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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 KK is idempotent semiring that has a zero 00 and in which for every non-zero element xx there is an element yy such that xy=1x \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

xy:x+y=y 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 *K^* given on the same set of elements as KK 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=yx^n = y has a solution for all xKx\in K and n=1,2,n=1,2,\ldots.

Theorem

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

p 0+p 1x 1+p 2x 2++p nx n=y p_0 + p_1 \cdot x^1 + p_2 \cdot x^2 + \ldots + p_n x^n = y

with y,p 0,,p nKy,p_0, \ldots, p_n \in K and n=1,2,n=1,2,\ldots has a solution in KK if and only if yp 0y \geq p_0. If p 0=0p_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,,p ny, 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} \{ 0, 1 \} with 01 0 \leq 1 and +== + = \cdot = \vee is commutative and algebraically closed but does not fulfill the assumptions of theorem . Indeed, the equation 1x=1 1 \cdot x = 1 does not have a unique solution.

References

In the reference the assumption that 0a=00 \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

Last revised on February 25, 2019 at 06:39:26. See the history of this page for a list of all contributions to it.