Residuated mapping

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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).


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.


[a1] T.S. Blyth, M.F. Janowitz, "Residuation theory" , Pergamon (1972)
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Residuated mapping. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article