In his seminal paper [2], Arens gave an example of a positive bilinear map P : l 1 × l 1 → R that is not regular. In this note we show that for Banach lattices E such a bilinear non-regular map E × E → R exists if and only if l 1 does not embed into E.

In this paper we study properties of bilinear maps of order bounded variation. Theorems of preservation of properties in passage to the triadjoint and the tensor product are presented.