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]

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.


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