Difference between revisions of "Residuated mapping"
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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: | 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: | ||
$$\phi(\sup A)=\sup\phi(A)$$ | $$\phi(\sup A)=\sup\phi(A)$$ | ||
| − | 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 [[ | + | 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]]. | + | 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
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) |
Residuated mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Residuated_mapping&oldid=34373