The Archimedean property follows immediately from the ordering in the semilattice L, since with this ordering we have f ( x ) \ le f ( y ) if and only if x ^ n = y z for some z and n > 0.
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He is noted for many theorems including : H�lder's inequality, the Jordan H�lder theorem, the theorem stating that every linearly ordered group that satisfies an Archimedean property is isomorphic to a subgroup of the additive analysis, including the theories of partial differential equations and function spaces.
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