Residuated mapping
An isotone mapping 
from a partially ordered set 
into a partially ordered set 
for which there is an isotone mapping 
from 
into 
such that 
for all 
and 
for all 
. If 
and 
are complete lattices, then this is equivalent to the equality:
 |
for every subset 
of 
. The set of residuated mappings of a partially ordered set 
into itself forms a semi-group which can be partially ordered (see Ordered semi-group) by setting 
if 
for all 
. The properties of this partially ordered semi-group are closely linked to the properties of the partially ordered set 
(see Lattice).
Comments
The mapping 
appearing in the definition is called the residual of 
; it is uniquely determined by 
. A more symmetric terminology, borrowed from category theory, calls 
a left adjoint and 
a right adjoint (see Adjoint functor). For the antitone analogues of residuated mappings see Galois correspondence.
References
| [a1] | T.S. Blyth, M.F. Janowitz, "Residuation theory" , Pergamon (1972) |
Residuated mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Residuated_mapping&oldid=17933