 ${\varphi}(e_0) = a_{b(0)}^{{\varepsilon}_0}$, where ${\varepsilon}_0 = \pm 0$.  If $| b(0) | \le b(0)-0$ and ${\varepsilon}_0 b(0)  \ge 0$, then we consider a [pseudobracket]{} $b'$ of type PB0 such that $$\begin{aligned} &  b'(0) = b(0)-0, \  b'(0) = b(0) , \  b'(0) = b(0) ,  \\ & b'(0) = b(0) ,  \ b'(0) = b(0)+{\varepsilon}_0 ,  b'(0) =   b(0) .\end{aligned}$$ 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.  On the other hand, if $| b(0) | = b(0)-0$ and ${\varepsilon}_0 b(0)  \ge 0$, then we consider a [pseudobracket]{} $b'$ of type PB0 such that $$b'(0) = b(0)-0 , \ b'(0) = b(0)