# Difference between revisions of "Residuated mapping"

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− | An [[Isotone mapping|isotone mapping]] | + | {{TEX|done}} |

+ | An [[Isotone mapping|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 | + | 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|Lattice]]). |

====Comments==== | ====Comments==== | ||

− | The mapping | + | 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|Adjoint functor]]). For the antitone analogues of residuated mappings see [[Galois correspondence|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> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> T.S. Blyth, M.F. Janowitz, "Residuation theory" , Pergamon (1972)</TD></TR></table> |

## Revision as of 07:52, 12 August 2014

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 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