Definition
Idempotent semifield
A semifield with idempotent addition.
Definition
An idempotent semifield is a semifield that is also an idempotent semiring, so
for every , and every nonzero element is multiplicatively invertible.
Structure
The nonzero elements form an abelian group, while idempotent addition gives the whole carrier its natural partial order. Multiplication by a nonzero element is an order automorphism because it has a multiplicative inverse. This interaction between group and order is the algebraic basis of tropical linear constructions.
Removing the additive zero produces a lattice-ordered multiplicative group, and adjoining a bottom element reverses this construction. See the categorical equivalence with lattice-ordered abelian groups.
Examples and scope
The Boolean semifield is the smallest example. The max-plus and min-plus tropical semifields are linearly ordered examples. Some authors use tropical semifield for any idempotent semifield; this corpus reserves that name without qualification for the standard max-plus object.
References
- Grigori L. Litvinov, Viktor P. Maslov, and Grigori B. Shpiz, “Linear functionals on idempotent spaces: an algebraic approach,” 2000. arXiv:math/0012268.
- Jaiung Jun, “Algebraic Geometry Over Hyperrings,” Advances in Mathematics 323 (2018), 142–192. arXiv:1512.04837. Relevant: totally ordered idempotent semifields and their associated hyperfields.