# Difference between revisions of "Residuated mapping"

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− | + | {{TEX|done}}{{MSC|06A}} | |

− | + | 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 | + | $$\phi(\sup A)=\sup\phi(A)$$ |

+ | |||

+ | 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==== | ====Comments==== | ||

− | The mapping | + | 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> | + | <table> |

+ | <TR><TD valign="top">[a1]</TD> <TD valign="top"> T.S. Blyth, M.F. Janowitz, "Residuation theory" , Pergamon (1972)</TD></TR> | ||

+ | </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