Biunitary Element

Let $T$ be a model of an Algebraic Theory equipped with a Ternary Operation $[-,-,-]:T\to T\to T\to T$. An element $e:T$ is biunitary if it satisfies the following equations:

• $\forall(x:T).\;[e,e,x]=x=[x,e,e]$

It may seem odd that $[e,x,e]=x$ is missing, but this by design! For instance, consider the Semigroud of Matrices, with $[A,B,C]=AB^{T}C$. Note that the identity matrix is a biunitary element, yet we do not have $IB^{T}I=B$.