Definition 1

A subset Z of $\Omega (\widetilde{\Gamma})$ is said to be distinguished (relative to the pair $(\varphi , \psi )$) if it satisfies the following properties:

(i) For every arrow $z$ of $\widetilde{\Gamma}$, one has $(z, z)\in Z$ ;

(ii) For every $(z_1, . . . , z_n)\in$ Z, one has $(\overline{z}_n, . . . ,\overline{z}_2,\overline{z}_1)\in$ Z and $(z_n, z_1, . . . , z_{n-1})\in Z$ ;

(iii) Let $\mathbf{z}= (z_1, . . . , z_n)$ and $\mathbf{z}'= (z'_1, . . . , z'_m)$ be elements of Z such that $t(z_n) =o(z'_1)$. Put $\mathbf{z}\mathbf{z}'= (z_1, . . . , z_n, z'_1, . . . , z'_m)$. If two elements among $\mathbf{z},\mathbf{z}',\mathbf{z}\mathbf{z}'$ belong to Z, the same is true of the third;

(iv) For every arrow $f$ of H, put $\widetilde{\varphi}(f,1) =\widetilde{\psi}(f,-1) =$ $\varphi (f)$ and $\widetilde{\varphi}(f,-1) =\widetilde{\psi}(f,1) =\psi (f)$. Let $n$ be an integer $\geqslant$ 1 and $(f_1, \varepsilon_1), . . . ,(f_n, \varepsilon_n)$ a sequence of elements of Fl(H) $\times \{-1,1\}$ such that $\widetilde{\psi}(f_i, \varepsilon_i) =\widetilde{\varphi}(f_{i+1}, \varepsilon_{i+1})$ for $1\leqslant i < n$ and $\widetilde{\psi}(f_n, \varepsilon_n) =\widetilde{\varphi}(f_1, \varepsilon_1)$; let $a_i$ denote the origin of $f_i$ and $b_i$ its term. In order that $((a_1, \varepsilon_1), . . . ,(a_n, \varepsilon_n))$ belong to Z, it is necessary and sufficient that $((b_1, \varepsilon_1), . . . ,(b_n, \varepsilon_n))$ belong to Z.

The intersection in $\Omega (\widetilde{\Gamma})$ of every family of distinguished subsets (relative to $(\varphi , \psi )$) is again distinguished. In particular, there exists a smallest distinguished subset containing a given subset Z of $\Omega (\widetilde{\Gamma})$.