aking both $b_0$, $b_0$ out of $B$ and putting $b'$ in $B$ is a [*merger* ]{} operation over $B$.  We will say that additions, extensions, and mergers, as defined above, are [*[elementary operation]{}s*]{} over brackets and [bracket system]{}s for $(W, {\Delta})$.  Assume that one [bracket system]{}  $B_\ell$ is obtained from another [bracket system]{}  $B_0$ by a finite sequence $\Omega$ of [elementary operation]{}s and $B_0,  B_0, \dots, B_\ell$ is the corresponding to $\Omega$ sequence of [bracket system]{}s. 