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An [[Isotone mapping|isotone mapping]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081540/r0815401.png" /> from a partially ordered set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081540/r0815402.png" /> into a partially ordered set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081540/r0815403.png" /> for which there is an isotone mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081540/r0815404.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081540/r0815405.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081540/r0815406.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081540/r0815407.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081540/r0815408.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081540/r0815409.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081540/r08154010.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081540/r08154011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081540/r08154012.png" /> are complete lattices, then this is equivalent to the equality:
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{{TEX|done}}{{MSC|06A}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081540/r08154013.png" /></td> </tr></table>
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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 lattice]]s, then this is equivalent to the equality:
  
for every subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081540/r08154014.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081540/r08154015.png" />. The set of residuated mappings of a partially ordered set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081540/r08154016.png" /> into itself forms a semi-group which can be partially ordered (see [[Ordered semi-group|Ordered semi-group]]) by setting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081540/r08154017.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081540/r08154018.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081540/r08154019.png" />. The properties of this partially ordered semi-group are closely linked to the properties of the partially ordered set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081540/r08154020.png" /> (see [[Lattice|Lattice]]).
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$$\phi(\sup A)=\sup\phi(A)$$
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081540/r08154021.png" /> appearing in the definition is called the residual of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081540/r08154022.png" />; it is uniquely determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081540/r08154023.png" />. A more symmetric terminology, borrowed from category theory, calls <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081540/r08154024.png" /> a left adjoint and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081540/r08154025.png" /> a right adjoint (see [[Adjoint functor|Adjoint functor]]). For the antitone analogues of residuated mappings see [[Galois correspondence|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

2010 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=17933
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article