1974
DOI: 10.1090/s0002-9904-1974-13585-8
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Representation of partially ordered linear algebras

Abstract: In [2] and [3] a condition on partially ordered linear algebras (pola's) is defined, and it is shown that Dedekind cr-complete polas satisfying this condition have many of the properties of function spaces. Using a theorem of H. Nakano we can show, even without the hypothesis that the pola is Dedekind cr-complete, that any such pola is isomorphic to a pola of continuous, almost-finite, extended-real-valued functions. If A is a pola with multiplicative identity 1 the condition mentioned is:Pi. If x e A and x^. … Show more

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“…Any dsc-pola A with a multiplicative DP map A always admits a function with the logarithm (root) property which we explain as follows. In [6] it was proved that A, is algebraically and order isomorphic to an algebra S of continuous, almost-finite, extended-real-valued functions defined on a compact Hausdorff space. Hence, we may identify A, with S. It is clear then for 1 < z E Ax, ln(z) and Vz also belong to Ax.…”
mentioning
confidence: 99%
“…Any dsc-pola A with a multiplicative DP map A always admits a function with the logarithm (root) property which we explain as follows. In [6] it was proved that A, is algebraically and order isomorphic to an algebra S of continuous, almost-finite, extended-real-valued functions defined on a compact Hausdorff space. Hence, we may identify A, with S. It is clear then for 1 < z E Ax, ln(z) and Vz also belong to Ax.…”
mentioning
confidence: 99%