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semigroup; reductive semigroup; semigroup compactification; enveloping semigroup; $E$- subcompactification
For extending the notion of $E$-algebra, as defined in [2], we present an example of an m-admissible algebra which is not an $E$ - algebra. Then we define $E$-subcompactification and $E{\mathcal F}$-subcompactification to study the universal $E$-subcompactification and the universal $E{\mathcal F}$-subcompactification from the function algebras point of view.
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