The order bidual of almost F-algebras and D-algebras

The algebraic structure of the second order dual of a lattice ordered algebra is studiedon.It is shown that the order continuous order bidual of an Archimedean almost f-algebraequipped with respect to the Arens multiplication is again an almost f-algebra and thesecond order dual of an Archimedean almost f-algebra is also an almost f-algebra.It is proved that the order continuous order bidual of an Archimedean f-algebra is anf-algebra and the second order dual of an Archimedean f-algebra is again an f-algebrawith respect to the Arens multiplication.It is obtained that the order continuous order bidual of an Archimedean d-algebra is ad-algebra with respect to the Arens multiplication. It is shown that if the d-algebra iscommutative or has positive squares then the second order dual is again a d-algebrawith respect to the Arens multiplication and the second order dual is commutative andits squares are positive.