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Partial Order Equivalence Relation Axiom Category Continuation Of Exponent Small Transitive Antisymmetric Relation Between Objects Reflexive

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Quasi Order

Definition

A pre-order on a set is a relation $\lesssim$ on satisfying the following two axioms:

: reflexivity: $s \lesssim s$ for all $s \in S$ , and
: transitivity: If $s \lesssim t$ and $t \lesssim u$ , then $s \lesssim u$ ; for all $s,t,u \in S$ .

Partial order induced by a pre-order

Given such a relation, define a new relation $s\sim t$ on by

$\displaystyle s\sim t \hbox{ if and only if } s\lesssim t \hbox{ and } t \lesssim s.$

Then $\sim$ is an equivalence relation on

, and $\lesssim$ induces a partial order $\leq$ on the set $S/\sim$ of equivalence classes of $\sim$ defined by

$\displaystyle [s] \leq [t] \hbox{ if and only if } s \lesssim t,$

where

and

denote the equivalence classes of

and

. In particular, $\leq$ does satisfy antisymmetry, whereas $\lesssim$ may not.

Pre-orders as categories

A pre-order $\lesssim$ on a set can be considered as a small category, in the which the objects are the elements of and there is a unique morphism from to if $x\lesssim y$ (and none otherwise).

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