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Inverse semigroup homomorphisms via partial group actions

Published online by Cambridge University Press:  17 April 2009

Benjamin Steinberg
Benjamin Steinberg
Affiliation:
Departmento de Matemática Pura, Faculdade de Ciências, da Universidade do Porto, 4099–002 Porto, Portugal e-mail: [email protected]
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Abstract

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This papar constructs all homomorphisms of inverse semigroups which factor through an E-unitary inverse semigroup; the construction is in terms of a semilattice component and a group component. It is shown that such homomorphisms have a unique factorisation βα with α preserving the maximal group image, β idempotent separating, and the domain I of β E-unitary; moreover, the P-representation of I is explicitly constructed. This theory, in particular, applies whenever the domain or codomain of a homomorphism is E-unitary. Stronger results are obtained for the case of F-inverse monoids.

Special cases of our results include the P-theorem and the factorisation theorem for homomorphisms from E-unitary inverse semigroups (via idempotent pure followed by idempotent separating). We also deduce a criterion of McAlister–Reilly for the existence of E-unitary covers over a group, as well as a generalisation to F-inverse covers, allowing a quick proof that every inverse monoid has an F-inverse cover.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 64 , Issue 1 , August 2001 , pp. 157 - 168
Copyright
Copyright © Australian Mathematical Society 2001

References

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