SimpleTropical

This is an implementation of tropical (min-plus) arithmetic in Julia.

Numbers

The tropical numbers consist of the real numbers and infinity. The SimpleTropical module defines the Tropical type (which is a subtype of Number).

julia> using SimpleTropical

julia> x = Tropical(3.5)
Tropical(3.5)

julia> y = Tropical{Int}(4)
Tropical(4)

Tropical infinity is available via TropicalInf:

julia> TropicalInf
Tropical(∞)

Conversion to real numbers

To convert a Tropical number to a real number, use real:

julia> a = Tropical(5)
Tropical(5)

julia> a+2
Tropical(2)

julia> real(a)+2
7

Arithmetic

The + operation is defined as the min of the two values and * as the sum:

julia> x+y
Tropical(3.5)

julia> x*y
Tropical(7.5)

The identity element for + is TropicalInf and the identity element for * is Tropical(0):

julia> x + TropicalInf
Tropical(3.5)

julia> x * Tropical(0)
Tropical(3.5)

No elements in tropical arithmetic have additive inverses, but they do have multiplicative inverses (except for infinity):

julia> inv(x)
Tropical(-3.5)

julia> inv(TropicalInf)
ERROR: AssertionError: TropicalInf is not invertible

Exponentiation by integers works:

julia> x^10
Tropical(35.0)

julia> x^-2
Tropical(-7.0)

Tropical division is permitted, except one cannot divide by infinity:

julia> Tropical(3) / Tropical(2)
Tropical(1)

julia> Tropical(3) / Tropical(Inf)
ERROR: AssertionError: TropicalInf is not invertible

Tropical subtraction is undefined and therefore forbidden.

⊕ and ⊗

The symbols ⊕ and ⊗ may be used instead of + and * for Tropical numbers. In addition, these may be used on Real numbers with the result being the appropriate Tropical number.

julia> 5 ⊕ 2
Tropical(2)

julia> 5 ⊗ 2
Tropical(7)

Note: In the REPL, the symbol ⊕ is created by typing \oplus and then pressing TAB. Likewise, ⊗ is created as \otimes followed by TAB.

Identity elements: zero and one

The Julia function zero normally returns the number zero because that is the identity element for addition; likewise, one returns the number one because that is the identity element for multiplication.

In tropical arithmetic, these identity elements are Tropical(∞) and Tropical(0), respectively. Therefore we define the zero and one functions to return these values.

julia> a = Tropical(3)
Tropical(3)

julia> zero(a)
Tropical(∞)

julia> zero(Tropical)
Tropical(∞)

julia> one(a)
Tropical(0)

julia> one(Tropical)
Tropical(0)

Likewise, the functions zeros and ones return an array of the appropriate values:

julia> zeros(Tropical,3)
3-element Vector{Tropical}:
 Tropical(∞)
 Tropical(∞)
 Tropical(∞)

julia> ones(Tropical,3)
3-element Vector{Tropical}:
 Tropical(0)
 Tropical(0)
 Tropical(0)

Predicates

Use isinf(X) to test if a tropical number is infinity.

julia> isinf(x)
false

julia> isinf(TropicalInf)
true

The usual comparison operators == and !== work as expected:

julia> Tropical(3.0) == Tropical(3)
true

julia> Tropical(3.1) != Tropical(3//1)
true

Likewise, standard ordering comparisons work as expected:

julia> Tropical(π) < Tropical(3)
false

julia> Tropical(π) > Tropical(3)
true

julia> Tropical(5) < TropicalInf
true

Display Style

By default, Tropical numbers are displayed in the form Tropical(xxx) such as Tropical(-3) or Tropical(∞). This behavior can be changed using the function long_tropical_show.

• long_tropical_show(true) gives the default behavior.
• long_tropical_show(false) makes Tropical numbers appear as ordinary real numbers.

julia> long_tropical_show(false)
false

julia> 5 ⊕ 2
2

julia> 5 ⊗ 2
7

julia> typeof(ans)
Tropical{Int64}

Calling long_tropical_show() without any arguments returns the current state for showing Tropical numbers: true for the default behavior and false for the short display.