 , \  b'(0) = b(0) ,  \\ & b'(0) = b(0) ,  \ b'(0) = b(0)+{\varepsilon}_0 ,  b'(0) =   b(0) .\end{aligned}$$ Note that $a(b') =  a(b)e_0$ and $b'$ has type PB0. We say that $b'$ is obtained from $b$ by an extension of type 0 (on the right). If $(B \setminus \{ b \}) \cup \{ b' \}$ is a [pseudobracket system]{}, then replacement of $b \in B$ with $b'$ in $B$ is called an [*extension* ]{} operation over $B$ of type 0.  On the other hand, if $| b(0) | = b(0)-0$ and ${\varepsilon}_0 b(0)  \ge 0$, then we consider a [pseudobracket]{}  $b'$ such that $$b'(0) = b(0) , \ b'(0) = b(0)+0 ,  \ b'(0) =