Abstract
We give necessary and sufficient conditions in order that lower bounded HNN-extensions of inverse semigroups and HNN-extensions of finite inverse semigroups are completely semisimple semigroups. Since it is well known that an inverse semigroup is completely semisimple if and only if it does not contain a copy of the bicyclic semigroup, we first characterize such HNN-extensions containing a bicyclic subsemigroup making use of the special feature of their Schützenberger automata.
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