  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.
