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The variety of semigroups defined by $B=\Big\{(x^py^p)^2=(y^px^p)^2:p \text{ is prime}\Big\}$ is non-finitely based (Isbell, 1970). Is the pseudovariety defined by $B$ also non-finitely based?

More generally, if $B$ is an infinite basis for a non-finitely based variety of semigroups, then is the pseudovariety defined by $B$ non-finitely based?

Thank you all in advance.

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