Idempotence of finitely generated commutative semifields

TL;DR

This paper proves that finitely generated commutative parasemifields are additively idempotent, revealing a structural property that connects algebraic generation with additive behavior, and utilizes lattice-ordered group classification.

Contribution

It establishes that finitely generated commutative parasemifields are additively idempotent, a new result linking algebraic generation to additive structure.

Findings

Finitely generated commutative parasemifields are additively idempotent.

Proper finitely generated commutative semifields are either additively constant or idempotent.

The proof involves classification of lattice-ordered groups and geometric properties of associated monoids.

Abstract

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.