The adjoint of a bilinear operation

Abstract

is an extension of m. Recall that X, Y, Z are naturally embeddable in X-, Y--, Z-resp. Moreover, certain properties, such as associativity, when m has them, are transmitted to m*** (this makes sense only when Y=Z=X). On the other hand, the transmission of commutativity (which makes sense when Y=X) was left open, and will be considered in this paper. This question of commutativity can be generalized as follows. If m satisfies 1.1-1.3, one can define the transposed operation