Question

Show that the tropical semiring (N ∪ {[infinity]}, +, 0, min, [infinity])

satisfies the axioms of a semiring.

Answer 1

To show that the tropical semiring (N ∪ {∞}, +, 0, min, ∞) satisfies the axioms of a semiring, we need to verify that it satisfies the properties of addition, multiplication, zero, and one.

Step-by-step explanation:

To show that the tropical semiring (N ∪ {∞}, +, 0, min, ∞) satisfies the axioms of a semiring, we need to verify that it satisfies the properties of addition, multiplication, zero, and one.

1. Addition: Addition in the tropical semiring is defined as the maximum of two numbers, represented as max(a, b). It is commutative and associative, and has an identity element of -∞.
2. Multiplication: Multiplication in the tropical semiring is defined as the minimum of two numbers, represented as min(a, b). It is commutative and associative, and has an identity element of ∞.
3. Zero: The element 0 is the identity element for addition, and its maximum with any number a is equal to a.
4. One: The element 1 is the identity element for multiplication, and its minimum with any number a is equal to a.