Difference between revisions of "Residuated mapping"
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\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=35431