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Abstract

We characterize pairs $\mathfrak {n}$, $\mathfrak {m}$ of cardinals with the property that there exist an Archimedean linear lattice $X$ and an order interval in $X$ such that $\mathfrak {n}$ is its cardinality while $\mathfrak {m}$ is the cardinality of the set of its extreme points. We also present analogous results, complete or partial, in the case where $X$ is additionally required to be nonatomic, atomic, Dedekind $\sigma $-complete, hyper-Archimedean, or to be a $C(K)$-space, where $K$ is a compact space.