Semifield

In mathematics, a semifield is an algebraic structure with two binary operations, addition and multiplication, which is similar to a field, but with some axioms relaxed. 
Overview.
The term semifield has two conflicting meanings, both of which include fields as a special case.
Primitivity of semifields.
A semifield D is called right (resp. left) primitive if it has an element w such that the set of nonzero elements of D* is equal to the set of all right (resp. left) principal powers of w.
Examples.
We only give examples of semifields in the second sense, i.e. additive semigroups with distributive multiplication. Moreover, addition is commutative and multiplication is associative in our examples.