Residuation and Representation

Abstract

A very natural way to express the properties of matrices implied by Theorem 8–9 is to use the language of residuation theory, as set out in [42], for example.We recall that a function f: S → T where S, T are given partially ordered sets, is called residuated if there exists a function f ^*: T → S such that the following hold:

• R1 f is isotone and f ^* is isotone

• R2
  $$\begin{matrix} \left( i \right)f\left( {{f}^{*}}\left( t \right) \right)\le tforallt\varepsilon T\\ \left( ii \right){{f}^{*}}\left( f\left( s \right) \right)\ge sforalls\varepsilon S\\\end{matrix}$$

  (10-1)