Spread-Theoretic Dual of a Semifield

Abstract

Given a finite pre-semifield $(S,+,\circ )$, the dual pre-semifield $%(S,+,\ast )$ has multiplication $a\ast b=b\circ a$. In this article, acharacterization is given of the dual pre-semifield in terms of theassociated spreads. This is then used to give a new proof that shows thatself-dual pre-semifield spreads when transposed and dualized constructself-transpose semifield spreads. When the self-dual pre-semifield isactually commutative then the transpose-dual spread is symplectic. We alsogive a spread-only description of the six semifields arising from a givensemifield.