  Semifields are semirings in which every nonzero element has a multiplicative
inverse. A rough classification uses the characteristic of the semifield, that
is the isomorphism type of the semifield generated by the two neutral elements.
For every characteristic, we provide a structure theorem that reduces the
classification of semifields to the classification of better-known algebraic
structures. Every semifield of characteristic $p$ is actually a field. There is
an equivalence between semifields of characteristic one and lattice-ordered
groups. Strict semifields of characteristic zero are quotients of cancellative
semifields and there is an equivalence between concellative strict semifields
and a particular class of partially ordered rings.
