Formal Methods in Manufacturing

2.3.5 RESIDUATION THEORY EQUATIONS

Multiplication in idempotent semirings does not necessarily admit an inverse. However, a pseudo inversion of mappings defined over ordered sets is provided by the so-called residuation theory [3,4]. Since dioids are defined on (partially) ordered sets, it is possible to use residuation theory to determine the greatest solution (with respect to the natural order of the dioid) of inequality $f\left(a\right)\underset{\_}{\prec }b$ . Let $\left(\mathcal{D},\oplus ,\otimes \right)$ and $\left(\mathcal{C},\oplus ,\otimes \right)$ be dioids, then the following applies:

Definition 2.17 (Residuated mapping) An isotone mapping $f:\mathcal{C}\to \mathcal{D}$ is said to be residuated, if the inequality $f\left(a\right)\underset{\_}{\prec }b$ has a greatest solution in $\mathcal{D}$ for all $b\in \mathcal{C}$ .

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Theorem 2.1 ([4]) An isotone mapping $f:\mathcal{D}\to \mathcal{C}$ is residuated if and only if there exists a unique ...