Inverse $H$-semigroups and $t$-semisimple inverse $H$-semigroups

Abstract:

An $H$-semigroup is a semigroup such that both its right and left congruences are two-sided. A semigroup is $t$-semisimple provided the intersection of all its maximal modular congruences is the identity relation. We prove that a semigroup is an inverse $H$-semigroup if and only if it is a semilattice of disjoint Hamiltonian groups. Using the set $E$ of idempotents of $S$ as the semilattice, we show that an inverse $H$-semigroup $S$ is $t$-semisimple if and only if for each pair of groups ${G_e},{G_f}$, in the semilattice, with $f \geqq e$ in $E$, the homomorphism ${\varphi _{f,e}}$ on ${G_f}$, into ${G_e}$, defined by $a{\varphi _{f,e}} = ae$, is a monomorphism; and for each $e$ in $E$, for each $a \ne e$ in ${G_e}$, there exists a subsemigroup ${T_p}$ of $S$ such that $a \notin {T_p}$ and, for each $f$ in $E$, ${T_p} \cap {G_f} = {H_f}$, where ${H_f} = {G_f}$ or ${H_f}$ is a maximal subgroup of prime index $p$ in ${G_f}$.