Flows, skew pairs and partial flows in regular semigroups

This work connected flows on regular semigroups with skew pairs of idempotents via the bipartite category $\beta (S)$, where arrows $e\stackrel{s}{\to }f$ exist if and only if some inverse $s^{\prime }$ satisfies $e=s{s}^{\prime }$ and $f=s^{\prime }s$. Partial flows satisfying $aba=a$ and $bab=b$ were introduced. Using Luangchaisam and Changphas' theorem, explicit partial flows were constructed for strong, left regular, and right regular skew pairs, with domain sizes $1$, $2$ and $2$, respectively. Flow monoids were classified for groups ( $\mathrm{\Phi }(G)\cong G$), rectangular bands, Brandt semigroups ( $\mathrm{\Phi }(B_n)\cong T_n$), inverse semigroups, and completely simple semigroups, where the sandwich matrix twists composition. The discrete skew pair case remains open.