Formal Methods in Manufacturing

2.3 MATHEMATICAL BACKGROUND

This section provides the algebraic background necessary to obtain a linear representation of the firing instants in a TEG. This section is rather technical and does not claim to be a complete presentation. For a more exhaustive description of the mathematical background, the interested reader is referred to [2].

2.3.1 ORDERED SETS

Before introducing the basic notion of idempotent semirings (or dioids), the fundamentals of ordered sets will be recalled.

Definition 2.1 (Order relation) A binary relation $\underset{\_}{\prec }$ on a set $\mathcal{C}$ is an order relation if the following properties hold $\forall a,b,c\in \mathcal{C}$ :

• Reflexivity: $a\underset{\_}{\prec }a$

• Anti-symmetry: $(a\underset{\_}{\prec }b$ and $b\underset{\_}{\prec }a)\Rightarrow a=b$

• Transitivity: $(a\underset{\_}{\prec }b$ and $b\underset{\_}{\prec }c)\Rightarrow a\underset{\_}{\prec }c$

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Definition 2.2 (Ordered set) A set ...