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 G.
Convex effect algebras have an action of the real unit interval [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
D. Foulis and M. Bennett. "Effect algebras and unsharp quantum logics", Found. Phys., 24(10):1331–1352, 1994.[better source needed]
Frank Roumen, "Cohomology of effect algebras" arXiv:1602.00567
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. S2CID 16707878.
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. S2CID 115468918.
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. S2CID 117445132.
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. S2CID 116763476.
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.
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.
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