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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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

Author: Mary Joel Jordan
Journal: Trans. Amer. Math. Soc. 163 (1972), 75-84
MSC: Primary 20.93
MathSciNet review: 0284529
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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}$.

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Keywords: Semigroup, inverse semigroup, $ H$-semigroup, $ t$-semisimple semigroup, Hamiltonian group, semilattice, maximal congruence, modular congruence
Article copyright: © Copyright 1972 American Mathematical Society