Examples
We only give examples of semifields in the second sense, i.e. additive semigroups with distributive multiplication. Moreover, addition is commutative and multiplication is associative in our examples.
- Positive real numbers with the usual addition and multiplication form a commutative semifield.
- Rational functions of the form f /g, where f and g are polynomials in one variable with positive coefficients form a commutative semifield.
- Max-plus algebra, or the tropical semiring, (R, max, +) is a semifield. Here the sum of two elements is defined to be their maximum, and the product to be their ordinary sum.
- If (A,≤) is a lattice ordered group then (A,+,·) is an additively idempotent semifield. The semifield sum is defined to be the sup of two elements. Conversely, any additively idempotent semifield (A,+,·) defines a lattice-ordered group (A,≤), where a≤b if and only if a + b = b.
Read more about this topic: Semifield
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