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Difference between revisions of "Residuated mapping"

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(also: residual mapping)
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{{TEX|done}}{{MSC|06A}}
 
{{TEX|done}}{{MSC|06A}}
 
''residual mapping''
 
  
 
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:
 
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:
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$$\phi(\sup A)=\sup\phi(A)$$
 
$$\phi(\sup A)=\sup\phi(A)$$
  
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====

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=37396
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article