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Difference between revisions of "Residuated mapping"

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\phi(\sup A)=\sup\phi(A)
  
for every subset A of P. The set of residuated mappings of a partially ordered set P into itself forms a [[semi-group]] which can be partially ordered (see [[Ordered semi-group|Ordered semi-group]]) by setting \phi\leq\psi if \phi(x)\leq\psi(x) for all x\in P. The properties of this partially ordered semi-group are closely linked to the properties of the partially ordered set P (see [[Lattice]]).
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for every subset A of P. The set of residuated mappings of a partially ordered set P into itself forms a [[semi-group]] which can be partially ordered (see [[Ordered semi-group]]) by setting \phi\leq\psi if \phi(x)\leq\psi(x) for all x\in P. The properties of this partially ordered semi-group are closely linked to the properties of the partially ordered set P (see [[Lattice]]).
  
  
  
 
====Comments====
 
====Comments====
The mapping \phi' appearing in the definition is called the residual of \phi; it is uniquely determined by \phi. A more symmetric terminology, borrowed from category theory, calls \phi a left adjoint and \phi' a right adjoint (see [[Adjoint functor]]). For the antitone analogues of residuated mappings see [[Galois correspondence]].
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The mapping \phi' appearing in the definition is called the ''residual mapping'' of \phi; it is uniquely determined by \phi. A more symmetric terminology, borrowed from category theory, calls \phi a left adjoint and \phi' a right adjoint (see [[Adjoint functor]]). For the antitone analogues of residuated mappings see [[Galois correspondence]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T.S. Blyth,  M.F. Janowitz,  "Residuation theory" , Pergamon  (1972)</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  T.S. Blyth,  M.F. Janowitz,  "Residuation theory" , Pergamon  (1972)</TD></TR>
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</table>

Latest revision as of 20:53, 8 January 2016

2020 Mathematics Subject Classification: Primary: 06A [MSN][ZBL]

An isotone mapping \phi from a partially ordered set P into a partially ordered set P' for which there is an isotone mapping \phi' from P' into P such that \phi'(\phi(x))\geq x for all x\in P and \phi(\phi'(x'))\leq x' for all x'\in P'. If P and P' are complete lattices, then this is equivalent to the equality:

\phi(\sup A)=\sup\phi(A)

for every subset A of P. The set of residuated mappings of a partially ordered set P into itself forms a semi-group which can be partially ordered (see Ordered semi-group) by setting \phi\leq\psi if \phi(x)\leq\psi(x) for all x\in P. The properties of this partially ordered semi-group are closely linked to the properties of the partially ordered set P (see Lattice).


Comments

The mapping \phi' appearing in the definition is called the residual mapping of \phi; it is uniquely determined by \phi. A more symmetric terminology, borrowed from category theory, calls \phi a left adjoint and \phi' 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)
How to Cite This Entry:
Residuated mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Residuated_mapping&oldid=35431
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article