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Journal of Mathematical Physics

The nth root of sequential effect algebras

Jun Shen, Junde Wu

Published Jun 1, 2010• 6 citations
Publisher

Abstract

In 2005, Gudder [Int. J. Theor. Phys. 44, 2219 (2005)] presented 25 problems of sequential effect algebras, the 20th problem asked: In a sequential effect algebra, if the square root of some element exists, is it unique? In this paper, we show that for each given positive integer n>1, there is a sequential effect algebra such that the nth root of its some element c is not unique, and the nth root of c is not the kth root of c (k<n). Thus, we answer the problem negatively.

DOI
10.1063/1.3431627

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In 2005, Gudder [Int. J. Theor. Phys. 44, 2219 (2005)] presented 25 problems of sequential effect algebras, the 20th problem asked: In a sequential effect algebra, if the square root of some element exists, is it unique? In this paper, we show that for each given positive integer n>1, there is a sequential effect algebra such that the nth root of its some element c is not unique, and the nth root of c is not the kth root of c (k<n). Thus, we answer the problem negatively.
Sequential effect algebra is an important model for studying quantum measurement theory. In 2005, Professor Gudder presented 25 open problems to motivate its study. The 20th problem asked: In a sequential effect algebra, if the square root of some element exists, is it unique ? We can strengthen the problem as following: For each given positive integer n>1n>1n>1, is there a sequential effect algebra such that the n-th root of its some element ccc is not unique and the n-th root of ccc is not the k-th root of ccc ($k
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.
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 σ\sigmaσ-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.
A quantum effect is an operator AAA on a complex Hilbert space HHH that satisfies 0≤A≤I0\leq A\leq I0≤A≤I, E(H){\cal E} (H)E(H) is the set of all quantum effects on HHH. In 2001, Professor Gudder and Nagy studied the sequential product A∘B=A1/2BA1/2A\circ B=A^{1/2}BA^{1/2}A∘B=A1/2BA1/2 of A,B∈E(H)A, B\in {\cal E}(H)A,B∈E(H). In 2005, Professor Gudder asked: Is A∘B=A1/2BA1/2A\circ B=A^{1/2}BA^{1/2}A∘B=A1/2BA1/2 the only sequential product on E(H){\cal E} (H)E(H)? Recently, Liu and Wu presented an example to show that the answer is negative. In this paper, firstly, we characterize some algebraic properties of the abstract sequential product on E(H){\cal E} (H)E(H); secondly, we present a general method for constructing sequential products on E(H){\cal E} (H)E(H); finally, we study some properties of the sequential products constructed by the method
Compression base effect algebras were recently introduced by Gudder [Demonstr. Math. 39, 43 (2006)]. They generalize sequential effect algebras [Rep. Math. Phys. 49, 87 (2002)] and compressible effect algebras [Rep. Math. Phys. 54, 93 (2004)]. The present paper focuses on atomic compression base effect algebras and the consequences of atoms being foci (so-called projections) of the compressions in the compression base. Part of our work generalizes results obtained in atomic sequential effect algebras by Tkadlec [Int. J. Theor. Phys. 47, 185 (2008)]. The notion of projection-atomicity is introduced and studied, and several conditions that force a compression base effect algebra or the set of its projections to be Boolean are found. Finally, we apply some of these results to sequential effect algebras and strengthen a previously established result concerning a sufficient condition for them to be Boolean.