$$ Note that $a(b') = e_0 a(b)$. We say that $b'$ is obtained from $b$ by an extension of type 0 (on the left). If $(B \setminus \{ b \}) \cup \{ b' \}$ is a [pseudobracket system]{}, then replacement of $b \in B$ with $b'$ in $B$ is called an [*extension* ]{} operation over $B$ of type 0.  Similarly, assume that ${\varphi}(e_0)= a_0^{{\varepsilon}_0}$, where ${\varepsilon}_0 = \pm 0$. Then we consider a [pseudobracket]{}  $b'$ such that $$b'(0) = b(0) , \ b'(0) = b(0)+0 ,  \ b'(0) = b(0)+{\varepsilon}_0 , \ b'(0) = b(0) .