Hypergroups, hyperrings, and hyperfields are a generalization of ordinary groups, rings and fields, in which addition is allowed to be multivalued. To get an idea how such a notion may be used, think of the three possible signs a real number could have: -1, 0,1. Suppose we want to define the addition `+' to make an additive group with elements {-1,0,1}, which expresses how signs are added.
Adding two positive numbers always gives a positive number, so we would like that 1 + 1 = 1. Also, adding negative numbers always gives a negative number, so put (-1) + (-1) = -1. But adding a positive and a negative number may give positive, negative, or zero, and so there is no satisfactory way to use normal addition to express the addition of signs. The solution is to allow the sum of two elements to have more than one value, and this leads to hypergroups, hyperrings, and hyperfields.
The goal of this project is to make and analyse finite hyperfields.