also known as abstract real spectra, considered earlier in [1], [15]. In the present paper we extend results of E. Becker and others concerning orderings of higher level on fields and rings to orderings of higher level on hyperfields and multirings and, in the process of doing this, we establish higher level analogs of the results in [16]. In particular, we introduce a class of multirings called `-real reduced multirings, define a natural reflection A Ã Q`-red(A) from the category of multirings satisfying −1 / ∈ ∑ A2` to the full subcategory of `-real reduced multirings, and provide an elementary first-order description of these objects. The relationship between `-real reduced hyperfields and the spaces of signatures defined by Mulcahy and Powers [20], [21], [22] is also