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arXiv:0906.1412 · Journal of Mathematical Physics

On fixed points of Lüders operation

Weihua Liu, Junde Wu

Published Oct 1, 2009• 15 citations
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Abstract

In this paper, we give a concrete example of a Lüders operation LA with n=3, such that LA(B)=B does not imply that B commutes with all E1, E2, and E3 in A, this example answers an open problem of Professor Gudder.
DOI
10.1063/1.3253574

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Let EEE be an effect algebra and ESE_SES​ be the set of all sharp elements of EEE. EEE is said to be sharply dominating if for each a∈Ea\in Ea∈E there exists a smallest element a^∈Es\widehat{a}\in E_sa∈Es​ such that a≤a^a\leq \widehat{a}a≤a. In 2002, Professors Gudder and Greechie proved that each σσσ-sequential effect algebra is sharply dominating. In 2005, Professor Gudder presented 25 open problems in International Journal of Theoretical Physics, Vol. 44, 2199-2205, the 3th problem asked: Is each sequential effect algebra sharply dominating? Now, we construct an example to answer the problem negatively.