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:
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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