Hypergroups and hyperfields in universal algebra

Abstract

Hypergroups are lifted to power semigroups with negation, yielding a method of transferring results from semigroup theory. This applies to analogous structures such as hypergroups, hyperfields, and hypermodules, and permits us to transfer the general theory espoused in [19] to the hypertheory.Definition 1.1. A monoid (A, •, 1) acts on a set S if there is a multiplication A × S → S satisfying 1s = s and (a 1 a 2 )s = a 1 (a 2 s) for all a i ∈ A and s ∈ S.A pre-semiring (A, •, +, 1) is a multiplicative monoid (A, •, 1 R ) also possessing the structure of an additive Abelian semigroup (A, +), on which (A, •) acts.A pre-semifield is a pre-semiring (A, •, +, 1) for which (A, •, 1) is a group. A premodule S over a monoid (A, •, 1) is an Abelian group (S, +, 0) on which A acts, also satisfying the condition: