Residuated 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 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)