ABSTRACT
Before we begin a systematic investigation, let us mention some examples of properties of ternary operations that in some way generalize the binary associative law. The simplest possible example of an operad generated by a ternary operation is the totally associative ternary operad. In notation of Chapter 6, it is the operad tAs(3)0 generated by one ternary generator subject to the following relations:
= = . (10.1)
This kind of associativity is exhibited by the so-called triadic groups; see [208] (the first systematic paper in English that established n-ary operations as an independent field of study). The following proposition shows that such algebras are intimately related to usual associative algebras. The first two parts of it are folklore, the third can be traced back to [50], where it is shown that n-ary associativity is intimately related to binary associativity.
