Not each sequential effect algebra is sharply dominating

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.

Journal of Mathematical PhysicsJun 1, 20106 citations

Sequential product on standard effect algebra {\cal E} (H)

Jun Shen, Junde Wu

arXiv:0905.0596Journal of Physics A Mathematical and TheoreticalAug 6, 200936 citations

Not each sequential effect algebra is sharply dominating

Shen Jun, Junde Wu

Published Jan 1, 2009• 0 citations

OpenAlex

Abstract

Let

E

be an effect algebra and

E_S

be the set of all sharp elements of

E

E

is said to be sharply dominating if for each

a\in E

there exists a smallest element

\widehat{a}\in E_s

such that

a\leq \widehat{a}

. In 2002, Professors Gudder and Greechie proved that each

\sigma

-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.

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Not each sequential effect algebra is sharply dominating

Jun Shen, Junde Wu

Let $E$ be an effect algebra and $E_S$ be the set of all sharp elements of $E$ . $E$ is said to be sharply dominating if for each $a\in E$ there exists a smallest element $\widehat{a}\in E_s$ such that $a\leq \widehat{a}$ . In 2002, Professors Gudder and Greechie proved that each $σ$ -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 $E$ be an effect algebra and $E_S$ be the set of all sharp elements of $E$. $E$ is said to be sharply dominating if for each $a\in E$ there exists a smallest element $\widehat{a}\in E_s$ such that $a\leq \widehat{a}$. In 2002, Professors Gudder and Greechie proved that each $σ$-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.

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Sequential product on standard effect algebra {\cal E} (H)

Jun Shen, Junde Wu

A quantum effect is an operator $A$ on a complex Hilbert space $H$ that satisfies $0\leq A\leq I$ , ${\cal E} (H)$ is the set of all quantum effects on $H$ . In 2001, Professor Gudder and Nagy studied the sequential product $A\circ B=A^{1/2}BA^{1/2}$ of $A, B\in {\cal E}(H)$ . In 2005, Professor Gudder asked: Is $A\circ B=A^{1/2}BA^{1/2}$ the only sequential product on ${\cal E} (H)$ ? 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 ${\cal E} (H)$ ; secondly, we present a general method for constructing sequential products on ${\cal E} (H)$ ; finally, we study some properties of the sequential products constructed by the methodA quantum effect is an operator $A$ on a complex Hilbert space $H$ that satisfies $0\leq A\leq I$, ${\cal E} (H)$ is the set of all quantum effects on $H$. In 2001, Professor Gudder and Nagy studied the sequential product $A\circ B=A^{1/2}BA^{1/2}$ of $A, B\in {\cal E}(H)$. In 2005, Professor Gudder asked: Is $A\circ B=A^{1/2}BA^{1/2}$ the only sequential product on ${\cal E} (H)$? 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 ${\cal E} (H)$; secondly, we present a general method for constructing sequential products on ${\cal E} (H)$; finally, we study some properties of the sequential products constructed by the method

arXiv:0905.0596Journal of Physics A Mathematical and TheoreticalAug 6, 200936 citations

Let

E

be an effect algebra and

E_S

be the set of all sharp elements of

E

E

is said to be sharply dominating if for each

a\in E

there exists a smallest element

\widehat{a}\in E_s

such that

a\leq \widehat{a}

. In 2002, Professors Gudder and Greechie proved that each

\sigma

Let

E

be an effect algebra and

E_S

be the set of all sharp elements of

E

E

is said to be sharply dominating if for each

a\in E

there exists a smallest element

\widehat{a}\in E_s

such that

a\leq \widehat{a}

. In 2002, Professors Gudder and Greechie proved that each

σ

n>1

, is there a sequential effect algebra such that the n-th root of its some element

c

is not unique and the n-th root of

c

is not the k-th root of

c

($k

A quantum effect is an operator

A

on a complex Hilbert space

H

that satisfies

0\leq A\leq I

{\cal E} (H)

is the set of all quantum effects on

H

. In 2001, Professor Gudder and Nagy studied the sequential product

A\circ B=A^{1/2}BA^{1/2}

A, B\in {\cal E}(H)

. In 2005, Professor Gudder asked: Is

A\circ B=A^{1/2}BA^{1/2}

the only sequential product on

{\cal 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

{\cal E} (H)

; secondly, we present a general method for constructing sequential products on

{\cal E} (H)

; finally, we study some properties of the sequential products constructed by the method