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 $$\omega$$ , $$\omega _k$$ , $$\omega _t$$ , $$(\omega _k)_t$$ , $$(\omega _t)_k$$ , $$((\omega _k)_t)_k$$ , $$((\omega _t)_k)_t$$ , $$\ldots$$ . The quotients $$\{S/\omega _k\}$$ , $$\{S/\omega _t\}$$ , $$\{S/(\omega _k)_t\}$$ , $$\{S/(\omega _t)_k\}$$ , $$\{S/((\omega _k)_t)_k\}$$ , $$\{S/((\omega _t)_k)_t\}$$ , $$\ldots$$ , as S runs over all inverse semigroups, form quasivarieties. This article explores the relationships among these quasivarieties.