In this paperwestudy the Arens triadjoints of some bilinear maps on vector lattices. In particular, we prove that, for Archimedean vector lattices A and B, the Arens triadjoint i) T*** : A" x A" ? B" of a positive orthosymmetric bilinear map T:A x A ? B is positive orthosymmetric, and ii) T***: A" x A" ? A" of a bi-orthomorphism T : A x A ? A is a bi-orthomorphism. These generalize results on the order bidual of f -algebras and almost f-algebras in [4].

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Abstract:In this paper we introduce a new concept of a d-bimorphism on a vector lattice and prove that, for vector lattices A and B, the Arens triadjoint T * * * :This generalizes the concept of d-algebra and some results on the order bidual of d-algebras.

In this paper we introduce a new concept, namely that of a quasi-orthomorphism, on a vector lattice, which generalizes the notion of an orthomorphism. We consider quasi-multipliers on -algebras and establish the relationship between quasi-orthomorphisms and quasi-multipliers. We prove that, for certain Banach algebras, the quasi-orthomorphisms Q Orth(A) form a Dedekind complete Banach f -algebra with a multiplicative identity. We conclude with a Kakutani type representation theorem for Q Orth(A).