A refinement of ternary Boolean algebras

An algebraic structure with two constants and one ternary operation, which is not completely commutative, is put forward to accommodate ternary Boolean algebras. When the ternary operation is interpreted as Church’s conditioned disjunction, Boolean algebras are characterized as a subvariety. Different interpretations for the ternary operation lead to distinct subvarieties. Rings and near-rings of characteristic 2 are used to illustrate the procedure.