Notation 4.3.3.1. Let $J$ be a linearly ordered set. We say that a subset $I \subseteq J$ is an initial segment of $J$ if it is closed downwards: that is, if, for every pair of elements $i \leq j$ in $J$, we have $(j \in I) \Rightarrow (i \in I)$. We will write $I \sqsubseteq J$ to indicate that $I$ is an initial segment of $J$.
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