semifield

There are different definitions of semifield.  We give three such which are not equivalent (http://planetmath.org/Biconditional).

Let $K$ be a set with two binary operations “$+$” and “$\cdot$”.

• Semifield  $(K,+,\cdot )$  is a semiring where all non-zero 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 “$+$”,  $K$ the multiplicative identity  $:=1$  and all equations  $ax=b$  and  $ya=b$  with  $a\ne 0$  have solutions $x$, $y$ in $K$.

• Semifield  $(K,+,\cdot )$  satisfies all postulates of field except the associativity of the multiplication “$\cdot$”.