Hyperfields
"Krasner, Marshall, Connes and Consani and the author came tohyperfields for different reasons, motivated by different mathematicalproblems, 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 mainstreamnotion is that a multivalued operation does not fit to the tradition ofset-theoretic terminology, which forces to avoid multivalued maps atany cost.I believe the taboo on multivalued maps has no real ground, andeventually will be removed. Hyperfields, as well as multigroups, hyperrings and multirings, are legitimate algebraic objects related in manyways to the classical core of mathematics. They provide elegant terminological and conceptual opportunities. In this paper I try to presentnew 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