Let 
and 
be vector spaces over a field 
, and let 
be a field automorphism
of 
. A map 
is 
-semilinear if
 |  |  |
(1)
|
 |  |  |
(2)
|
Here 
and 
.
Some authors more generally allow 
and 
to have different scalar fields and take 
to be a field homomorphism
between them.
When 
is the identity map, 
is a linear map. When 
and 
, the map 
is antilinear. Bijective
semilinear maps from 
to itself are called semilinear automorphisms or
semilinear transformations. Under composition, the bijective semilinear maps associated
with all field automorphisms form the general
semilinear group 
.
