Abstract
We consider the following question. Which varieties of completely regular semigroups have the weak or strong amalgamation property? For varieties not consisting entirely of completely simple semigroups, we can answer the question fully: such a variety has the weak [strong] amalgamation property if and only if it is a variety of orthodox normal bands of abelian groups. For varieties of completely simple semigroups with abelian subgroups, we can again answer the question fully: all these varieties have the strong amalgamation property. We outline the difficulties presented by the remaining varieties of completely simple semigroups.
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