In algebra, a max tropical semiring (also called a max-plus algebra) is the semiring
(ℝ ∪ -Inf, ⊕, ⊗) containing the ring of real numbers ℝ augmented by negative infinity,
the max function (returns the greater of two real numbers), and addition.

In max tropical algebra, x ⊕ y = max(x, y) and x ⊗ y = x + y. The identity for ⊕
is -Inf (the max of any number with -infinity is that number), and the identity for ⊗ is 0.

;Task:

* Define functions or, if the language supports the symbols as operators, operators for ⊕ and ⊗ that fit the above description. If the language does not support ⊕ and ⊗ as operators but allows overloading operators for a new object type, you may instead overload + and * for a new min tropical albrbraic type. If you cannot overload operators in the language used, define ordinary functions for the purpose. 

Show that 2 ⊗ -2 is 0, -0.001 ⊕ -Inf is -0.001, 0 ⊗ -Inf is -Inf, 1.5 ⊕ -1 is 1.5, and -0.5 ⊗ 0 is -0.5.

* Define exponentiation as serial ⊗, and in general that a to the power of b is a * b, where a is a real number and b must be a positive integer. Use either ↑ or similar up arrow or the carat ^, as an exponentiation operator if this can be used to overload such "exponentiation" in the language being used. Calculate 5 ↑ 7 using this definition.

* Max tropical algebra is distributive, so that

   a ⊗ (b ⊕ c) equals a ⊗ b ⊕ b ⊗ c, 

where ⊗ has precedence over ⊕. Demonstrate that 5 ⊗ (8 ⊕ 7) equals  5 ⊗ 8 ⊕ 5 ⊗ 7.

* If the language used does not support operator overloading, you may use ordinary function names such as tropicalAdd(x, y) and tropicalMul(x, y).


;See also
:;*[[https://en.wikipedia.org/wiki/Tropical_semiring#tropical_algebra Tropical algebra]]
:;*[[https://arxiv.org/pdf/1908.07012.pdf Tropical geometry review article]]
:;*[[https://en.wikipedia.org/wiki/Operator_overloading Operator overloading]]


