31 lines
1.9 KiB
Plaintext
31 lines
1.9 KiB
Plaintext
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]]
|
||
|
||
|