Sequential products on effect algebras

doi:10.1016/S0034-4877(02)80007-6

Reports on Mathematical Physics

Volume 49, Issue 1, February 2002, Pages 87-111

https://doi.org/10.1016/S0034-4877(02)80007-6 Get rights and content

Abstract

A sequential effect algebra (SEA) is an effect algebra on which a sequential product with natural properties is defined. The properties of sequential products on Hilbert space effect algebras are discussed. For a general SEA, relationships between sequential independence, coexistence and compatibility are given. It is shown that the sharp elements of a SEA form an orthomodular poset. The sequential center of a SEA is discussed and a characterization of when the sequential center is isomorphic to a fuzzy set system is presented. It is shown that the existence, of a sequential product is a strong restriction that eliminates many effect algebras from being SEA's. For example, there are no finite nonboolean SEA's, A measure of sharpness called the sharpness index is studied. The existence of horizontal sums of SEA's is characterized and examples of horizontal sums and tensor products are presented.

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We say that p is an automorphism with respect to the convex sequential product ○λ for some λ ∈ [0,1] if ϕ (A ○λ B) = ϕ (A) ○λ ϕ (B) for any A, B ∈ E (ℌ). If λ is 0 or 1, then the results are considered in e.g. [5–8]. Thus we may assume that ϕ is a convex sequential automorphism on E (ℌ) for a λ ∈ (0, 1).
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Sequential products on effect algebras

Abstract

Rep. Math. Phys.

Adv. Appl. Math.

Trans. Amer. Math. Soc.

Found. Phys.

Found. Phys.