Effect algebra

Effect algebras are algebraic structures of a kind introduced by D. Foulis and M. Bennett to serve as a framework for unsharp measurements in quantum mechanics.[1]

An effect algebra consists of an underlying set A equipped with a partial binary operation ⊞, a unary operation (−)^⊥, and two special elements 0, 1 such that the following relationships hold:[2]

• The binary operation is commutative: if a ⊞ b is defined, then so is b ⊞ a, and they are equal.
• The binary operation is associative: if a ⊞ b and (a ⊞ b) ⊞ c are defined, then so are b ⊞ c and a ⊞ (b ⊞ c), and (a ⊞ b) ⊞ c = a ⊞ (b ⊞ c).
• The zero element behaves as expected: 0 ⊞ a is always defined and equals a.
• The unary operation is an orthocomplementation: for each a ∈ A, a^⊥ is the unique element of A for which a ⊞ a^⊥ = 1.
• A zero-one law holds: if a ⊞ 1 is defined, then a = 0.

Every effect algebra carries a natural order: define a ≤ b if and only if there exists an element c such that a ⊞ c exists and is equal to b. The defining axioms of effect algebras guarantee that ≤ is a partial order.[3]

Examples

The motivating example of an effect algebra is the set of effects on a unital C*-algebra: the elements ${\displaystyle a\in {\mathfrak {A}}}$ satisfying ${\displaystyle 0\leq a\leq 1}$. The addition operation on ${\displaystyle a,b\in [0,1]_{\mathfrak {A}}}$ is defined when ${\displaystyle a+b\leq 1}$ and then a⊞b = a+b. The involution is given by ${\displaystyle a^{\perp }=1-a}$.

Other examples include any orthomodular poset (and thus any Boolean algebra).

Types of effect algebras

There are various types of effect algebras that have been studied.

• Interval effect algebras that arise as an interval ${\displaystyle [0,u]_{G}}$ of some ordered Abelian group ${\displaystyle G}$.
• Convex effect algebras have an action of the real unit interval ${\displaystyle [0,1]}$ on the algebra. A representation theorem of Gudder shows that these all arise as an interval effect algebra of a real ordered vector space.[4]
• Lattice effect algebras where the order structure forms a lattice.
• Effect algebras satisfying the Riesz decomposition property.[5]
• An MV-algebra is precisely a lattice effect algebra with the Riesz decomposition property.[6]
• Sequential effect algebras have an additional sequential product operation that models the Lüders product on a C*-algebra.[7]
• Effect monoids are the monoids in the category of effect algebras. They are effect algebras that have an additional associative unital distributive multiplication operation.[8]

References

1. D. Foulis and M. Bennett. "Effect algebras and unsharp quantum logics", Found. Phys., 24(10):1331–1352, 1994.
2. Frank Roumen, "Cohomology of effect algebras" arXiv:1602.00567
3. Roumen, Frank (2016-02-02). "Cohomology of effect algebras". Electronic Proceedings in Theoretical Computer Science. 236: 174–201. arXiv:1602.00567. doi:10.4204/EPTCS.236.12.
4. Gudder, Stanley (1999-12-01). "Convex Structures and Effect Algebras". International Journal of Theoretical Physics. 38 (12): 3179–3187. doi:10.1023/A:1026678114856. ISSN 1572-9575.
5. Pulmannova, Sylvia (1999-09-01). "Effect Algebras with the Riesz Decomposition Property and AF C*-Algebras". Foundations of Physics. 29 (9): 1389–1401. doi:10.1023/A:1018809209768. ISSN 1572-9516.
6. Foulis, D. J. (2000-10-01). "MV and Heyting Effect Algebras". Foundations of Physics. 30 (10): 1687–1706. doi:10.1023/A:1026454318245. ISSN 1572-9516.
7. Gudder, Stan; Greechie, Richard (2002-02-01). "Sequential products on effect algebras". Reports on Mathematical Physics. 49 (1): 87–111. doi:10.1016/S0034-4877(02)80007-6. ISSN 0034-4877.
8. Jacobs, Bart; Mandemaker, Jorik (2012-07-01). "Coreflections in Algebraic Quantum Logic". Foundations of Physics. 42 (7): 932–958. doi:10.1007/s10701-012-9654-8. ISSN 1572-9516.

External links

• Effect algebra in nLab