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# Tropical Semirings

Tropical Semirings

AUTHORS:

- Travis Scrimshaw (2013-04-28) - Initial version

class `sage.rings.semirings.tropical_semiring.``TropicalSemiring`(base, use_min=True)

  Bases: `sage.structure.parent.Parent`, `sage.structure.unique_representation.UniqueRepresentation`

  The tropical semiring.

  Given an ordered additive semigroup \(R\), we define the tropical semiring \(T = R \cup \{+\infty\}\) by defining tropical addition and multiplication as follows:

  \[a \oplus b = \min(a, b), \quad \quad a \odot b = a + b.\]

  In particular, note that there are no (tropical) additive inverses (except for \(\infty\)), and every element in \(R\) has a (tropical) multiplicative inverse.

  There is an alternative definition where we define \(T = R \cup \{-\infty\}\) and alter tropical addition to be defined by

  \[a \oplus b = \max(a, b).\]

  To use the \(\max\) definition, set the argument `use_min = False`.

  Warning

  `zero()` and `one()` refer to the tropical additive and multiplicative identities respectively. These are not the same as calling `T(0)` and `T(1)` respectively as these are not the tropical additive and multiplicative identities respectively.

  Specifically do not use `sum(...)` as this converts \(0\) to \(0\) as a tropical element, which is not the same as `zero()`. Instead use the `sum` method of the tropical semiring:

      sage: T = TropicalSemiring(QQ)

      sage: sum([T(1), T(2)]) # This is wrong
      0
      sage: T.sum([T(1), T(2)]) # This is correct
      1

  Be careful about using code that has not been checked for tropical safety.

  INPUT:

  - `base` – the base ordered additive semigroup \(R\)
  - `use_min` – (default: `True`) if `True`, then the semiring uses \(a \oplus b = \min(a, b)\); otherwise uses \(a \oplus b = \max(a, b)\)

  EXAMPLES:

      sage: T = TropicalSemiring(QQ)
      sage: elt = T(2); elt
      2

  Recall that tropical addition is the minimum of two elements:

      sage: T(3) + T(5)
      3

  Tropical multiplication is the addition of two elements:

      sage: T(2) * T(3)
      5
      sage: T(0) * T(-2)
      -2

  We can also do tropical division and arbitrary tropical exponentiation:

      sage: T(2) / T(1)
      1
      sage: T(2)^(-3/7)
      -6/7

  Note that “zero” and “one” are the additive and multiplicative identities of the tropical semiring. In general, they are not the elements \(0\) and \(1\) of \(R\), respectively, even if such elements exist (e.g., for \(R = \ZZ\)), but instead the (tropical) additive and multiplicative identities \(+\infty\) and \(0\) respectively:

      sage: T.zero() + T(3) == T(3)
      True
      sage: T.one() * T(3) == T(3)
      True
      sage: T.zero() == T(0)
      False
      sage: T.one() == T(1)
      False

  `Element`

    alias of `TropicalSemiringElement`

  `additive_identity`()

    Return the (tropical) additive identity element \(+\infty\).

    EXAMPLES:

        sage: T = TropicalSemiring(QQ)
        sage: T.zero_element()
        +infinity

  `gens`()

    Return the generators of `self`.

    EXAMPLES:

        sage: T = TropicalSemiring(QQ)
        sage: T.gens()
        (1, +infinity)

  `infinity`()

    Return the (tropical) additive identity element \(+\infty\).

    EXAMPLES:

        sage: T = TropicalSemiring(QQ)
        sage: T.zero_element()
        +infinity

  `multiplicative_identity`()

    Return the (tropical) multiplicative identity element \(0\).

    EXAMPLES:

        sage: T = TropicalSemiring(QQ)
        sage: T.one_element()
        0

  `one`()

    Return the (tropical) multiplicative identity element \(0\).

    EXAMPLES:

        sage: T = TropicalSemiring(QQ)
        sage: T.one_element()
        0

  `one_element`()

    Return the (tropical) multiplicative identity element \(0\).

    EXAMPLES:

        sage: T = TropicalSemiring(QQ)
        sage: T.one_element()
        0

  `zero`()

    Return the (tropical) additive identity element \(+\infty\).

    EXAMPLES:

        sage: T = TropicalSemiring(QQ)
        sage: T.zero_element()
        +infinity

  `zero_element`()

    Return the (tropical) additive identity element \(+\infty\).

    EXAMPLES:

        sage: T = TropicalSemiring(QQ)
        sage: T.zero_element()
        +infinity

class `sage.rings.semirings.tropical_semiring.``TropicalSemiringElement`

  Bases: `sage.structure.element.RingElement`

  An element in the tropical semiring over an ordered additive semigroup \(R\). Either in \(R\) or \(\infty\). The operators \(+, \cdot\) are defined as the tropical operators \(\oplus, \odot\) respectively.

  `lift`()

    Return the value of `self` lifted to the base.

    EXAMPLES:

        sage: T = TropicalSemiring(QQ)
        sage: elt = T(2)
        sage: elt.lift()
        2
        sage: elt.lift().parent() is QQ
        True
        sage: T.additive_identity().lift().parent()
        The Infinity Ring

  `multiplicative_order`()

    Return the multiplicative order of `self`.

    EXAMPLES:

        sage: T = TropicalSemiring(QQ)
        sage: T.multiplicative_identity().multiplicative_order()
        1
        sage: T.additive_identity().multiplicative_order()
        +Infinity

class `sage.rings.semirings.tropical_semiring.``TropicalToTropical`

  Bases: `sage.categories.map.Map`

  Map from the tropical semiring to itself (possibly with different bases). Used in coercion.

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