Relationships among quasivarieties induced by the min networks on inverse semigroups

A congruence on an inverse semigroup $S$ is determined uniquely by its kernel and trace. Denoting by $\rho_k$ and $\rho_t$ the least congruence on $S$ having the same kernel and the same trace as $\rho$, respectively, and denoting by $\omega$ the universal congruence on $S$, we consider the sequence $\omega$, $\omega_k$, $\omega_t$, $(\omega_k)_t$, $(\omega_t)_k$, $((\omega_k)_t)_k$, $((\omega_t)_k)_t$, $\cdots$. The quotients $\{S/\omega_k\}$, $\{S/\omega_t\}$, $\{S/(\omega_k)_t\}$, $\{S/(\omega_t)_k\}$, $\{S/((\omega_k)_t)_k\}$, $\{S/((\omega_t)_k)_t\}$, $\cdots$, as $S$ runs over all inverse semigroups, form quasivarieties. This article explores the relationships among these quasivarieties.