Definition:Lower Section/Class Theory

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Let $A$ be a class under a total ordering $\preccurlyeq$.

Let $L$ be a subclass of $A$ such that:

$\forall x \in L: \forall a \in A \setminus L: x \preccurlyeq a$

where $A \setminus L$ is the difference between $A$ and $L$.

Then $L$ is known as a lower section of $A$.

Also see

  • Results about lower sections can be found here.