You are here
Homesemifield
Primary tabs
semifield
There are different definitions of semifield. We give three such which are not equivalent.
Let $K$ be a set with two binary operations “$+$” and “$\cdot$”.

Semifield $(K,\,+,\,\cdot)$ is a semiring where all nonzero elements have a multiplicative inverse.

Semifield is the algebraic system $(K,\,+,\,\cdot)$, where $(K,\,+)$ is a group (identity $:=0$), the multiplication “$\cdot$” distributes over the addition “$+$”, $K$ contains the multiplicative identity $:=1$ and all equations $ax=b$ and $ya=b$ with $a\neq 0$ have solutions $x$, $y$ in $K$.

Semifield $(K,\,+,\,\cdot)$ satisfies all postulates of field except the associativity of the multiplication “$\cdot$”.
Related:
NonAssociativeAlgebra
Major Section:
Reference
Type of Math Object:
Definition
Parent:
Mathematics Subject Classification
16Y60 no label found12K10 no label found Forums
 Planetary Bugs
 HS/Secondary
 University/Tertiary
 Graduate/Advanced
 Industry/Practice
 Research Topics
 LaTeX help
 Math Comptetitions
 Math History
 Math Humor
 PlanetMath Comments
 PlanetMath System Updates and News
 PlanetMath help
 PlanetMath.ORG
 Strategic Communications Development
 The Math Pub
 Testing messages (ignore)
 Other useful stuff
 Corrections