Parasemifield

Definition

A parasemifield is a set $S$ with associative, commutative operations $+$ and $\cdot$ such that multiplication distributes over addition and $(S,\cdot,1)$ is an abelian group. No additive identity is required.

Thus a parasemifield differs from a semifield: a semifield has an additive zero $0$, which is absorbing for multiplication, and only its nonzero elements form a multiplicative group. Removing the additive zero from an idempotent semifield produces an idempotent parasemifield.