Idempotence of finitely generated commutative semifields

Abstrakt

We prove that a commutative parasemifield S is additively idempotent, provided that it is finitely generated as a semiring. Consequently, every proper commutative semifield T that is finitely generated as a semiring is either additively constant or additively idempotent.

As part of the proof, we use the classification of finitely generated lattice-ordered groups to prove that a certain monoid associated to the parasemifield S has a distinguished geometrical property called prismality.

Klíčová slova

• Commutative semiring
• parasemifield
• idempotent
• convex
• affine monoid