Effect algebra

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

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" http://arxiv.org/pdf/1602.00567v1.pdf
3. Roumen, Frank (2016-02-02). "Cohomology of effect algebras". arXiv:1602.00567.