# algebraic ternary operators

In the analysis of certain Natural Language Processing queries, there is a certain algebra that arises with a ternary operator over a group with the following two properties:

$$( ( a , b , c ) , d , e ) = ( a , (d , b , c) , (e , b , c ) )$$

$$x( a , b , c ) = (a x^{-1} , b x^{-1} , c x^{-1} )$$

and in general the ternary operator might not be symmetric under even permutations. (In fact I think it can be understood as a non-associative binary product but I don't have anything to back up that assertion other than a vague suspicion.)

Is there a name for these algebraic entities? Is there a reference I can read more about these?

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