nLab idempotent semifield

Idea

An idempotent semifield is an idempotent semiring that has a zero and in which every non-zero element is invertible. Or, in other words, a skewfield in which an element may not have a negative but is always idempotent with respect to addition.

Definition

An idempotent semifield $K$ is idempotent semiring that has a zero $0$ and in which for every non-zero element $x$ there is an element $y$ such that $x \cdot y = 1$. It is said to be commutative if the multiplication is commutative.

Properties as a lattice

As in the case of an idempotent semiring there is a partial order given by

$$x \leq y \;\;{:\!\!\Longleftrightarrow}\;\; x + y = y$$

which has a join given by addition. In case of an idempotent semifield this partial order also has a meet. To each semifield there is a dual semifield $K^*$ given on the same set of elements as $K$ and with the same zero, identity, and multiplication but with addition given by interchanging join and meet, i.e. the meet becomes the new addition on non-zero elements and the zero element behaves neutral.

Equations in semifields

A version of the fundamental theorem of algebra can be formulated for semifields: A semifield is said to be algebraically closed if the equation $x^n = y$ has a solution for all $x\in K$ and $n=1,2,\ldots$.

Theorem

In an algebraically closed commutative idempotent semifield $K$ with $0 \cdot a = 0$ for all $a \in K$ the equation

$$p_0 + p_1 \cdot x^1 + p_2 \cdot x^2 + \ldots + p_n x^n = y$$

with $y,p_0, \ldots, p_n \in K$ and $n=1,2,\ldots$ has a solution in $K$ if and only if $y \geq p_0$. If $p_0 = 0$, then the solution is unique and can be expressed in terms of the coefficients with help of radicals, multiplication, inverse and meet (i.e. a polynomial in the radicals of $y, p_0, \ldots, p_n$ over the dual semifield).

Examples

• The max-plus algebra $[-\infty, +\infty)$ with addition given by maximum and multiplication given by ordinary addition.

• The two element semifield $\{ 0, 1 \}$ with $0 \leq 1$ and $+ = \cdot = \vee$ is commutative and algebraically closed but does not fulfill the assumptions of theorem . Indeed, the equation $1 \cdot x = 1$ does not have a unique solution.

Related concepts

• rig

• rng (nonunital ring)

• ring, near-ring

• tropical geometry

• tropical semiring

• idempotent semiring

References

In the reference the assumption that $0 \cdot a = 0$ is not explicit (otherwise the base step in the induction in the proof of Proposition 2 therein doesn’t work).

• Shpiz, Solution of algebraic equations in idempotent semifields, Communications of the Moscow Mathematical Society (1960), russian english