Hyperfields
"Krasner, Marshall, Connes and Consani and the author came to hyperfields for different reasons, motivated by different mathematical problems, but we came to the same conclusion: the hyperrings and hyperfields are great, very useful and very underdeveloped in the mathematical literature. Probably, the main obstacle for hyperfields to become a mainstream notion is that a multivalued operation does not fit to the tradition of set-theoretic terminology, which forces to avoid multivalued maps at any cost. I believe the taboo on multivalued maps has no real ground, and eventually will be removed. Hyperfields, as well as multigroups, hyperrings and multirings, are legitimate algebraic objects related in many ways to the classical core of mathematics. They provide elegant terminological and conceptual opportunities. In this paper I try to present new evidences for this."
Hyperfields For Tropical Geometry I. Hyperfields And Dequantization, Oleg Viro, http://arxiv.org/abs/1006.3034
The hyperring of adèle classes, Alain Connes, Caterina Consani, http://arxiv.org/abs/1001.4260
