Knowlpedia

Definition

Idempotent semiring

A semiring whose addition satisfies a+a=a.

Definition

An idempotent semiring is a SS whose addition is idempotent:

a+a=afor every a∈S.a+a=a\qquad\text{for every }a\in S.

In the tropical-algebra pages, idempotent semirings are normally assumed commutative unless noncommutative multiplication is explicitly mentioned.

Order-theoretic meaning

Idempotent addition is a join operation for the a≤b  ⟺  a+b=ba\le b\iff a+b=b. Thus finite sums behave like finite suprema rather than like repeated counting. Multiplication distributes over these joins and is monotone in each variable.

Examples

The of a set, with union as addition and intersection as multiplication, is an idempotent . The Boolean and tropical semifields are further examples. An ordinary nonzero ring cannot be additively idempotent: a+a=aa+a=a and additive cancellation would force a=0a=0.

Terminology

The word dioid is not uniform: some authors use it for every idempotent semiring, while others add order-completeness assumptions. The explicit phrase “idempotent semiring” is used here.

References
  1. Grigori L. Litvinov, Viktor P. Maslov, and Grigori B. Shpiz, “Linear functionals on idempotent spaces: an algebraic approach,” 2000. arXiv:math/0012268. Relevant: the algebraic idempotent-semiring viewpoint.
  2. Grigori L. Litvinov, Viktor P. Maslov, and Grigori B. Shpiz, “Idempotent (Asymptotic) Mathematics and the Representation Theory,” 2002. arXiv:math/0206025.