Several generalizations of effect algebras exist, based on the needs of various logics; these may not have a clear relationship to the quantum foundations of effect algebras, but deal with informationally similar situations where the “resources” of the deductions rules are subject to certain restrictions. Separation algebras correspond to separation logic in the way effect algebras correspond to quantum logic, and are often used as models for memory allocation and program modularity, where the outcome of a program must be guaranteed with respect to specified memory states. Separation algebra therefore has a role in certified programming.

Definition

Definition

((Generalized) Separation Algebra)

A generalized separation algebra (GSA) is a partial monoid satisfying the following equations:

(1) $ab = ac \rightarrow b=c$,

(2) $ab=cb \rightarrow a=c$,

and the non-equational formula:

(3) $\forall a, b (\exists d : da=b \iff \exists c : ac = b)$

A separation algebra is a commutative GSA, which removes the axiom (3). The real interest of (3) lies in other order-theoretic properties.

This reveals effect algebras to be those separation algebras with the positivity property.

A Family of Partial Algebras

(…)

Examples

Aside from low-dimensional cases found in a model generator such as Mace, many natural examples which are not effect algebras arise from ordered bisemigroups.