Every Effect Algebra Can be Made into a Total Algebra

In our previous paper [1] we introduced the concept of a basic algebra, this being an algebra (A,⊕,¬, 0) of type (2, 1, 0) with the property that the underlying poset (A,≤) defined by x ≤ y if and only if ¬x ⊕ y = ¬0 is a bounded lattice and, for each a ∈ A, the mapping x 7 → ¬x ⊕ a is an antitone involution on the principal filter [a) = {x ∈ A | a ≤ x}. The name ‘basic algebra ’ is used because these algebras capture common features of many known structures such as Boolean algebras, orthomodular lattices, MV-algebras or lattice effect algebras. In [1] we paid special attention to lattice effect algebras, which were originally defined as partial algebras (E,+, 0, 1), but the presence of the join operation allows one to replace partial + by total ⊕. The intent of the present paper is to establish similar results for general effect algebras in the context of commutative directoids; we shall prove that every effect algebra (E,+, 0, 1) can be made into a total algebra (E,⊕,¬, 0) in such a way that two elements are compatible in (E,+, 0, 1) if and only if they commute in (E,⊕,¬, 0)