## Sets

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# Sets

Definition 3.1   a pre-ordered set is a set with the property that, between any two objects there is at most one arrow. This entails that there exists a binary relation R between the objects of the pre-ordered set such that the following holds:

1. aRa $\Longrightarrow$ reflexivity

2. if aRb and bRc then aRc $\Longrightarrow$ transitivity

Definition 3.2   a poset is a pre-ordered set with the extra property of being antisymmetric: $pRq\hspace{.2in}and\hspace{.2in}qRp\hspace{.2in}\Rightarrow p=q$

Cecilia Flori 2007-01-02